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The problem of pointwise adaptive estimation of the drift coefficient of a multivariate diffusion process is investigated. We propose an estimator which is sharp adaptive on scales of Sobolev smoothness classes. The analysis of the exact risk asymptotics allows to identify the impact of the dimension and other influencing values—such as the geometry of the diffusion coefficient—of the prototypical drift estimation problem for a large class of multidimensional diffusion processes. We further sketch generalizations of our results to arbitrary diffusions satisfying suitable Bernstein-type inequalities.
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- 1.
Aït-Sahalia, Y.: Closed-form likelihood expansions for multivariate diffusions. Ann. Stat. 36(2), 906–937 (2008)
- 2.
Bakry, D., Cattiaux, P., Guillin, A.: Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 245(3), 727–759 (2008)
- 3.
Butucea, C.: Exact adaptive pointwise estimation on Sobolev classes of densities. ESAIM: Prob. Stat. 5, 1–31 (2001)
- 4.
Cattiaux, P., Chafaï, D., Guillin, A.: Central limit theorems for additive functionals of ergodic Markov diffusions processes. ALEA, Lat. Am. J Probab. Math. Stat. 9(2), 337–382 (2012)
- 5.
Dalalyan, A.S.: Sharp adaptive estimation of the drift function for ergodic diffusions. Ann. Stat. 33(6), 2507–2528 (2005)
- 6.
Dalalyan, A.S., Kutoyants, Y.A.: Asymptotically efficient trend coefficient estimation for ergodic diffusion. Math. Methods Stat. 11, 402–427 (2002)
- 7.
Dalalyan, A.S., Reiß, M.: Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case. Probab. Theory Relat. Fields 137(1), 25–47 (2007)
- 8.
Guillin, A., Léonard, C., Wu, L., Yao, N.: Transportation-information inequalities for Markov processes. Probab. Theory Relat. Fields 144(3–4), 669–695 (2009)
- 9.
Itō, K., McKean, H.P.: Diffusion Processes and their Sample Paths. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, vol. 125. Springer, Berlin (1965)
- 10.
Klemelä, J., Tsybakov, A.B.: Sharp adaptive estimation of linear functionals. Ann. Stat. 29(6), 1567–1600 (2001)
- 11.
Klemelä, J., Tsybakov, A.B.: Exact constants for pointwise adaptive estimation under the Riesz transform. Probab. Theory Relat. Fields 129, 441–467 (2004)
- 12.
Lepski, O.V.: One problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35, 459–470 (1990)
- 13.
Lezaud, P.: Chernoff and Berry–Esséen inequalities for Markov processes. ESAIM: Probab. Stat. 5, 183–201 (2001)
- 14.
Liptser, R.S., Shiryaev, A.N.: Statistics of Random Processes. General Theory of Applications of Mathematics: Stochastic Modelling and Applied Probability, vol. 1, 2nd edn. Springer, Berlin (2001)
- 15.
Metafune, G., Pallara, D., Rhandi, A.: Global properties of invariant measures. J. Funct. Anal. 223(2), 396–424 (2005)
- 16.
Parzen, E.: On estimation of a probability density function and mode. Ann. Math. Stat. 33(3), 1065–1076 (1962)
- 17.
Qian, Z., Zheng, W.: A representation formula for transition probability densities of diffusions and applications. Stoch. Process. Appl. 111(1), 57–76 (2004)
- 18.
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Grundlehren der mathematischen Wissenschaften, vol. 293, 3rd edn. Springer, Berlin (1999)
- 19.
Royer, G.: An initiation to logarithmic Sobolev inequalities. Collection SMF: Cours spécialisés. American Mathematical Society (2007)
- 20.
Spokoiny, V.G.: Adaptive drift estimation for nonparametric diffusion model. Ann. Stat. 28(3), 815–836 (2000)
- 21.
Strauch, C.: Sharp adaptive drift estimation and Donsker-type theorems for multidimensional ergodic diffusions. Ph.D. thesis, Universität Hamburg (2013)
- 22.
Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Grundlehren der mathematischen Wissenschaften, vol. 233. Springer, Berlin, New York (1979)
- 23.
Tsybakov, A.B.: Pointwise and \(\sup \)-norm sharp adaptive estimation of functions on the Sobolev classes. Ann. Stat. 26(6), 2420–2469 (1998)
- 24.
van de Geer, S.: Empirical Processes in M-Estimation. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2000)
- 25.
van der Vaart, A.W., Wellner, J.W.: Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer, New York (1996)
The author is grateful to her Ph.D. advisor, Angelika Rohde, and Enno Mammen for constant encouragement and constructive advise. She also would like to thank two anonymous referees for helpful comments which led to a substantial improvement of the paper. This work was partially supported by the DFG Priority Program SPP 1324 (Project: RO 3766/2-1).
Appendix A: Preliminaries
We first give a result which allows to deduce the exact asymptotics for pointwise estimation of the drift component \(b^j\) from exact results on estimating \(b^j\rho _{b}\), \(j\in \{1,\ldots ,d\}\). In particular, it allows to identify \(D(\beta ,L;\rho _b,\sigma _0)\) as defined in (1.3) as the optimal normalizing factor for estimating the \(j\)-th component of \(b\in \varPi (c_1,c_2,\sigma _0\mathbf {Id})\). For the detailed derivation of the exact lower bound, we refer to Theorem 2.5.7 in Strauch [21].
Lemma 1
There exist two positive constants \(C_1,C_2\) (depending only on \(c_1\), \(c_2\) and \(\sigma )\) such that the invariant density \(\rho _{b}\) satisfies \(\rho _{b}(x) \le C_1\mathrm {e}^{-C_2\Vert x\Vert ^2}\), \(x \in \mathbb {R}^d\), for any \(b\in \varPi (c_1,c_2)\).
If \(b\in \widetilde{\varPi }(c_1,c_2)\) and if \(\rho _{b} \in {\mathcal {S}}(\beta +1,L')\), for some \(\beta >d/2\) and \(L' > 0\), then there exists an invariant density estimator \(\widehat{\rho }_T\) such that$$\begin{aligned} \mathbf {E}_b|\widehat{\rho }_T(x)-\rho _{b}(x)|^{2} \le K_1 T^{-\frac{\beta +1-(d/2)}{\beta +1}} \exp \left( -K_2\Vert x\Vert \right) \!, \qquad x \in \mathbb {R}^d, \end{aligned}$$(7.1)
where the constants \(K_1,K_2\) depend only on \(L',c_1,c_2\) and \(\sigma \).
The pointwise upper bound on \(\rho _b\) is an immediate consequence of the results of Metafune et al. [15] who study global regularity properties of invariant measures of divergence-form operators. Their results also hold in our specific framework since we restrict attention to the case of constant, uniformly elliptic diffusion part. Denote by \(\lambda _{\max }\) the largest eigenvalue of \(a\). Due to Corollary 2.5 in Metafune et al. [15], \(\mathrm{(P_1) }\) implies that \(\exp (\eta \Vert x\Vert ^2) \in L^1(\mu _b)\) for \(\eta < c_1\left( 2\lambda _{\max }\right) ^{-1}\). Since \(\left\| b(x)\right\| \le c_2(1+\Vert x\Vert )\lesssim \exp (\Vert x\Vert )\), Theorem 6.1 in Metafune et al. [15] applies and yields the assertion.
Let \(G=G_{\beta +1}:\mathbb {R}^d\rightarrow \mathbb {R}\) be the kernel with Fourier transform$$\begin{aligned} \phi _G(\lambda ) = \int _{\mathbb {R}^d} \mathrm {e}^{\mathrm{i }\lambda ^\top y} G(y)\mathrm {d}y = \frac{1}{1+\Vert \lambda \Vert ^{2(\beta +1)}}, \quad \lambda \in \mathbb {R}^d, \end{aligned}$$
and define the invariant density estimator$$\begin{aligned} \widehat{\rho }_T(x)=\widehat{\rho }_{T,h}(x) := \frac{1}{T h^d} \int _0^T G\left( \frac{X_u-x}{h}\right) \mathrm {d}u, \quad x \in \mathbb {R}^d. \end{aligned}$$
The bandwidth \(h = h_T \searrow 0\) is to be specified later. For bounding the stochastic error, note that, using (4.3),$$\begin{aligned} \mathbf {E}_b \big |\widehat{\rho }_T(x)-\mathbf {E}_b \widehat{\rho }_T(x)\big |^2&= \frac{1}{T^2h^{2d}}\ \mathrm{Var }_b\Bigg (\int _0^T G\left( \frac{X_u-x}{h}\right) \mathrm {d}u\Bigg ) \\&\le \frac{C}{T h^{2d}} \int _{\mathbb {R}^d} G^2\left( \frac{y-x}{h}\right) \rho _{b}(y)\mathrm {d}y. \end{aligned}$$
Taking into account the regularity properties of \(G\), a multidimensional version of Theorem 1A in Parzen [16] yields$$\begin{aligned} \mathbf {E}_b \big |\widehat{\rho }_T(x)-\mathbf {E}_b \widehat{\rho }_T(x)\big |^2 \le \frac{C}{Th^d} \rho _{b}(x)\Vert G\Vert _{L^2(\mathbb {R}^d)}^2 (1+o_T(1)). \end{aligned}$$
It remains to treat the bias term. Note that, using in particular Cauchy–Schwarz,$$\begin{aligned} \big |\mathbf {E}_b \widehat{\rho }_T(x)-\rho _{b}(x)\big |&= (2\pi )^{-d} \left| \int _{\mathbb {R}^d}\phi _{\rho _{b}}(\lambda )\left\{ \left( 1+\Vert h\lambda \Vert ^{2\beta }\right) ^{-1}-1\right\} \mathrm {e}^{-\mathrm{i }\lambda ^\top x}\mathrm {d}\lambda \right| \\&\le h^{\beta + 1}\left( (2\pi )^{-d}\int _{\mathbb {R}^d}\left| \phi _{\rho _{b}} (\lambda )\right| ^2\Vert \lambda \Vert ^{2(\beta +1)}\mathrm {d}\lambda \right) ^{1/2}\\&\quad \times \left( (2\pi )^{-d}\int _{\mathbb {R}^d}\frac{\left\| h\lambda \right\| ^{2(\beta +1)}}{\left( 1+\Vert h\lambda \Vert ^{2(\beta +1)}\right) ^2} \mathrm {d}\lambda \right) ^{1/2}\\&\le L' ~ (2\pi )^{-d/2}\left( \int _{\Vert y\Vert \le 1}\frac{\mathrm {d}y}{(1+\Vert y\Vert ^{2\beta })^2} + \int _{\Vert y\Vert >1} \frac{\mathrm {d}y}{\Vert y\Vert ^{2\beta }}\right) ^{1/2}\\&\quad \times h^{\beta +1-d/2} =: M h^{\beta +1-d/2}. \end{aligned}$$
Specifying \(h = h_T \sim \left( \frac{C\rho _{b}(x)}{M^2T}\right) ^{\frac{1}{2(\beta +1)}}\) and using the upper bound on \(\rho _b(x)\) from part (a), we obtain (7.1). \(\square \)
Denote by \(N_{[\,]}\left( \varepsilon ,\mathcal {F},L^2(\mu _b)\right) \) the \(\varepsilon \)-entropy with bracketing, that is, the smallest number of \(\varepsilon \)-brackets (in \(L^2(\mu _b)\)) which are required to cover \(\mathcal {F}\) (cf. van der Vaart and Wellner [25], Definition 2.1.6).
Lemma 2
Let \(b\in \varPi (c_1,c_2,\sigma )\), and suppose that \(X\) satisfies (PI). Fix \(j \in \{1,\ldots ,d\}\), and assume that there exists some positive constant \(B\) such that, for any bounded measurable \(f\in L^2(\mu _b)\),$$\begin{aligned} \max \bigg \{\sup _{x \in \mathrm{supp}(f)} |b^j(x)|,\sup _{x \in \mathrm{supp}(f^2)} |b^j(x)|^2\bigg \} \le B. \end{aligned}$$(7.2)
Then Assumption (BI) is satisfied.
Let \(\mathcal {F}\subset L^2(\mu _b)\) be some class of measurable functions \(f: \mathbb {R}^d \rightarrow \mathbb {R}\), and assume that, for some positive constants \(K\) and \(M\), it holds$$\begin{aligned} \sup _{f \in \mathcal {F}} \Vert f\Vert _\infty \le K, \quad \sup _{f \in \mathcal {F}}\Vert f\Vert _{L^2(\mu _b)} \le M. \end{aligned}$$
Grant Assumptions (BI) and (SG). Then, for arbitrary \(T >0\) and any positive \(r\) satisfying, for some positive constants \(K_1\) and \(K_2\),$$\begin{aligned} \frac{K_1}{\sqrt{T}}\int _0^1 \max \left\{ \sqrt{\log N_{[\ ]}(\varepsilon ,\mathcal {F},L^2(\mu _b))},1\right\} \mathrm {d}\varepsilon ~ \le ~ r ~ \le ~ \frac{K_2M^2}{K}, \end{aligned}$$
there exist some positive constants \(C_1\) and \(C_2\) such that
(a) Letting, for \(r,T>0\),
Theorem 1.1 in Lezaud [13] implies that
Using the spectral gap assumption, we get, for any \(T>0\), \(g \in L^2(\mu _b)\),
Consequently, in view of (7.2),
and \(\Vert fb^j\Vert _\infty \le B \Vert f\Vert _\infty \). Plugging these estimates into (7.4), we obtain the asserted inequality.
(b) Under the given assumptions, Bernstein's inequality for continuous martingales can be used to show that there exists some constant \(\widetilde{C}_B\) such that, for any \(r,T>0\),
To see this, write \(\mathsf {q}_T(r) \le \mathsf {p}_T(r/2) + \mathsf {p}'_T(r/2)\), for \(\mathsf {p}_T(\cdot )\) introduced in (7.3) and
Letting \(M_t(f) := \int _0^t f(X_u)\sum _{k=1}^d \sigma _{jk}~ \mathrm {d}W_u^k\), \(t \ge 0\), and denoting by \(\langle M\rangle _\cdot \) the quadratic variation of the martingale \(M\), Bernstein's inequality for continuous martingales (see p. 154 in Revuz and Yor [18]) gives
Theorem 1.1 in Lezaud [13] then can be used to show that
The inequality (7.5) now follows for \(\widetilde{C}_B := 4\max \big \{2c_P,2c_Pa_{jj}^2,c_Pa_{jj},1\big \}\). In view of (7.5), a uniform exponential inequality in the spirit of Theorem 5.11 in van de Geer [24] is available. Indeed, Theorem 5.11 in van de Geer [24] appears as a special case of the uniform inequality for martingales in van de Geer [24]'s Theorem 8.13, and the proof of Theorem 8.13 continues to hold in the diffusion setting if the Bernstein inequality for martingales in van de Geer [24]'s Corollary 8.10 is replaced with the Bernstein-type deviation inequality (BI). \(\square \)
For the proof of the following Lemma, we refer to the proofs of Proposition 1 in Klemelä and Tsybakov [10] and of Lemma 10 in Tsybakov [23].
Lemma 3
Let \(\beta >d/2\), and, for \(\mathbb {I}_\beta \), \(\widetilde{K}_\beta (\cdot )\) and \(\mathrm {b}= \mathrm {b}(\beta )\) defined according to (1.4), (4.7) and (4.8), respectively, let \(K_\beta ^*(x):= \mathbb {I}_\beta ^{-1} \mathrm {b}^{-\beta +d/2} \ \widetilde{K}_\beta (\mathrm {b}x)\).
(cf. Proposition 1 in Klemelä and Tsybakov [10]) It holds$$\begin{aligned} \widetilde{K}_\beta (0) = (2\pi )^{-d} \int _{\mathbb {R}^d} \left( 1+\Vert \lambda \Vert ^{2\beta }\right) ^{-1}\mathrm {d}\lambda = \frac{2\beta }{d}\ \mathbb {I}_\beta ^2, \end{aligned}$$
and, for \(K_\beta (x) = \mathrm {b}^d \widetilde{K}_\beta (\mathrm {b}x)\), \(\big \Vert K_\beta \big \Vert _{L^2(\mathbb {R}^d)} = \mathbb {I}_\beta \left( \frac{2\beta - d}{d}\right) ^{\frac{\beta +d/2}{2\beta }} = \mathbb {I}_\beta \ \mathrm {b}^{\beta +d/2}\).
(cf. Lemma 10 in Tsybakov [23]) For fixed \(\delta \in (0,1)\), there exists some compactly supported modification \(\overline{K}_\beta \) of \(K_\beta ^*\) which enjoys the following properties,$$\begin{aligned} \big \Vert \overline{K}_\beta \big \Vert _{L^2(\mathbb {R}^d)}&\le 1-\delta /2, \end{aligned}$$(K1)$$\begin{aligned} \eta _\beta (\overline{K}_\beta )&\le 1-\delta /2, \end{aligned}$$(K2)$$\begin{aligned} (1-\delta /2) K_\beta ^*(0)&\le \overline{K}_\beta (0) \le K_\beta ^*(0). \end{aligned}$$(K3)
Appendix B: Proofs
B.1: Lower bound
Proof (of Theorem 1)
Let \(\psi _{\beta ,L} = \psi ^j_{\beta ,L} := \psi _{T,\beta } C_j(\beta ,L;\rho _{b},\sigma )\), for \(\psi _{T,\beta }\) and \(C_j(\beta ,L;\rho _{b},\sigma )\) defined in (3.1) and (3.3), respectively. To enlighten notation, the dependence on the coordinate \(j \in \{1,\ldots ,d\}\) will be mostly suppressed in the sequel.
(I) Construction of the hypotheses. Let \(L \in [L_*,L^*]\), fix some nondegenerate \(\mathbb {R}^{d\times d}\)-matrix \(\sigma \), let \(a:=\sigma \sigma ^\top \), and consider some positive density function
Fix \(\delta _{0} \in (0,1/2)\), \(c_1,c_2>0\), and assume that \(\rho \) is such that the function
satisfies \(\rho _{T,0}(x_0) \ge \rho _T^*\), for any \(x\) with \(\Vert x\Vert \) large enough,
and for any \(x\in \mathbb {R}^d\),
Consequently, \(a\nabla (\log \rho _{T,0})/2=:b_{T,0}\in \varPi (c_1,c_2,\sigma )\). In particular, the SDE \(\mathrm {d}X_t = b_{T,0}(X_t)\mathrm {d}t +\sigma ~ \mathrm {d}W_t\), \(t\ge 0\), admits a strong solution with Lebesgue continuous invariant measure and invariant density \(\rho _{T,0}\). Define further
For \(\beta >d/2\), consider \(\widetilde{K}_\beta \) and \(\mathrm {b}= \mathrm {b}(\beta )\) as introduced in (4.7) and (4.8), respectively, and denote again \(K_\beta ^*(x)=\mathbb {I}_\beta ^{-1} \mathrm {b}^{-\beta + d/2} ~ \widetilde{K}_\beta (\mathrm {b}x)\). Lemma 3 implies that
Denote by \(\overline{K}_\beta \) the compactly supported modification of \(K_\beta ^*\) from Lemma 3 satisfying (K1), (K2), and (K3) for \(\delta = \delta _0\). Define the function \(g_{T,\beta _*}: \mathbb {R}^d \rightarrow \mathbb {R}\) such that, for any \(k \in \{1,\ldots ,d\}\),
and consider the hypothesis \(g_{T,1}\), defined as \(g_{T,1}:= \sum _{k=1}^d a_{jk}\partial _k\rho _{T,1}/2\). The function \(b_{T,1}:\mathbb {R}^d\rightarrow \mathbb {R}^d\) is taken as \(b_{T,1} := a\nabla (\log \rho _{T,1})/2\). Note that, for \(T\) large enough,
Plugging in the respective definitions of the hypotheses, it can be shown that \(g_{T,0}\in {\mathcal {S}}(\beta _T,L)\), \(g_{T,1} \in {\mathcal {S}}(\beta _*,L)\) and \(b_{T,1}\in \varPi (c_1,c_2,\sigma )\). The above definitions of the hypotheses further imply that \(\rho _{T,0}\in {\mathcal {S}}(\beta _T+1,L'), \rho _{T,1}\in {\mathcal {S}}(\beta _*+1,L')\) and
Summing up, \(\rho _{T,0}\in \varSigma _T(\beta _T,L)\) and \(\rho _{T,1}\in \varSigma _T(\beta _*,L)\).
(II) A version of Theorem 6(i) in Tsybakov [23]. The central ingredient of the proof is a special case of Theorem 6(i) in Tsybakov [23]. It will be applied in the following situation: Denote by \(\mathbf {E}_i = \mathbf {E}_{b_{T,i}}\) expectation under the measure \(\mathbf {P}_i = \mathbf {P}_{b_{T,i}}\) associated with the hypothesis \(b=b_{T,i}\), \(i \in \{0,1\}\), and note that
where \(Q_T := \psi _{\beta _*,L}\psi _{\beta _T,L}^{-1}\), \(\widehat{T}_T:= \psi _{\beta _*,L}^{-1}\big (\widehat{g}_T(x_0)-g_{T,0}(x_0)\big )\), and
The proof of the following lemma is completely along the lines of the proof of Theorem 6(i) in Tsybakov [23].
Lemma 4
(Theorem 6(i) in Tsybakov [23]) Consider \(Q_T\), \(\widehat{T}_T\) and \(\theta _1\) as introduced above, and assume that \(\theta _1 \in \mathbb {R}\) satisfies
If \(\mathbf {P}_0,\mathbf {P}_1\) are such that \(\mathbf {P}_0 \ll \mathbf {P}_1\) and, for \(\tau > 0\) and \(\alpha \in (0,1)\) fixed,
where the infimum is taken over all \(\widehat{T}_T= \psi _{\beta _*,L}^{-1}\big (\widehat{g}_T(x_0)-g_{T,0}(x_0)\big )\).
We proceed with verifying (A1) and (A2). Note first that
Since, for \(T\) large enough, \(\psi _{\beta _*,L}^{-1}\int _{\mathbb {R}^d}g_{T,\beta _*}(y) \mathrm {d}y \le \delta _0/2\), this implies
Denote by \(Y\) the solution of the SDE \(\mathrm {d}Y_t = b_{T,1}(Y_t)\mathrm {d}t+\sigma ~\mathrm {d}W_t\). In order to verify (A2), note that the specifications on pp. 296–297 in Liptser and Shiryaev [14] imply that the likelihood ratio under \(\mathbf {P}_1\) is given by
To proceed, set
denote \(g := (b_{T,0} -b_{T,1})^\top a^{-1}(b_{T,0} -b_{T,1})\), and consider the following stationary sequence of random variables,
Since \(g \in L^1(\mathbf {P}_1)\), it follows from the ergodic theorem that, for any \(t>0\),
In particular, this implies by means of the martingale CLT that, for some standard Brownian motion \(W\),
Denoting by \([s]\) the integer part of \(s\) and considering an arbitrary sequence \(\gamma (s)\rightarrow _{s\rightarrow \infty }0\), it holds
Choosing \(t\equiv 1\) in (8.8) and passing to the continuous-time case, we obtain
It is verified by straightforward algebra that the definition of the hypotheses \(b_{T,0} \) and \(b_{T,1}\) entails that
The definition of \(g_{T,\beta _*}\) further implies that, using in particular (8.1),
The last inequality follows from (K1). Thus, plugging in the definition of \(h_{T,\beta _*}\) [see (8.2)] and using (8.3),
It can be shown by analogous arguments that the terms
are asymptotically negligible. Thus, for \(c\) defined in (8.7) and whenever \(\delta _{0}\) is small and \(T\) is large enough,
Consequently, for \(\tau := \exp \left( -\frac{\left( 1-\delta _{0}^2/4\right) d\log T}{2\beta _*}\right) \), it holds a.s.
The verification of (A2) is accomplished by means of a tightness argument. Consider some sequence of probability measures \((\mathbf {P}_n)_{n\ge 1}\) on some measurable space, converging weakly to some probability measure \(\mathbf {P}\). Tightness of \(\mathbf {P}_n\) implies that, for any sequence \(\gamma _n\rightarrow -\infty \),
Thus, \(\lim _{m\rightarrow \infty }\sup _{n\in \mathbb {N}}\mathbf {P}_n\left( (-\infty ,\gamma _m)\right) = 0\) and \(\lim _{m\rightarrow \infty }\inf _{n\in \mathbb {N}}\mathbf {P}_n\left( (-\infty ,\gamma _m)\right) =1\). In particular,
In the current framework, this last assertion implies that, for
one has [plugging in (8.6)]
For large enough \(T\) and fixed \(\tau >0\) (where \(\delta _{0}\in (0,1)\) can be chosen arbitrarily small), we thus obtain
(III) Completion of the proof. In view of (8.5) and (8.9), Lemma 4 gives
Since, for \(C := C_j(\beta _*,L;\rho _{T,0},\sigma )/ C_j(\beta _T,L;\rho _{T,0},\sigma )\),
we have
As \(T \rightarrow \infty \), \(\tau Q_T^2 \rightarrow \infty \). Choosing \(\delta _{0} > 1/A\) for \(A\) large enough to ensure \(\delta _{0} < 1/2\), it holds
Taking now \(A\rightarrow \infty \), the assertion follows. \(\square \)
B.2: Upper bound
Proof (of Theorem 2)
Let \(\beta \in \left[ \beta _*,\beta _T\right] \), \(L\in \left[ L_*,L^*\right] \), \(\beta ' \in (d/2,\beta ]\), \(c_1\in (0,\infty ]\), \(c_2>0\), \(\sigma \) some nondegenerate \(\mathbb {R}^{d\times d}\)-matrix, \(L'>0\), and fix \(j \in \{1,\ldots ,d\}\).
Denote by \(\gamma _{Ti}\), \(i \in \mathbb {N}\), functions of \(T\) such that \(\lim _{T\rightarrow \infty }\gamma _{Ti} = 0\). For \(\psi _{T,\beta }\) and \(C_j(\beta ,L;\rho _{b},\sigma )\) introduced in (3.1) and (3.3), respectively, recall that \(\psi _{\beta ,L} = \psi _{\beta ,L}^j = \psi _{T,\beta } C_j(\beta ,L;\rho _{b},\sigma )\). To enlighten notation, the dependence on the coordinate \(j\) again will be mostly suppressed. Denote by \(\widetilde{T}(\beta )\) the effective noise level under adaptation, defined as
Consider the following deterministic counterparts of the bandwidth \(\widehat{h}_{T,\beta '}\) and the thresholding sequence \(\widehat{\eta }_{T,\beta '}\),
Define \(\overline{\delta }_T := (\log T)^{-1}\), and introduce the random event
and the associated deterministic set \({\mathcal {H}}_{T,\beta '} \!=\! {\mathcal {H}}^j_{T,\beta '} \!:=\! \bigg \{h : \Big |\big (h/h_{T,\beta '}\big )^{\widetilde{\beta '}-d/2}\!-\!1\Big |\!\le \! \overline{\delta }_T\bigg \}\). Note that there exists a positive constant \(c_0\) such that
Denote the kernel estimator of \((b^j\rho _{b})(x_0)\) with deterministic bandwidth \(h \in {\mathcal {H}}_{T,\beta '}\) by
and set \(g_{T,\beta '}(x_0) := g_{T,\beta '}\big (x_0,h_{T,\beta '}\big )\). Define
and let \(d_T(\beta ):= \sqrt{(d\log T)/\beta }\) such that \(\mathrm{s }_T(\beta ) d_T(\beta ) = \eta _{T,\beta }\). For \(\beta ' \le \beta \), introduce the auxiliary sequence
Lemma 5
(Bound on the bias) Consider the estimator \(g_{T,\beta '}(x_0,h)\) defined in (8.11), and let
For any \({\mathcal {H}}_{T,\beta '} \ni h >0\),
The bound on the bias in (8.12) is proven by standard arguments and relies in particular on exploiting the scaling properties of the Fourier transform of \(K_{\beta ',h}\). The remaining assertions are Lemma 1(ii), (iii) in Klemelä and Tsybakov [11].\(\square \)
The principal importance of exponential bounds on the stochastic error of estimators considered in the adaptive procedure was already indicated in the introduction. The Bernstein-type deviation inequality (BI) and its implication (BI+), the basic uniform exponential inequality stated in Lemma 2, can be applied to derive more specific bounds on the stochastic error of the estimators \(g_{T,\beta }\) defined according to (8.11). The following function classes are defined analogously to Butucea [3],
For \(h \in H_{T,\beta }\), let
Lemma 6
Grant Assumptions (BI) and (SG+). For any \(\beta ' > d/2\), the stochastic error \(Z_{T,\beta '}(\cdot )\) defined according to (8.13) has the following properties:
For any \(u \in \big [\tau _T(\beta '), \ R_1 \mathrm{s }_T(\beta ')\sqrt{\log T}\big ]\), \(R_1>0\) an absolute constant, there exist some sufficiently small \(\gamma >0\), independent of \(\beta '\), and some universal constant \(c^{\prime }_1>0\) such that$$\begin{aligned} \mathbf {P}_b\Bigg (\sup _{h \in H_{T,\beta '}}\big |Z_{T,\beta '}(h)\big | > u\Bigg ) \le c^{\prime }_1 \exp \left( -\frac{1}{2} \bigg (\frac{u(1-\gamma )}{\mathrm{s }_T(\beta ')}\bigg )^2\right) + o\big (T^{-1}\big ).\qquad \quad \end{aligned}$$(8.14)
For any \(u\in \big [R_1 \mathrm{s }_T(\beta ')\sqrt{\log T}, \ R_2\big ]\), \(R_1, R_2 >0\) absolute constants, it holds, for some absolute constants \(c^{\prime }_2, c^{\prime }_3 >0\),$$\begin{aligned} \mathbf {P}_b\Bigg (\sup _{h \in H_{T,\beta '}}\big |Z_{T,\beta '}(h)\big |> u\Bigg ) \le c^{\prime }_2\exp \left( -c^{\prime }_3\bigg (\frac{u}{\mathrm{s }_T(\beta ')}\bigg )^2\right) . \end{aligned}$$
Assume that \(\beta ' < \beta \). Then, uniformly in \(\beta \in {\mathcal {B}}_T\),$$\begin{aligned}&\sup _{\mathop {\beta '<\beta }\limits ^{\beta '\in \mathcal {B},}} m \sup _{b\in \widetilde{\varPi }(c_1,c_2)} \sup _{\rho _b \in \varSigma _T(\beta ,L)}\psi _{\beta ,L}^{-2}\, \\&\quad \times \,\mathbf {E}_b\left[ \bigg (\sup _{h \in H_{T,\beta '}} \big |Z_{T,\beta '}(h)\big |\bigg )^2 \, {1\!\!1}\bigg \{\sup _{h\in H_{T,\beta '}}\big |Z_{T,\beta '}(h)\big | > \tau _T(\beta ')\bigg \}\right] \rightarrow 0,\\&\sup _{\mathop {\beta '<\beta }\limits ^{\beta '\in \mathcal {B},}} m\sup _{b\in \widetilde{\varPi }(c_1,c_2)} \sup _{\rho _b \in \varSigma _T(\beta ,L)}\psi _{\beta ,L}^{-2}\, \\&\quad \times \, \mathbf {E}_b\left[ \bigg (\sup _{h \in H_{T,\beta '}} \big |Z_{T,\beta '}(h)\big |\bigg )^2 \ {1\!\!1}\bigg \{\sup _{h\in H_{T,\beta '}}\big |Z_{T,\beta '}(h)\big | > \sqrt{\mathrm{s }_T(\beta ')\psi _{\beta ,L}}\bigg \}\right] \rightarrow 0. \end{aligned}$$
The assertions are analogue to the statements in Lemma 4.3, Lemma 4.5 and Theorem 4.6 in Butucea [3]. For deriving the inequality (8.14) with the specific factor \(1/2\) in the exponent in the current diffusion framework, we however have to go into greater detail. Throughout the proof, \(D_1, D_2, \ldots \) denote positive constants. For fixed \(\delta ' \in (0,1)\) and arbitrary \(u,T>0\), write
and, denoting \(M_t(K) := \int _0^t K(X_u)\sum _{r=1}^d \sigma _{jr}\mathrm {d}W_u^r\), \(t>0\),
For any \(u \le R_1 \mathrm{s }_T(\beta ')\sqrt{\log T}\), we have
such that, since \(\beta '>d/2\),
Furthermore, the enhanced spectral gap assumption (SG+) gives
Thus, for \(T\) sufficiently large, \(C_B\Big (\varsigma _b\big (K_{\beta ,{h_{T,\beta '}}}\big )+ \delta ' u\ \big \Vert K_{\beta ,h_{T,\beta '}}\big \Vert _\infty \Big ) \le \mathrm{s }_T^2(\beta ') T.\) The Bernstein-type deviation inequality (BI) therefore implies that
For bounding \(\mathsf {t}_2\) from above, we first use Bernstein's inequality for continuous martingales which gives, for any \(h>0\),
By means of (BI) and using again that \(\beta '>d/2\), it can be shown that
Adding the upper bounds (8.15), (8.16) and (8.17), we obtain, for some small \(\gamma >0\),
Consider the sequence \(\delta _{T1} := \beta _T\overline{\delta }_T\sqrt{\log T} = (\log \log T)^\delta (\log T)^{-1/2} \rightarrow 0\), and note that, for any \(u>0\),
Since \(u(1-\delta _{T1}) \le u \le R_1 \mathrm{s }_T(\beta ')\sqrt{\log T}\), (8.18) gives an upper bound on the latter summand. For \(T\) large enough, it further holds
Taking into account that
and since
the uniform exponential inequality (BI+) implies that
Summing the upper bounds due to (8.18) and (8.19), we obtain (8.14).
The inequality stated in \(\mathrm{(b) }\) follows as an application of (BI+) with \(K := h_{T,\beta '}^{-d} K_{\max }\) and
Finally, part \(\mathrm{(c) }\) is proven similarly to Theorem 4.6 in Butucea [3] by noting that there exists some positive constant \(R\) such that \(\sup _{h \in H_{T,\beta '}}\big |Z_{T,\beta '}(h)\big | \le R h_{T,\beta '}^{-d}\). A suitable decomposition of
and uniform exponential bounds on the corresponding integrands as they follow from parts \(\mathrm{(a) }\) and \(\mathrm{(b) }\) of this lemma then yield the assertions. \(\square \)
The next lemma contains a decomposition of the normalizing factor \(\psi _{\beta ,L}\) and some relations which are needed later in the proof. For \(h_{T,\beta }\) defined according to (8.10), denote
Lemma 7
Let \(\beta \in [d/2,\infty )\), \(L\in \left[ L_*,L^*\right] \), and denote \(\nu = (\beta ,L)\). It then holds
Furthermore, there exist positive constants \(D_1, \ldots , D_5\), depending only on \(\beta _*, L_*, L^*, d\) and \(\sigma \), such that
and, for \(\beta ' \in [d/2,\infty )\), \(\beta ' < \beta \),
The proof of the decomposition is comparable to the derivation of relation (68) on p. 461 in Klemelä and Tsybakov [11]; for details, see the proof of Lemma 2.6.6 in Strauch [21]. Assertions (8.22) and (8.23) follow analogously to the proof of the relations (44)–(46) in Lemma 4 in Klemelä and Tsybakov [11] (pp. 453–454). For the proof of (8.24), we refer to the proof of Lemma 3.5 in Butucea [3]. \(\square \)
Main part of the proof of the upper bound. Define \(\beta ^-=\beta ^-(\beta )\) by
where \(\beta _T^+ := (\log \log T)^{\delta '}\), for some \(\delta ' \in (\delta ,1)\). We follow the standard approach and decompose the risk successively. Assume that \(b\in \widetilde{\varPi }(c_1,c_2,\sigma )\), let \(\nu = (\beta ,L)\), and set
For ease of notation, we usually suppress the dependence of the risk on the coordinate \(j\).
(I) We first consider the case \(\widehat{\beta }_T^j \ge \beta ^-\), and we show that
Define \(\overline{\beta }= \overline{\beta }(\beta )\) via the equation \(\left( \frac{\log T}{L^2 T}\right) ^{1/(2\beta )} = \left( \frac{\log T}{T}\right) ^{1/(2\overline{\beta })}\). Let \(\beta ^+ \in \mathcal {G}\) be the largest grid point \(\le \overline{\beta }\), and assume that \(T\) is large enough for ensuring \(\beta ^- < \beta ^+\). Denote \(\mathcal {G}_1 = \mathcal {G}_1(\beta ) := \left\{ \beta ' \in \mathcal {G}\mid \beta ^-\le \beta ' \le \beta ^+\right\} \), \(\mathcal {G}_2 = \mathcal {G}_2(\beta ):=\left\{ \beta '\in \mathcal {G}\mid \beta ^+ < \beta ' \le \beta _T\right\} \), and rewrite
Let \(\beta ' \in \mathcal {G}_1 = \mathcal {G}_1(\beta )\) and \(\rho _b\in \varSigma _T(\beta ,L)\), and assume that \(T\) is so large that \(\widetilde{\beta }(\beta ,\beta ')=\beta \). Using Lemma 5, the facts that \(\mathcal {H}_{T,\beta '}\subset H_{T,\beta '}\), that \(\beta '\le \overline{\beta }\) and the definition of \(\overline{\beta }\), it can be shown that
The following arguments are along the lines of the proof of the upper bound in Klemelä and Tsybakov [11] (see pp. 461–463). For any \(\beta ' \in \mathcal {G}_1\), there exists some positive constant \(C\) such that
Since \(\Lambda (\beta ,\beta ')\) is uniformly continuous in \(\beta ,\beta ' \in [\beta _*,\infty )\), this implies that \(\Lambda (\beta ,\beta ') \le 1 +\gamma _{T1}\). Furthermore, for any \(\beta ' \in \mathcal {G}_1\), \(\beta \in \left[ \beta _*,\beta _T\right] \), it holds \(\mathrm {b}_{\beta ,\beta '}\le \mathrm {b}_{\beta ,\beta }\ (1+\gamma _{T2})\). Consequently, for any \(\beta ' \in \mathcal {G}_1\), \(\rho _b\in \Sigma _T(\beta ,L)\),
Similar arguments (also see the derivation of line (54) on p. 1591 in Klemelä and Tsybakov [10]) yield
Recall the definition of the stochastic error \(Z_{T,\beta }(\cdot )\). Whenever \(\widehat{\beta }_T^j = \beta ' \in \mathcal {G}_1\) and the event \(A_{T,\beta '}\) holds, the above arguments imply that
The last line holds true since \(\mathcal {H}_{T,\beta '} \subset H_{T,\beta '}\) and in view of the decomposition of the normalizing factor \(\psi _\nu \) according to (8.21). If \(\widehat{\beta }_T^j \ge \beta ^+\), the definition of the estimator \(\widehat{\beta }_T^j\) according to (4.10) implies that \(\big | \widehat{g}_{T,\widehat{\beta }_T^j}^j(x_0) - \widehat{g}_{T,\beta ^+}^j(x_0)\big | \le \widehat{\eta }_{T,\beta ^+}\).
Therefore, if \(\widehat{\beta }_T^j = \beta ' \in \mathcal {G}_2\), it holds on \(A_{T,\beta ^+}\),
In view of the decomposition (8.21), this last line implies that
Thus, using (8.29) and (8.30),
The terms \(\mathsf {p}_1(\cdot ), \ldots , \mathsf {p}_4(\cdot )\) are now considered separately. Note first that, for any \(\beta ' \in \mathcal {G}_1\),
Since (8.26) holds for any \(\beta ' \in \mathcal {G}_1\), it can be shown by means of (8.22) and (8.23) that
The summand in (8.31) is bounded from above by the sum of the terms
Part (c) of Lemma 6 entails that the latter term tends to zero, uniformly in \(\beta ' \in \mathcal {G}_1\). Therefore,
Recall that the cardinality \(m\) of the grid \(\mathcal {G}\) satisfies
By construction, \(\beta ^+ \in \mathcal {G}_1\), such that \(\mathsf {p}_3(\beta ')\) is upper-bounded analogously. Consequently,
For \(\mathsf {p}_2(\cdot )\) and any \(\beta ' \in \mathcal {G}_1\), there exists some universal constant \(c_0\) such that
The regularity conditions on the bandwidth and the kernel used for defining \(\widehat{\rho }_T(x_0)\) ensure that, for any \(\beta ' \in (d/2,\beta ]\) and some sufficiently small constant \(\alpha >0\) fixed,
This implies that \(\mathsf {p}_2(\beta ')\) is exponentially small, for any \(\beta ' \in \mathcal {G}_1\), such that \(\sum _{\beta ' \in \mathcal {G}_1}\mathsf {p}_2(\beta ') \rightarrow 0.\) Analogously, it follows that \(\sum _{\beta ' \in \mathcal {G}_2}\mathsf {p}_4(\beta ') \rightarrow 0\), completing finally the verification of (8.25).
(II) It is proven now that
Let \(\beta ' \in \mathcal {G}_T\), and assume that the event \(A_{T,\beta '}\) holds. In view of the definition of the stochastic error \(Z_{T,\beta '}\) in (8.13) and taking into account Lemma 5, it holds, whenever \(\rho _b\in \varSigma _T(\beta ,L)\),
Then, using the definition of \(\mathrm {b}_{T,\beta '}\) in (8.20),
The term \(g_1(\nu )\) is bounded from above by exploiting the fact that the probability to underestimate the value of \(\beta \) by \(\widehat{\beta }_T^j\) substantially is small, whenever \(\rho _b\in \Sigma _T(\beta ,L)\). Recall that \(m\) is the cardinality of the grid \(\mathcal {G}\).
Lemma 8
(Probability of undershooting) Let \(\beta \in [\beta _*,\infty )\), \(\beta ' \in \mathcal {G}\), \(\beta ' < \beta ^-\), \(L\in \left[ L_*,L^*\right] \), and \(\nu = (\beta ,L)\). Then there exists some constant \(K\), depending only on \(\beta _*,L_*,L^*,d\) and \(\sigma \) such that, for any \(b\in \widetilde{\varPi }(c_1,c_2,\sigma )\),
The proof substantially relies on applications of Lemma 2. Since the basic arguments are similar to those used in the proof of Lemma 4.8 in Butucea [3] and the proof of Lemma 5 in Klemelä and Tsybakov [11], we do not include the proof but refer to Strauch [21]. \(\square \)
Let \(\beta \in \left[ \beta _*,\beta _T\right] \), \(\beta ' \in \mathcal {G}\), \(L \in \left[ L_*,L^*\right] \), \(\nu = (\beta ,L)\). By means of Lemma 8 and using relation (8.24) in Lemma 7, we obtain
In view of the upper bound on the cardinality \(m\) of the grid \(\mathcal {G}= \mathcal {G}_T\) in (8.32), it follows that \(\lim _{T\rightarrow \infty }\sup _{\nu \in \mathcal {B}_T}g_1(\nu ) = 0,\) and the first assertion in Lemma 6(c) immediately gives \(\lim _{T\rightarrow \infty }\sup _{\nu \in \mathcal {B}_T}g_2(\nu ) = 0\). Note finally that, for any \(\beta ' \in \mathcal {G}\),
Consequently, taking into account (8.33), it holds, independent both of \(\beta \) and \(\beta '\),
thus completing the proof of (8.34). \(\square \)
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Strauch, C. Exact adaptive pointwise drift estimation for multidimensional ergodic diffusions. Probab. Theory Relat. Fields 164, 361–400 (2016). https://doi.org/10.1007/s00440-014-0614-4
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- Ergodic diffusion
- Minimax drift estimation
- Exact constants in nonparametric smoothing
- Sharp adaptivity
- Pointwise risk
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An extensive plan which could create large numbers of jobs in Cork has been launched by Hardware Association Ireland (HAI). The plan, entitled 'Rebuilding Our Future', aims to kick-start the local economy in Cork post-Covid by encouraging home retrofitting and renovations.
The plan would be a major boost for local businesses, construction workers and tradespeople, especially in areas outside Dublin. It involves serious incentives and supports for making homes more energy efficient and will support existing local businesses and tradespeople by allowing homeowners to claim tax back on home renovation works.
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About a month before President Bush is to make his first state visit to China, reports have surfaced that the U.S.-made presidential plane for Chinese President Jiang Zemin has been found to be bugged with at least 20 listening devices.
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Shigurui onlineA film about fierce battles for life and death. When the ruler longs for a bloody spectacle, the fighters in the tournament are ordered to fight with military weapons. Tadanaga Tokugawa - the cruel daimyo Surugi is obsessed with death, he sees a majestic meaning in it. The tournament will lead to big losses, but the ruler doesn't care about that. Spectators gather for a terrible performance, where two fighters crippled in past battles are to perform: the one-armed Gennosuke Fujiki and the blind Irako Seigen. They left the school of Iwamoto Kogen, now the two longtime rivals must meet again in the samurai tournament ...
The rather well-known Japanese animation studio Madhouse has been operating for about a quarter of a century, and during this period it has managed to exhibit many works for the audience. From my point of view, this project really deserves praise. The series is compiled for only one season and is not aimed at a wide audience. There are age restrictions here. It is worth noting that the series has a versatile design and is sharpened identically to the legendary weapons of the samurai who have sunk into history and history. The cruelty and overt brutality of the film, most likely, can only be compared with the cruelty of the characters and mores of that time. The magnificent graphic performance and excellent study of ethnic musical themes provide the viewer with an undeniable opportunity to fully immerse themselves in the serial atmosphere. For a person who is not aware of the history, culture and traditions of Japan, it is recommended to use auxiliary materials as an explanation during the viewing period, without which a lot of on-screen actions may be misinterpreted, or not completely realized. From my point of view, in this work there is no clear distribution and formulation of evil and good, as well as the main character is absent here, representing one of the parties. All heroic images are thoroughly worked out and form a single whole, in which the raging spirit of that era is embodied, leaving the viewer the opportunity to make an independent choice. This anime is not intended for all viewers. Initially, it should not be viewed by adolescents. This fact is justified not even by the presence here of scenes of obvious violence, demonstrated with possible details of the anatomical plan and all kinds of horrors, because such things can often be seen on TV in our time. That is why it will be difficult for teenagers to understand what is happening. In this case, the viewer is not presented with an animated film, which tells about the brave and high-spirited samurai, constantly moving, saving various princesses and enjoying life. This is a viscous, slow, full of pauses, designed for a meaningful and thoughtful perception by those people who have made up their minds and are able to make up their minds with others.
The well-known MadHouse studio presents to the audience the following project, which does not fall under the existing stereotypes and is rather controversial. It can't help but amaze how he stands out from the cohesive ranks of other samurai action movies and dramas that have been heralded in the genre. It is so distinctive that it is not possible to compare it with other anime, since the event is futile and thankless. It is as controversial as it is unique. Identical to a samurai duel, there are no unnecessary elements.Romance, humor, languidly melancholic bisonens, pompousness and excess pathos here would look at least blasphemous in relation to the conceptual orientation of the entire series. Any stroke on the background of the picture, any lightning-fast movement and any static type of frame, here has a certain meaning. In appearance, the anime turned out to be so complete that I would like to believe in the above statement. 1629 year. The beginning of the Edo era. From the Suruga province, the Tokugawa Tadanaga daimyo decides to hold a tournament using real weapons, which violates all existing foundations. Such an innovation should lead to the death of 16 swordsmen out of 22 participating in the competition. On this episode with a historical background, you can finish. Initially, the viewer sees the preparations for this tournament and the only duel where, on the basis of an unthinkable coincidence, a blind swordsman with a damaged foot and a one-armed warrior meet. This fact is simply inconceivable, and therefore the audience does not hesitate in their own expressions, defining everything that happens as an insult to the tournament as such, and the duel, in their opinion, is not worthy to be called that. Meanwhile, the shortcomings only hardened the rival spirit, and from the outside it might seem that problems to the soldiers are not at all an obstacle, and they themselves are dangerous and sinister opponents. But the unusualness of the fight lies not only in the shortcomings of the opponents - they know each other and feel a burning hatred for one another. Naturally, this fight will be deadly and will be the end of their long confrontation. From this, the story begins, but here, it ends. Further, another long history, preceding this battle, will be demonstrated. It is noteworthy that this anime initially lacks good or bad. Also, the main character is missing here, which at least somehow stood out. A pair of samurai who have their own life paths. Each of them has its own bushido, and no matter how strange it may seem, but also their own goals and ideals. They are completely different. With the passage of time, their rivalry transforms into confrontation and enmity, then reborn into hatred. It was not the circumstances that turned out that way, it was just the way it had to happen. It makes no sense to give preference to someone, but each viewer will be able to choose a certain hero for himself. The series is notable for its incredible cruelty. Naturally, the anatomical details can hardly be called accurate, since the spine in its cross section is not round, and when cutting off the head of the trachea, it is not possible to see. Meanwhile, this fact in no way diminishes the achievements of the animators, since the pulled out eyes and guts did their job completely. When you watch anime, you involuntarily start to frown. The fact that the authors focus on outright cruelty can be seen from the initial episode. And the episode of the first series, when, as an argument for his master, he removes the intestines from his stomach, immediately dispels all doubts. Meanwhile, it is impossible not to notice the relevance of bloodiness, since it is an integral part of the samurai life of that era. The conversation concerns only a part of that period, the atmosphere of which is successfully conveyed to us by the authors. And the atmosphere turned out to be just excellent. Dull, gloomy and fantastic art, where only red stands out as bright. In fact, the musical accompaniment fits perfectly, although it is still not worth listening to the soundtrack separately. In addition, the authors purposefully use the old Japanese language. They wanted to demonstrate to the audience a detailed and accurate image of those times and that life. We can confidently say that their idea was completely successful. Fights of samurai are presented separately, which are completely devoid of pretentious attacks and pirouettes, bombastic dialogues and ranting. It may seem that static shots in themselves carry so much tension and dynamics that the thought involuntarily arises: "This is how the real fight should be". In this respect, anime stands out quite strongly, which is virtually identical to the genre of Japanese cinema "chambara". In films, this is how fights are presented - just a few strikes, of which the first is often decisive. For connoisseurs, viewing will also be interesting - the first ten centimeters is the main part of the defeat of the katana, which in this case is observed and demonstrated repeatedly. Still, the anime turned out to be ambiguous. Naturally, the details and details are great fun. The life atmosphere of the dojo conveyed is great. I always want to define this anime as an incomparable composition. Meanwhile, there is always a certain "but", and in the present series it is quite significant. It is enclosed directly in the plot guide. In this respect, the authors made a mistake. The turn of the story and its narration is unhurried and to some extent prolonged, there is a lack of intrigue. Many viewers, most likely, will not be able to find the culmination, and the ending can completely cross out the impressions received when watching. Be that as it may, but summing up the above, it is worth noting that the anime turned out to be wonderful in terms of the uniqueness of the entourage and content. In this case, it is not the story of a couple of people that is considered, but a fragment of their life. Naturally, this series of samurai demonstrates from a newly appeared, in fact, documentary position, and in such a genre as a samurai thriller, he made a very serious leap. I cannot recommend it to absolutely everyone for viewing, since the spectacle is presented in a slightly different form than the manga and directors have already taught us. Here, not only the plot captures the spirit, but also its successful combination with the atmosphere. The pleasure that you get when viewing, when immersed in the aura and the world of the picture, is not possible to compare with something. If, nevertheless, it was decided to look, then it should be done especially carefully on the initial series, since it is very easy to get confused in events and names. If someone watches the series in full, then in any case, some emotions will form, since there can be virtually no one left indifferent after watching.


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Audiobook Genres
The One Man: The Riveting and Intense Bestselling WWII Thriller
Unabridged Audio Book
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Andrew Gross
Edoardo Ballerini
12 Hours 49 Minutes
Macmillan Audio
August 2016
Audio Book Summary
'Actor Ballerini turns the scene into a fascinating radio drama...These conversational moments, delicately crafted by Gross and splendidly performed by Ballerini, have a profound effect on the novel's equally well-enacted, action-filled, breathless escape sequence.' - Publishers Weekly
1944. Physics professor Alfred Mendel and his family were trying to flee Paris when they were caught and forced onto a train, along with thousands of other Jewish families. At the other end of the long, torturous train ride, Alfred is separated from his family and sent to the men's camp, where all of his belongings are tossed on a roaring fire. His books, his papers, his life's work. The Nazis have no idea what they have just destroyed. And without that physical record, Alfred is one of only two people in the world with his particular knowledge. Knowledge that could start a war, or end it.
Nathan Blum works behind a desk at an intelligence office in Washington, DC, but he longs to contribute to the war effort in a more meaningful way, and he has a particular skill set the U.S. suddenly needs. Nathan is fluent in German and Polish, and he proved his scrappiness at a young age when he escaped from the Krakow ghetto. Now, the government wants him to take on the most dangerous assignment of his life: Nathan must sneak into Auschwitz, on a mission to find and escape with one man.
This historical thriller from New York Times bestseller Andrew Gross is a deeply affecting, unputdownable series of twists and turns through a landscape at times horrifyingly familiar but still completely compelling.
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Really good
The hotel it's a good option if you're staying a couple of days in Rio do Janeiro. The rooms are a bit small, but the staff is amazing and really sweet, and everything is aesthetically pleasing! The breakfast could be improved, but it was really good! It's also very well located


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E-mail: firstname.lastname@example.com
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The NHRA is set to embark on an ambitious-but-tenuous return from the coronavirus-centered interruption, but evidently it will be doing so without the four-car John Force Racing team.
Just five days before the start of the E3 Spark Plugs Nationals at Lucas Oil Raceway—located no more than 2 miles from the expansive John Force Racing shop in Brownsburg, Indiana—the team let the Monday entry deadline pass without the names of 151-time winner and 16-time champion Force and Robert Hight on the Funny Car list and without the names of 2017 champion Brittany Force and Austin Prock on the Top Fuel list.
Force has not announced he has parked his team for the remainder of the year, despite rumors of sponsorship defections or suspensions.
Team spokeswoman Sara Slaughter said Monday, "We aren't making any sort of announcement or commenting on any of the numerous rumors that are out there."
However, Jeffrey Young, vice president of marketing and communications for the NHRA Mello Yello Drag Racing Series, confirmed that the rumors were, in fact, fact. JFR missed the deadline and is not expected to race this weekend.
"The John Force Racing team is not registered for the E3 Spark Plugs NHRA Nationals," Young told Autoweek on Monday evening after the 4 p.m. ET entry deadline. "We look forward to their return in the future."
At this point, the news has moved past rumor to fact, unless some financial miracle occurs overnight and the NHRA bends the deadline rule. For now, the live Fox TV broadcast of the July 11-12 race will be missing the Force contingent.
With the JFR news, the show will go on without the most successful drag racer ever and without reigning and three-time Funny Car champion Hight, who recorded the fastest speed in the sport's history (339.87 mph). The show will go on without Brittany Force, the 2017 Top Fuel champion who owns both the elapsed-time and speed records in the sport's headliner category (3.623 seconds, 338.17 mph in 1,000 feet). And it will go on without 2019 NHRA Rookie of the Year Prock in the Top Fuel mix.
Young said, however, that the entry lists are healthy for Indianapolis. A total of 18 cars are entered in the Top Fuel class, 18 in Funny Car and 22 in Pro Stock.
A rumor had floated throughout the industry for several weeks that Chevrolet, Auto Club of Southern California and the conglomerate Montana Brand/Rocky Mountain Twist/Flav-R-Pac have suspended financial support for JFR for the final 16 events of the year. JFR, through its PR representative, has remained mum about the team's 2020 plans.
Just this past week, NHRA president Glen Cromwell told Autoweek the sanctioning body is "in a healthy position to weather the pandemic, whether it's 2020 or 2021.
"I think every business has had to make changes to make sure they manage the balance sheet properly. We've made those changes and adjustments. We're in a very comfortable and confident place that we're here for many years to come," Cromwell said.
"We're in a healthy position. We have run models for the best-case and worst-case scenarios, both for 2020 and 2021," he said. "We have to be a responsible sanctioning body, make sure that we remain stable. The tracks, the race teams, our partners, our sponsors, our race fans rely on the NHRA to be here for many years to come. Our leaders in our company have run financial models in every way possible to make sure that we remain healthy. We're very comfortable with where we are."
Force, whom Cromwell called "a very smart businessman," weighed all his options, too, and concluded he wasn't in a healthy position.
Earlier during the shutdown, Force had said, "I'm going to do my best to keep it going."
Despite his attorney's advice to "shut it down quick. You're insane," Force borrowed heavily from his rainy-day corporate savings to meet payroll at a time when revenue had stopped coming into the company that has, or had, about 100 employees split between his dual headquarters near Indianapolis and at Yorba Linda, California.
"Sponsors, you can't expect 'em to go on paying with no racin'. So I'm going to carry this thing myself," Force said at the time. "I put away money for what I call a rainy day. I had years of making profit, building buildings, buying land, selling land. I made a lot of money with my six corporations. In the middle of it, I put away for a rainy day. And now I'm spending that money back.
"There's only so far I can go. I ain't going to spend until I'm broke at my age," Force, 71, said. "I set up a program that we can recover, even if it goes to Pomona of next year (the February 2021 Winternationals in Southern California)."
With his latest revision to that plan and to ensure he can return next year, Force appears to be weighing a hiatus.
The last time Force missed a race was at Houston in April 2018, 45 events ago. That broke a string of 221 races. Earlier, he shocked the drag-racing world when he failed to qualify at the spring Las Vegas event in 2007, halting his NHRA-record of 395 straight appearances.
Two-time and current Top Fuel champion Steve Torrence and dad Billy added their names to the entry list Monday. The Kilgore, Texas, tandem keeps its car and crew at Brownsburg in the racing-heavy area nicknamed "Nitro Alley."


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Depleting Savings, Racking Up Debt
People living in rural states like Vermont, where it is not unusual to commute 100 miles to work and back five days a week, are hit hard by skyrocketing fuel prices. With gas prices hovering around $4 a gallon, many Vermonters are falling deeper into debt and saving less for the future.
"The increased price of gasoline is taking a serious bite out of the paychecks of middle class families, many of whom are already working longer hours for lower wages," said Sen. Bernie Sanders, who has written to President Obama to ask for the immediate resignation of federal regulators who won't enforce a new law to stop excessive speculation in oil markets.
The New York Times reported Wednesday that federal commodities regulators filed a civil lawsuit against commodities traders that early in 2008 "tried to hoard nearly two-thirds of the available supply of a crucial American market for crude oil, then abruptly dumped it and improperly pocketed $50 million."
A Vermont Bernie Buzz poll earlier this month found:
- Nearly four in 10 motorists, 38 percent, said $4 a gallon fuel is forcing them to either deplete their savings account or save less for retirement. Meanwhile, nearly one in five, or 19 percent, said they are racking up more credit card debt to pay the increasing cost of a fill-up. Just 8 percent said they were not affected by rising gas prices.
- Asked why Vermont gas prices had risen 35 percent in one year and nearly 30 cents in one month, more than eight in 10 people, or 82 percent, pointed to Wall Street speculators and profiteering by big oil companies.
The Full Poll Results:
What do you think is the main reason gasoline prices have gone up recently?
(Total responses: 1060)
How have rising gas prices most affected you?


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What could give more fun than hosting an amazing party while also making a good deed? Today Princess College is going to make a grand Charity Ball! Yay! Elsa and Belle are super-excited about it. Jasmine and Ariel are ready to join their BFF's. And this lovely four want to show their best. First of Princess Charity Day is a great game that you can play for free at in friv2021.org. Regardless Princess Charity Day game, we provide many cool Friv 2021 games to play for free.🎮 Instructions
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Artworks Ad Agency
Artworks Ad Agency
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Artworks Ad Agency
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- Killeen
- Texas
- 76541
- Tel: 254-634-5110
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Artworks Ad Agency are one of the Advertising Agencies located at 300 W Central TX Expy, in the Texas city of Killeen. Artworks Ad Agency can be contacted on 254-634-5110.
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Borgore has been playing out this new track in his sets and it's a great single to have on your radar heading to EDC: Aaron Jackson's debut release on Buygore called "Rotate." Set to play his first EDC Las Vegas (and get married) this coming weekend, Jackson has plenty for fans to get excited about and will be earning plenty of new ones with this new single and his upcoming set. Check out his latest mixtape for a preview of his live skills.
Rotate is a free download – stream below and grab your copy! This is a funky house jam with classic breaks and a friendly summer tempo.
Make sure to check out Buygore's pages for more new music. Links below:
Aaron Jackson: Facebook, Twitter, Soundcloud, Instagram, YouTube.
Buygore: Website, Facebook, Twitter, Soundcloud.


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Justin J. McShane, Esq. If you are looking for a highly-experienced DUI trial lawyer in Pennsylvania, look no further than Attorney Justin J. McShane. Based ... more
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text: LON:SSPGSSP Group Share Price, Forecast & News Adding Choose a watchlist: Adding You have already added five stocks to your watchlist. Upgrade to MarketBeat Daily Premium to add more stocks to your watchlist. Adding SSP Group PLC Please log in to your account or sign up in order to add this asset to your watchlist. Log In and Add GBX 239.40 +7.20 (+3.10 %) (As of 08/11/2020 04:00 PM ET) Add Compare Share Today's Range 231.40Now: GBX 239.40▼ 239.8650-Day Range 205.60MA: GBX 241.70▼ 276.8052-Week Range 2.45Now: GBX 239.40▼ 725Volume613,791 shsAverage Volume1.08 million shsMarket Capitalization£1.29 billionP/E Ratio17.47Dividend Yield5.47%BetaN/A ProfileAnalyst RatingsAdvanced ChartCompetitorsDividendEarningsInsider TradesHeadlines SSP Group plc operates food and beverage outlets. The company operates outlets at airports, railway stations, motorway service areas, hospitals, and sports stadia and shopping areas. It operates approximately 450 brands through a portfolio of 2,500 outlets, including coffee shops, sandwich bars, bakeries, and casual and fine-dining restaurants, as well as convenience and retail outlets in travel locations in approximately 30 countries in the United Kingdom, Europe, North America, the Asia Pacific, and the Middle East. The company was incorporated in 2006 and is based in London, the United Kingdom.Read More Beat the Market (BTM) Rank™ 4.4 Analyst's Opinion Consensus RatingSSP Group has received a consensus rating of Buy. The company's average rating score is 2.83, and is based on 10 buy ratings, 2 hold ratings, and no sell ratings.Price Target Upside/DownsideAccording to analysts' consensus price target of GBX 470, SSP Group has a potential upside of 96.3% from its current price of GBX 239.40.Amount of Analyst CoverageSSP Group has been the subject of 8 research reports in the past 90 days, demonstrating strong analyst interest in this stock. Previous Next 2.6 Community Rank Outperform VotesSSP Group has received 473 "outperform" votes. (Add your "outperform" vote.)Underperform VotesSSP Group has received 427 "underperform" votes. (Add your "underperform" vote.)Community SentimentSSP Group has received 52.56% "outperform" votes from our community.MarketBeat's community ratings are surveys of what our community members think about SSP Group and other stocks. Vote "Outperform" if you believe SSPG will outperform the S&P 500 over the long term. Vote "Underperform" if you believe SSPG will underperform the S&P 500 over the long term. You may vote once every thirty days. Previous Next 1.7 Dividend Strength Dividend LeadershipSSP Group is a leading dividend payer. It pays a dividend yield of 5.47%, putting its dividend yield in the top 25% of dividend-paying stocks.Dividend GrowthSSP Group does not have a long track record of dividend growth.Dividend CoverageThe dividend payout ratio of SSP Group is 87.59%. Payout ratios above 75% are not desirable because they may not be sustainable. Previous Next 0.0 Company Ownership Insider Buying vs. Insider SellingIn the past three months, SSP Group insiders have sold more of their company's stock than they have bought. Specifically, they have bought GBX 0 in company stock and sold £945,000 in company stock. Previous Next 0.6 Earnings and Valuation Price to Earnings Ratio vs. the MarketThe P/E ratio of SSP Group is 17.47, which means that it is trading at a more expensive P/E ratio than the market average P/E ratio of about 17.10.Price to Earnings Ratio vs. SectorThe P/E ratio of SSP Group is 17.47, which means that it is trading at a less expensive P/E ratio than the Services sector average P/E ratio of about 31.05.Price to Book Value per Share RatioSSP Group has a P/B Ratio of 2.81. P/B Ratios below 3 indicates that a company is reasonably valued with respect to its assets and liabilities. Previous Next Analyst Opinion: 4.4Community Rank: 2.6Dividend Strength: 1.7Insider Behavior: 0.0Valuation: 0.6 5 -4 -3 -2 -1 -Overall Beat the Market Rank™1.86 out of 5 stars Industry, Sector and Symbol Stock Exchange LON Industry Restaurants Sub-IndustryN/A SectorServices Current SymbolLON:SSPG Previous Symbol CUSIPN/A CIKN/A Webwww.foodtravelexperts.com Phone+44-20-75433300Debt Debt-to-Equity Ratio442.10 Current Ratio0.61 Quick Ratio0.57Price-To-Earnings Trailing P/E Ratio17.47 Forward P/E Ratio0.09 P/E GrowthN/A Sales & Book Value Annual Sales£2.75 billion Price / Sales0.47 Cash FlowGBX 74.45 per share Price / Cash Flow3.22 Book ValueGBX 85.30 per share Price / Book2.81Profitability EPS (Most Recent Fiscal Year)N/A Net IncomeN/A Net MarginsN/A Return on EquityN/A Return on AssetsN/AMiscellaneous Employees37,549 Outstanding Shares537,596,032Market Cap£1.29 billion Next Earnings DateN/A OptionableNot Optionable GBX 239.40 +7.20 (+3.10 %) (As of 08/11/2020 04:00 PM ET) 30 days | 90 days | 365 days | Advanced Chart Receive SSPG News and Ratings via Email Sign-up to receive the latest news and ratings for SSPG and its competitors with MarketBeat's FREE daily newsletter. SSP Group (LON:SSPG) Frequently Asked Questions How has SSP Group's stock been impacted by COVID-19 (Coronavirus)? SSP Group's stock was trading at GBX 415.50 on March 11th, 2020 when Coronavirus (COVID-19) reached pandemic status according to the World Health Organization (WHO). Since then, SSPG shares have decreased by 42.4% and is now trading at GBX 239.40. View which stocks have been most impacted by Coronavirus. Do Wall Street analysts recommend investors buy shares of SSP Group? 12 Wall Street analysts have issued "buy," "hold," and "sell" ratings for SSP Group in the last year. There are currently 2 hold ratings and 10 buy ratings for the stock, resulting in a consensus recommendation of "Buy." View analyst ratings for SSP Group. How were SSP Group's earnings last quarter? SSP Group PLC (LON:SSPG) issued its quarterly earnings results on Wednesday, June, 3rd. The company reported ($7.50) earnings per share (EPS) for the quarter, missing the Zacks' consensus estimate of ($4.90) by $2.60. View SSP Group's earnings history. How often does SSP Group pay dividends? What is the dividend yield for SSP Group? SSP Group announced a dividend on Wednesday, November 20th. Stockholders of record on Thursday, March 5th will be paid a dividend of GBX 6 per share on Friday, March 27th. This represents a yield of 0.95%. The ex-dividend date is Thursday, March 5th. This is a positive change from SSP Group's previous dividend of GBX 5.80. The official announcement can be seen at this link. View SSP Group's dividend history. What price target have analysts set for SSPG? 12 analysts have issued 12-month price targets for SSP Group's shares. Their forecasts range from GBX 325 to GBX 680. On average, they expect SSP Group's share price to reach GBX 470 in the next year. This suggests a possible upside of 96.3% from the stock's current price. View analysts' price targets for SSP Group. Has SSP Group been receiving favorable news coverage? News headlines about SSPG stock have trended neutral on Tuesday, according to InfoTrie Sentiment Analysis. The research group identifies negative and positive press coverage by analyzing more than six thousand news and blog sources. The firm ranks coverage of public companies on a scale of -5 to 5, with scores nearest to five being the most favorable. SSP Group earned a coverage optimism score of 0.2 on InfoTrie's scale. They also gave media coverage about the company a news buzz of 0.0 out of 10, indicating that recent press coverage is extremely unlikely to have an impact on the company's share price in the next few days. View the latest news about SSP Group. Who are some of SSP Group's key competitors? Some companies that are related to SSP Group include Domino's Pizza Enterprises (DMP), Whitbread (WTB), Greene King (GNK), Domino's Pizza Group (DOM), EI Group (EIG), J D Wetherspoon (JDW), Collins Foods (CKF), Mitchells & Butlers (MAB), MTY Food Group (MTY), A and W Revenue Royalties Income Fund (AW.UN), YOUS BREW/PAR VTG FPD 0.125 (YNGA), Marston's (MARS), Restaurant Group (RTN), Pizza Pizza Royalty (PZA) and Fuller, Smith & Turner (FSTA). View all competitors. What other stocks do shareholders of SSP Group own? Based on aggregate information from My MarketBeat watchlists, some companies that other SSP Group investors own include Aviva (AV), Melrose Industries (MRO), American Airlines Group (AAL), Royal Dutch Shell (RDSB), SIG (SHI), TUI (TUI), Barclays (BARC), Delta Air Lines (DAL), International Consolidated Airlns Grp (IAG) and IQE (IQE). Who are SSP Group's key executives? SSP Group's management team includes the following people: Ms. Kate E. Swann FCA, CEO & Exec. Director (Age 54)Mr. Jonathan Davies, CFO & Exec. Director (Age 57)Ms. Sarah John, Director of Investor RelationsMs. Helen Byrne, Gen. Counsel & Company Sec. (Age 63)Ms. Rachel Howarth, HR Director & MD of Netherlands What is SSP Group's stock symbol? SSP Group trades on the London Stock Exchange (LON) under the ticker symbol "SSPG." How do I buy shares of SSP Group? Shares of SSPG and other U.K. stocks can be purchased through online brokerage accounts that support trading on the London Stock Exchange (LSX). Some U.K. companies can be purchased through major U.S. brokerages in the form of American Depository Receipts (ADRs), which are placeholder equities held in a trust by a bank that represent shares of a foreign stock. What is SSP Group's stock price today? One share of SSPG stock can currently be purchased for approximately GBX 239.40. How big of a company is SSP Group? SSP Group has a market capitalization of £1.29 billion and generates £2.75 billion in revenue each year. SSP Group employs 37,549 workers across the globe. What is SSP Group's official website? The official website for SSP Group is www.foodtravelexperts.com. How can I contact SSP Group? SSP Group's mailing address is 169 Euston Road, LONDON, NW1 2AE, United Kingdom. The company can be reached via phone at +44-20-75433300. This page was last updated on 8/11/2020 by MarketBeat.com Staff 30 Days of MarketBeat All Access for $1.00 Sign up for MarketBeat All Access to gain access to MarketBeat's full suite of research tools: Best-in-Class Portfolio Monitoring View the latest news, buy/sell ratings, SEC filings and insider transactions for your stocks. Compare your portfolio performance to leading indices and get personalized stock ideas based on your portfolio. Stock Ideas and Recommendations Get daily stock ideas top-performing Wall Street analysts. Get short term trading ideas from the MarketBeat Idea Engine. View which stocks are hot on social media with MarketBeat's trending stocks report. Advanced Stock Screeners and Research Tools Identify stocks that meet your criteria using seven unique stock screeners. See what's happening in the market right now with MarketBeat's real-time news feed. Export data to Excel for your own analysis. Start Your Risk-Free Trial Subscription Here Questions? Get Started with MarketBeat — Read Your Daily NewsletterCheck Out My MarketBeatWatch Our Tutorial VideosAdd a Stock to Your WatchlistSet Up an Instant AlertView a Company's Profile PageView Our Financial CalendarsView Today's Analyst Ratings


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2007 Mercedes-Benz E-Class 4dr Sdn 3.5L RWD
Internet Price: $5,995
3.5L DOHC SFI 24-valve V6 engine
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Dealership Information
K O Enterprises of Columbia
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- Black gauges w/white markings
- Center console-inc: wood-trimmed rolltop cover, enclosed storage, removable dual front cupholders
- COMAND cockpit management/data system-inc: large color-LCD screen, (10) soft key, controls for AM/FM/weatherband radio, CD changer
- Driver/front passenger dual stage air bags w/front passenger adaptive feature
- Dual auto heated pwr mirrors-inc: memory, integrated turn signals, auto- dimming driver mirror
- Electrically heated rear window w/auto shut off based on time/exterior temp
- Folding rear armrest w/enclosed storage, dual cup holders
- Front halogen foglamps
- Front passenger Occupant Classification System (OCS)
- Front/rear stabilizer bars, gas shock absorbers, coil springs
- Harman/kardon Logic7 digital surround sound system-inc: (12) speakers, 7.1 channel amplifier, multichannel digital surround sound technology, digital dynamic volume control
- HomeLink compatible universal garage door opener
- Illuminated entry system w/locator lighting
- Independent 4-link front/5-arm multilink rear suspension
- Instrumentation-inc: fuel/coolant temp electronic bar-graph, tachometer, digital touch shift/mode selection display, quartz clock
- Leather-trimmed shift knob
- Mercedes-Benz Maintenance System Plus display
- Multifunction display in instrument cluster-inc: trip computer, exterior temp gauge, digital speedometer, audio status, reminder/malfunction indicator, driver-programmable settings, resettable trip meter
- Neutral-tinted glass
- Night security illumination
- Outboard rear universal lower anchors & tethers for children (LATCH)
- PRE-SAFE-inc: electrically tensioning front seat belts, front passenger seat adjustment for favorable airbag/seat belt effectiveness, closing of sunroof
- Pwr windows w/1-touch express down/up
- Rain-sensing windshield wipers w/heated windshield washer reservoir
- Rear 3-point seat belts w/outboard emergency tensioning device, adaptive belt force limiter, outboard automatic height adjustment
- Rear wheel drive
- Side impact airbags
- Single red rear foglamp
- Tele Aid GPS stolen vehicle recovery system w/remote door unlock, automatic collision notification alert
- Touch turn signals (flashes (3) times w/1-touch)
- Variable-focus halogen headlamps-inc: projector low beams, selectable light- sensing headlamps, automatic lamp substitution
- Vehicle-speed-sensitive pwr rack & pinion steering
- Window curtain airbags
We put forth our best effort to accurately describe every listing, however sometimes human error or the vehicle decoding system might have additional features listed or omitted. Please verify with an associate to ensure that all features, descriptions and price are accurately represented. Pricing does not include Sales Tax, Tag Fee's. Please feel free to contact us at (803) 779-3779 or visit us at 3820 RIVER DRIVE, COLUMBIA SC 29201...


text: Timeline from NYRR Run Center featuring FRNY 1982 Pride Run NYRR Run Center timeline at 320 West 57th Street. Mike McMahon 2017 June 1 FRNY 36th Year Retrospective Countdown Video (1979-2015) The Video counts down from 1979 to 2015 showing photographs from those years with FRNY members. The video includes names and photographs of leading members, including: Steve Gerben, Guy Zelenak, Brent Nicholson Earle, Mickey Zacuto, Sue Foster, Inger-Jo Berger, Lenore Beaky, Lee Gorman, Kelsey Louie and Mike Orzechowski. The video also includes a list of all past presidents with years they held office. Video was exhibited on Awards Night (January 30, 2016). Katrina Amaro 2016 January [created] 1979-2015 Music is copyrighted. If you are the copyright owner and object to the use of your music in this video available on this website, please contact firstname.lastname@example.com. High resolution version of video available on FRNY H&A hard drive as 36 YEAR RETROSPECTIVE 2016 version with COUNTDOWN_.mov video/H264 en-US Video New York, New York Pride Run finish line, 1982 Runner crossing the finish line Front Runners New York [photographer unknown] s_v13afvm9qnl2508.jpg Unknown 1982 s_v13afvm9qnl2507.jpg Unknown 1982 s_v13afvm9qnl2506.jpg Unknown 1982 s_v13afvm9qnl2505.jpg Unknown 1982 s_v13afvm9qnl2504.jpg Unknown 1982 s_v13afvm9qnl2503.jpg Unknown 1982 s_v13afvm9qnl2502_b.jpg Tom Cook? in Rosie Ruiz Track Club shirt Unknown 1982 Queens, New York Flushing Meadows Park s_v13afvm9qnl2501.jpg Corporate Challenge Race? Unknown 1982 s_v13afvm9qnl2500.jpg Corporate Challenge Race? Unknown 1982 s_v13afvm9qnl2499.jpg Unknown 1982 s_v13afvm9qnl2498.jpg Unknown 1982 s_v13afvm9qnl2497.jpg Unknown 1982 s_v13afvm9qnl2496.jpg Stan Moore (left) Unknown 1982 s_v13afvm9qnl2495.jpg Steve Gerben (center) Unknown 1982 s_v13afvm9qnl2494.jpg Stan Moore (left) Unknown 1982 s_v13afvm9qnl2493.jpg Unknown 1982 s_v13afvm9qnl2492.jpg Picnic Tom Cook (in red), Steve Gerben (in black watch) Unknown 1982 s_v13afvm9qnl2491.jpg Picnic Bill Dolan (in white headband), Marty King (in white hat), Tom Cook (in red shirt) Unknown 1982 s_v13afvm9qnl2490.jpg Unknown 1982 s_v13afvm9qnl2489.jpg Unknown 1982 s_v13afvm9qnl2488.jpg Unknown 1982 s_v13afvm9qnl2487.jpg Unknown 1982


text:
Outlook 1 Door Accent Cabinet by Greyleigh Best reviews
If you are interested in Outlook 1 Door Accent Cabinet by Greyleigh Best reviews Yes you see this. on-line shopping has currently gone a long way it's modified just how consumers and entrepreneurs work today. It hasn't drained the thought of Outlook 1 Door Accent Cabinet shopping after a physical store, nevertheless it gave the consumers an alternative solution way to buy and a larger market that offers bigger savings Thursday-August-6-2020
Outlook 1 Door Accent Cabinet by Greyleigh Best reviews
Clutter wreaking havoc in your home? Don't throw in the towel, that creates more laundry. Pick up this rectangular accent cabinet instead. Crafted of solid and manufactured wood, in a solid color finish; this cabinet features four interior adjustable shelves that offer plenty of storage for books, movies, knickknacks, and so much more. Made in the USA, this piece measures 70.984'' H x 22.598'' W x 16.772'' D and includes a tip over restraint device. A five-year product warranty is also...
- Base Material: Engineered Wood
- Top Material: Engineered Wood
- Number of Drawers: 1
- Tipover Restraint Device Included: Yes
- Overall: 70.984'' H x 22.598'' W x 16.772'' D
Outlook 1 Door Accent Cabinet Greyleigh
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review for Outlook 1 Door Accent Cabinet by Greyleigh Best reviews
The cost of the sofa also offers much to perform while using quality of fabric used by manufacturing the sofa set. Demands and availability of an sofa plays an important role to ascertain it price.
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text:
CATHOLIC SOCIAL AND COMMUNITY SERVICES, INC. is a business entity formed in Mississippi and is a Non Profit Corporation pursuant to local business registration regulations. Assigned the registration number 404596, according to the government registry, it is Good Standing.
Good Standing updated recently more like this →
Non Profit Corporation
Fullam, Dominick Registered Agent
JAMES MCGOUGH incorporator
JOSEPH L HOWZE incorporator
KEVIN BAMBRICK incorporator
Last checked at 2020-05-17 21:32:28 UTC


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The Answer movie
Video Credit: Teaser Trailer
Published on July 11, 2020 - Duration: 02:27s
The Answer movie
The Answer movie trailer - Plot synopsis: The Answer follows the life of a notorious gangster, Sozo Mathis who through circumstance finds himself at a cross road in his life.
At this pivotal point in his life he meets Pastor and Detective Alex Morgan.
Through intervention and guidance Alex directs Sozo in a direction that brings change to an entire community, but most importantly brings, "The Answer."
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