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A Minimal Contrast Estimator for the Linear Fractional Stable Motion

In this paper we present an estimator for the three-dimensional parameter (σ, α,H) of the linear fractional stable motion, where H represents the self-similarity parameter, and (σ, α) are the scaling and stability parameters of the driving symmetric Lévy process L. Our approach is based upon a minimal contrast method associated with the empirical characteristic function combined with a ratio type estimator for the selfsimilarity parameter H . The main result investigates the strong consistency and weak limit theorems for the resulting estimator. Furthermore, we propose several ideas to obtain feasible confidence regions in various parameter settings. Our work is mainly related to [16, 18], in which parameter estimation for the linear fractional stable motion and related Lévy moving average processes has been studied.


Introduction
During the last sixty years fractional stochastic processes have received a great deal of attention in probability, statistics and integration theory. One of the most prominent examples of a fractional model is the (scaled) fractional Brownian motion (fBm), which gained a lot of popularity in science since the pioneering work of Mandelbrot and van Ness [17]. The scaled fBm is the unique zero mean Gaussian process with stationary increments and self-similarity property. As a building block in stochastic models it found numerous applications in natural and social sciences such as physics, biology and economics. From the statistical perspective the scaled fBm is fully determined by its scale parameter σ > 0 and the self-similarity parameter (or Hurst index) H ∈ (0, 1). Nowadays, the estimation of (σ, H) is a well understood problem. We refer to [10] for efficient estimation of the Hurst parameter H in the low frequency setting, and to [5,8,14] for the estimation of (σ, H) in the high frequency setting, among many others. In more recent papers [2,15] statistical inference for the multifractional Brownian motion has been investigated, which accounts for the time varying nature of the Hurst parameter.
This paper focuses on another extension of the fBm, the linear fractional stable motion (lfsm). The lfsm (X t ) t≥0 is a three-parameter statistical model defined by where x + = x ∨ 0 denotes the positive part and we set x a + = 0 for all a ∈ R, x ≤ 0. Here (L t ) t≥0 is a symmetric α-stable Lévy process with α ∈ (0, 2) and scale parameter σ > 0, and H ∈ (0, 1) represents the Hurst parameter. In some sense the lfsm is a non-Gaussian analogue of fBm. The process (X t ) t≥0 has symmetric α-stable marginals, stationary increments and it is self-similar with parameter H. We remark that the class of stationary increments self-similar processes becomes much larger if we drop the Gaussianity assumption, cf. [20,23], but the lfsm is one of its most famous representatives due to the ergodicity property. Linear fractional stable motions are often used in natural sciences, e.g. in physics or internet traffic, where the process under consideration exhibits stationarity and selfsimilarity along with heavy tailed marginals, see e.g. [12] for the context of Ethernet and solare flare modelling.
The limit theory for statistics of lfsm, which is indispensable for the estimation of the parameter ξ = (σ, α, H), turns out to be of a quite complex nature. First central limit theorems for partial sums of bounded functions of Lévy moving average processes, which in particular include the lfsm, have been discussed in [21] and later extended in [22] to certain unbounded functions. In a more recent work [4] the authors presented a rather complete asymptotic theory for power variations of stationary increments Lévy moving average processes. Finally, the results of [4] have been extended to general functions in [3], who demonstrated that the weak limit theory crucially depends on the Appell rank of the given function and the parameters of the model (all functions considered in this paper have Appell rank 2). More specifically, they obtained three different asymptotic regimes, a normal and two stable ones, depending on the particular setting. It is this phase transition that depends on the parameter (α, H) which makes the statistical inference for lfsm a rather complicated matter.
Since the probabilistic theory for functionals of lfsm was not well understood until the recent work [3,4], the statistical literature on estimation of lfsm is rather scarce. The articles [1,22] investigate the asymptotic theory for a wavelet-based estimator of H when α ∈ (1, 2). In [4,26] the authors use power variation statistics to obtain an estimator of H, but this method also requires the a priori knowledge of the lower bound for the stability parameter α. The work [11] suggested to use negative power variations to get a consistent estimator of H, which applies for any α ∈ (0, 2), but this article does not contain a central limit theorem for this method. The paper [18] was the first instance, where estimation of the full parameter ξ = (σ, α, H) has been studied in low and high frequency settings. Their idea is based upon the identification of ξ through power variation statistics and the empirical characteristic function evaluated at two different values.
In this paper we aim at extending the approach of [18] by determining the asymptotic theory for the minimal contrast estimator of the parameter ξ = (σ, α, H), which is based upon the comparison of the empirical characteristic function with its theoretical counterpart under low frequency sampling. Indeed, the choice of the two evaluation points for the empirical characteristic function in [18] is rather ad hoc and we will show in the empirical study that the minimal contrast estimator exhibits better finite sample properties and robustness in various settings. Similarly to [18], we will show that the weak limit theory for our estimator has a normal and a stable regime, and the asymptotic distribution depends on the interplay between the parameters α and H. At this stage we remark that the minimal contrast approach has been investigated in [16] in the context of certain Lévy moving average models, which do not include the lfsm or its associated noise process, but only in the asymptotically normal regime. Another important contribution of our paper is the subsampling procedure, which provides confidence regions for the parameters of the model irrespectively of the unknown asymptotic regime.
The article is organized as follows. In Section 2 we introduce the necessary notation and formulate a new weak limit result related to [18], which is central to their parameter estimation for the linear fractional stable motion. The aforementioned theorem will be our starting point, where the aim is to extend the convergence of the finite dimensional distributions to convergence of integral functionals appearing in the minimal contrast method. Section 3 introduces the estimator and presents the main results of strong consistency and asymptotic distribution. Section 4 is devoted to a simulation study, which tests the finite sample performance of the minimal contrast estimator. We also discuss the parametric bootstrap and the subsampling method that are used to construct feasible confidence regions for the true parameters of the model. All proofs are collected in Section 5 and all larger tables are in Section 6.

Notation and recent results
We start out with introducing the main notation and statistics of interest. We consider low frequency observations X 1 , X 2 , . . . , X n from the lfsm (X t ) t≥0 introduced in (1.1). We denote by ∆ r i,k X (i, k, r ∈ N) the kth order increment of X at stage i and rate r, i.e.
The order k plays a crucial role in determining the asymptotic regime for statistics that we introduce below. We let the function h k,r : R → R denote the kth order increment at rate r of the kernel in (1.1), specifically We note that ∆ r i,k X = R h k,r (i − s) dL s . For less cumbersome notation we drop the index r if r = 1, so ∆ i,k X := ∆ 1 i,k X and h k = h k,1 . Throughout this paper we write θ = (σ, α) and ξ = (σ, α, H). The main probabilistic tools are statistics of the type where f : R → R is a Borel function satisfying E[|f (∆ r rk,k X)|] < ∞. We will specifically focus on two classes of functions, namely f p (x) = |x| p for p ∈ (−1, 1) and δ t (x) = cos(tx) for t ≥ 0. They correspond to power variation statistics and the real part of the empirical characteristic function respectively, and we use the notation ϕ n (t) = V n (δ t , k, 1) and ψ n (r) = V n (f p , k, r).
We note that Birkhoff's ergodic theorem implies the almost sure convergence An important coefficient in our context is The rate of convergence and the asymptotic distribution of statistics defined at (2.2) crucially depend on whether the condition k > H + 1/α is satisfied or not. Hence, we define the normalised versions of our statistics as Here m p,k = E[|∆ k,k X| p ], which is finite for any p ∈ (−1, α), and ϕ ξ is given at (2.3). Note that E[|∆ r rk,k X| p ] = r H m p,k , which explains the centring of W 1 and S 1 . It turns out that the finite dimensional limit of the statistics (W 1 n , W 2 n ) is Gaussian while the corresponding limit of (S 1 n , S 2 n ) is β-stable (see Theorem 2.1 below). We now introduce several notations to describe the limiting distribution. We start with the Gaussian case. For random variables (2.5) Next, for p ∈ (−1, 1) \ {0}, we introduce the constant where Γ denotes the Gamma function. Now, for each t ∈ R + , we set whenever the above integrals are finite. As it has been shown in [18], the following identities hold for any p ∈ (−1/2, 1/2) \ {0} with p < α/2 and t ∈ R + : and Cov(|X| p , exp(itY )) = Σ 12 (g, h; t).
Obviously, the quantities Σ 11 (g, h) and Σ 12 (g, h; t) will appear in the asymptotic covariance kernel of the vector (ψ n (r), ϕ n (t)) in the normal regime. Now, we introduce the necessary notations for the stable case. First, we define the functions and set Next, we introduce the functions κ 1 : N → R + and κ 2 : R → R − via In the final step we will need to define two Lévy measures ν 1 on (R + ) 2 and ν 2 on R + that are necessary to determine the asymptotic distribution of (S 1 n , S 2 n ). Let us denote by ν the Lévy measure of the symmetric α-stable Lévy motion L (i.e. ν(dx) = c(σ)|x| −1−α dx) and define the mappings τ 1 : R → (R + ) 2 and τ 2 : R → R + via Then, for any Borel sets A 1 ⊆ (R + ) 2 and A 2 ⊆ R + bounded away from (0, 0) and 0, respectively, we introduce (2.10) In the weak limit theorem below we write Z n L−f −−→ Z to denote the convergence of finite dimensional distributions, i.e. the convergence in distribution for any d ∈ N and t j ∈ R + . The following theorem is key for statistical applications.
(i) If k > H + 1/α then as n → ∞ where the quantity Σ ij has been introduce at (2.7). Moreover, the Gaussian process W exhibits a modification (denoted again by W ), which is locally Hölder continuous of any order smaller than α/4.
The finite dimensional asymptotic distribution demonstrated in Theorem 2.1 is a direct consequence of [18,Theorem 2.2], which even contains a more general multivariate result. However, the smoothness property of the limiting Gaussian process W and the particular form of the limit of S 2 n have not been investigated in [18]. Both properties are crucial for the statistical analysis of the minimal contrast estimator.
We observe that from a statistical perspective it is more favourable to use Theorem 2.1(i) to estimate the parameter ξ = (σ, α, H), since the convergence rate √ n in (i) is faster than the rate n 1−1/β in (ii). However, the phase transition happens at the point k = H + 1/α, which depends on unknown parameters α and H. This poses major difficulties in statistical applications and we will address this issue in the forthcoming discussion.

Main results
In this section we describe our minimal contrast approach and present the corresponding asymptotic theory. Before stating our the main result we define a power variation based estimator of the parameter H ∈ (0, 1). Since the increments of the process (X t ) t≥0 are strongly ergodic (cf. [7]), we deduce by Birkhoff's ergodic theorem the almost sure convergence for any p ∈ (−1, α). In particular, we have that consequently yielding a consistent estimator H n (p, k) of H as for any p ∈ (−1, α). The idea of using negative powers p ∈ (−1, 0) to estimate H, which has been proposed in [11] and applied in [18], has the obvious advantage that it does not require knowledge of the parameter α. From Theorem 2.1(i) and the δ-method applied to the function v p (x, y) = 1 p (log 2 (x) − log 2 (y)) we immediately deduce the convergence for k > H + 1/α, where M 1 is a centred Gaussian random variable. Similarly, when k < H + 1/α, we deduce the convergence from Theorem 2.1(ii).
Below we denote by ∇ θ the gradient with respect to the parameter θ and similarly by ∇ 2 θ the Hessian. We further write ∂ z f for the partial derivative of f with respect to z ∈ {σ, α, H}. The main theoretical result of this paper is the following theorem.
is the true parameter of the linear fractional stable motion (X t ) t≥0 at (1.1) and that the weight function w ∈ L 1 (R + ) is continuous.
(iii) If k < H + 1/α then In principle, the statement of Theorem 3.1 follows from (3.12), (3.13) and an application of the implicit function theorem. For general infinite dimensional functionals of our statistics we would usually need to show tightness of the process (W 2 n (t)) t≥0 (or (S 2 n (t)) t≥0 ), which is by far not a trivial issue. However, in the particular setting of integral functionals, it suffices to show a weaker condition that is displayed in Proposition 5.6. Indeed, this is the key step of the proof.

Simulations, parametric bootstrap and subsampling
The theoretical results of Theorem 3.1 are far from easy to apply in practice. There are a number of issues, which need to be addressed. First of all, since the parameters H 0 and α 0 are unknown, we do not know whether we are in the regime of Theorem 3.1(ii) or (iii). Furthermore, even if we could determine whether the condition k > H 0 + 1/α 0 holds or not, the exact computation or a reliable numerical simulation of the quantities defined in (2.7) and (2.8) seems to be out of reach. Below we will propose two methods to overcome these problems. In the setting where the lfsm (X t ) t≥0 is continuous, which corresponds to the condition H 0 − 1/α 0 > 0, we will see that it suffices to choose k = 2 to end up in the normal regime of Theorem 3.1(ii). The confidence regions are then constructed using the parametric bootstrap approach. In the general setting we propose a novel subsampling method which, in some sense, automatically adapts to the unknown limiting distribution.
For comparison reasons we include the estimation of the parameter H 0 using H n (p, k) defined at (3.11), even though its properties have already been studied in [18]. Moreover, we pick our weight function in the class of Gaussian kernels: In particular we can use Gauss-Hermite quadrature, see [24], to estimate the integral This procedure is based on a number of weights, unless otherwise stated we pick 12 weights.
To produce the simulation study we generate observations from the lfsm using the recent R package rlfsm, which implements an algorithm based on [27]; this package already includes the implementation of the minimal contrast estimator. To compute the estimator a minimization argmin has to be carried out. For this purpose we use [19], which in particular entails picking a starting point for the algorithm. For σ no immediate choice exists, so we simply pick σ = 2, while α = 1 seems obvious.

Empirical bias and variance
In this section we will check the bias and variance performance of our minimal contrast estimator. First, we consider the empirical bias and standard deviation, which are summarized in Tables 5 and 6 for n = 1.000 and Tables 7 and 8  At this stage we recall that due to Theorem 3.1(iii) we obtain a slower rate of convergence when H 0 + 1/α 0 > 2; the stable regime is indicated in bold in Tables 5-8. This explains a rather bad estimation performance for α 0 = 0.4. The effect is specifically pronounced for the parameter σ, which has the worst performance when α 0 = 0.4. This observation is in line with the findings of [18], who concluded that the scale parameter σ is the hardest to estimate in practice. Also the starting point σ = 2 of the minimisation algorithm, which is not close to σ 0 = 0.3, might have a negative effect on the performance. The estimation performance for values α 0 > 0.4 is quite satisfactory for all parameters, improving from n = 1.000 to n = 10.000. We remark the superior performance of our method around the value α = 1, which is explained by the fact that α = 1 is the starting point of the minimisation procedure.
For comparison, we display the bias and standard deviation of our estimator for k = 1 based on n = 10.000 observations in Tables 9 and 10, where the stable regime is again highlighted in bold. We see a better finite sample performance for α 0 = 0.4, but in most other cases we observe a larger bias and standard deviation compared to k = 2. This is explained by slower rates of convergence in the setting of the stable regime and k = 1.
We will now compare the minimal contrast estimator with the estimator proposed in [18]. Their approach is based upon computing H n (p, k) and inverting the map θ → ϕ θ,Hn(p,k) (t 1 ), ϕ θ,Hn(p,k) (t 2 ) (4. 16) for t 1 = 1, t 2 = 2, where the theoretical characteristic function is replaced by the empirical one. The resulting estimator is denoted by ( σ low , α low , H n (p, k)). The first comparison is between tables 7 and 8 for the minimal contrast estimator with tables 11 and 12, all of which are based on at least 1.000 Monte Carlo repetitions and the same parameter choices. We see that the minimal contrast estimator outperforms the old estimator for values α ≤ 0.8, where the latter is completely unreliable in most cases. But for larger values of α the old estimator might be slightly better. However, the results for the estimation of the scale parameter σ in the case α = 0.4 and H 0 ∈ {0.2, 0.4} are hard to interpret, since σ low often delivers values that are indistinguishable from 0 yielding a small variance and relatively small bias for σ 0 = 0.3.
Another point is the instability of the old estimator. Table 13, which is based on at least 1.200 simulations, shows the rate at which the estimators fail to return a value α ∈ (0, 2). In this regard the minimal contrast estimator is far superior in most cases and actually in the case k = 1 the estimator almost never fails, although we dispense with the simulation results. We remark that in theory the minimal contrast estimator should never return α's not in the interval [0, 2]. However, we apply the minimisation procedure from [19] that does not allow constrained optimisation. One could instead use e.g. the procedure from [6]; we choose the first method as it is hailed as very robust.
An additional advantage of the minimal contrast estimator is that it allows incorporation of a priori knowledge of the parameters σ and α, using the weight function but also the starting point for the minimisation algorithm.

Bootstrap inference in the continuous case
In this section we only consider the continuous case, which corresponds to the setting H 0 − 1/α 0 > 0. In this setup H 0 ∈ (1/2, 1) and α 0 ∈ (1, 2) must hold. We can in particular choose k = 2 to ensure that Theorem 3.1(ii) applies, thus yielding the faster convergence rate √ n. We are interested in obtaining feasible confidence regions for all parameters, but, as we mentioned earlier, the computation or reliable numerical approximation of the asymptotic variance in Theorem 3.1(ii) is out of reach. Instead we propose the following parametric bootstrap procedure to estimate the confidence regions for the true parameters.
(3) Compute new estimators ξ To test this we repeated the above procedure for 500 Monte Carlo simulations with N = 200. Tables 1 and 2 report acceptance rates for √ n(ξ n − ξ 0 ) for an approximate 95 %confidence interval. We observe a good performance for all estimators with the exception of σ and n = 1.000 in Table 1.

Subsampling method in the general case
In contrast to the continuous setting H 0 − 1/α 0 > 0, there exists no a priori choice of k in the general case, which ensures the asymptotically normal regime of Theorem 3.1(ii). This problem was tackled in [18] via the following two stage approach. In the first step they obtained a preliminary estimator α 0 n (t 1 , t 2 ) of α 0 by the empirical inversion of (4.16) for k = 1, t 1 = 1 and t 2 = 2. In the second step they defined the random number and computed the estimator ( σ low , α low , H n (p, k)) based on k =k. They showed that the resulting estimator is √ n-consistent and derived the associated weak limit theory. However, this approach does not completely solve the original problem, since they obtained four different convergence regimes according to whether 1 > H 0 + 1/α 0 or not, and whether α −1 0 ∈ N or not. Nevertheless, we apply their idea to propose a new subsampling method to determine feasible confidence regions for the parameters of the model. For our procedure it is crucial that the convergence rate is known explicitly and the weak convergence of the involved statistics is insured. We proceed as follows: (1) Given observations X 1 , . . . , X n computek from (4.17) and construct the minimal contrast estimator ξ n = (σ n , α n , H n (p,k)).
(4) Compute the 97.5% and 2.5% quantiles for each of the distribution functions Let us explain the intuition behind the proposed subsampling procedure. First of all, similarly to the theory developed in [18], the minimal contrast estimator ξ n obtained through a two step method described in the beginning of the section leads to four different limit regimes for √ n(ξ n − ξ 0 ) (although we leave out the theoretical derivation here). Using this knowledge we may conclude that, for each l = 1, . . . , L, n/L(ξ (l) n − ξ 0 ) has the same (unknown) asymptotic distribution as the statistic √ n(ξ n − ξ 0 ) as long as n/L → ∞. Since the true parameter ξ 0 is unknown, we use its approximation ξ n , which has a much faster rate of convergence than n/L when L → ∞. Finally, the statistics constructed on different blocks are asymptotically independent, which follows along the lines of the proofs in [18]. Hence, the law of large numbers implies that the proposed subsampling statistics converge to the unknown true asymptotic distributions when L → ∞ and n/L → ∞.
In Tables 3 and 4 we report the empirical 95%-confidence regions for the parameters of the model using the subsampling approach. We perform 500 Monte Carlo simulations and choose n = 12.5 × L 2 .  Table 3 shows a satisfactory performance for all estimators, while the results of Table 4 are quite unreliable for the parameters σ and α. The reason for the latter finding is the suboptimal finite sample performance of the estimators in the case of (α, H) = (1.8, 0.8), which is displayed in Tables 5 and 6.
We conclude this section by remarking the rather satisfactory performance of our estimator in the continuous setting H 0 − 1/α 0 > 0. On the other hand, when using the subsampling method in the general setting, a further careful tuning seems to be required. In particular, the choice of the weight function w and the group number L plays an important role in estimator's performance. We leave this study for future research.

Proofs
We denote by C a finite, positive constant which may differ from line to line. Moreover, any important dependence on other constants warrants a subscript. To simplify notations we set H n = H n (p, k).

Proof of Theorem 2.1(i)
As we mentioned earlier, the convergence of finite dimensional distributions has been shown in [18,Theorem 2.2], and thus we only need to prove the smoothness property of the limit W . We recall the definition of the quantity U g,h at (2.5) and start with the following lemma.
In particular, it holds that Next, we define for each l ∈ Z and recall the following lemma from [18]. To prove that the process W is locally Hölder continuous of any order smaller than α/4, we use Kolmogorov's criterion. Since W is a Gaussian process it suffices to prove that for each T > 0 there exists a constant C T ≥ 0 such that This is performed in a similar fashion as in [16,Section 4.1]. First, we reduce the problem. Using cos(tx) = (exp(itx) + exp(−itx))/2 and the symmetry of the distribution of X, we observe the identity Cov(cos(t∆ j,k X), cos(s∆ j+l,k X)) = In the following we focus on the first term in the above decomposition (the second term is treated similarly). More specifically, we will show the inequality (5.18) for the quantity r(t, s), which is given as Moreover, since r(t, t) + r(s, s) − 2r(t, s) ≤ |r(t, t) − r(t, s)| + |r(s, s) − r(t, s)| it is by symmetry enough to prove that For l ∈ Z decompose now as follows: l (t, s).
Applying the second inequality of Lemma 5.1 and the mean value theorem we deduce the estimate: Again by the mean value theorem we find that The last term can be rewritten as As α ∈ (0, 2) we have |x α − y α | ≤ |x 2 − y 2 | α/2 for all x, y ≥ 0. In particular It then follows that where µ l is the quantity defined as It remains to prove that l∈Z ρ l < ∞ and l∈Z µ l < ∞. (5.21) The first claim is a direct consequence of Lemma 5.2. The second convergence can equivalently be formulated as where we used the assumption k > H + 1/α.

Proof of Theorem 2.1(ii)
We recall that the asymptotic distribution of the vector (S 1 n (1), S 1 n (2)) and its asymptotic independence of S 2 n (t) have been shown in [18, Theorem 2.2]. Hence, we only need to determine the functional form of the limit of the statistic S 2 n (t). In the following we will recall a number of estimates and decompositions from [18, Theorem 2.2], which will be also helpful in the proof of Theorem 3.1(iii). We start out with a series of estimates on the function Φ 2 t given at (2.8), but for a general scale parameter η > 0. Let Φ t,η denote the function where Y is an SαS distributed random variable with scale parameter η. We obviously have the representation Φ t,η (x) = (cos(xt) − 1) exp(−|ηt| α ).

(5.22)
The next lemma gives some estimates on the function Φ t,η . t,η (x) denote the vth derivative at x ∈ R. Then there exists a constant C > 0 such that for all t ≥ 0 it holds that (iv) For any x, y > 0 and a ∈ R then
If |x − y| > 1 we simply bound the latter by t 2 g η (t). If |x − y| ≤ 1, then by the mean value theorem there exists a number s with |x − s| ≤ |x − y| such that
(iv): Let a, x and y be given. Observe that The last equality implies that F (a, x, y) ≤ Cg η (t). The first equality implies that F (a, x, y) ≤ Cg η (t)ty. Reversing the order of integration we get a similar expression as the first equality with x replaced by y. Hence, F (a, x, y) ≤ Cg η (t)tx. Lastly, using (i) on the first integral yields F (a, x, y) ≤ Cg η (t)t 2 xy. Splitting into the four cases completes the proof.
We will consider the asymptotic decomposition of the statistic S 2 n (t) given in [3, Section 5] (see also [4,18]). We set Define for each s ≥ 0 we define the σ-algebras We also set for all n ≥ k, i ∈ {k, . . . , n} and t ≥ 0 . Then the following decomposition holds: Using the inequality 2 − x − 1/x < 0 for all x > 1 on β > 1 it follows by picking ε > 0 small enough that the first term in (5.23) is asymptotically negligible. Decomposing the second term and using arguments as in the equations (5.30), (5.31) and (5.38) in [4] we obtain the following result.
Proposition 5.5. For any ε > 0 there exist an r > 1, an r ′ > β ∨ r and a constant C > 0 such that for all n in N sup Using again the inequality 2 − x − 1/x < 0 for all x > 1, it follows immediately that the second term in (5.23) is asymptotically negligible. Hence, S n (t) is asymptotically equivalent to the statistic S 2 n (t), and it suffices to analyse its finite dimensional distribution.

Finite dimensional convergence and integral functionals
Let (Y n ) n≥1 and Y be stochastic processes index by R + with paths in L 1 (R + ). We will give simple sufficient conditions for when the implication holds true. Such a result is obviously required to obtain Theorem 3.1 from Theorem 2.1. Before we state these conditions we remark that the question has already been studied in the literature. As an example Theorem 22 in Appendix I of [13] gives two sufficient conditions for (5.27) to hold, but the second condition is a Hölder type criteria, which is not easily verifiable in our setting. Moreover, the theorem only deals with integration over bounded sets. The article [9] studies this question in general, but the conditions of e.g. Lemma 1 therein are too abstract even though we are in the case of a finite measure (the one induced by the weight function w). What can be deduced from [9] is that some kind of uniform integrability (with respect to the product measure) is sufficient for (5.27).
To formulate the lemma, define for each n, m, l ∈ N the intermediate random variables Proof. Note that for each n, m, l ∈ N we have the decomposition ∞ 0 Y n u du = X n,m,l + (X n,l − X n,m,l ) + ∞ l Y n u du.
The continuity of Y implies immediately that Y m,l a.s.
The assumptions in (5.28) then imply the convergence

Proof of Theorem 3.1(i) and (ii)
The strong consistency result of Theorem 3.1(i) is an immediate consequence of (3.11) and ϕ n − ϕ ξ 0 w,2 a.s.
− − → 0, where the latter follows from (2.3) and the dominated convergence theorem. Hence, we are left to proving Theorem 3.1(ii).
Recall the definition of the function F : is an open neighbourhood of (σ 0 , α 0 ) bounded away from (0, 0). Now, the minimal contrast estimator at (3.15) can be obtained using the criteria which is satisfied at (ϕ ξ 0 , ξ 0 ). Denote by ζ(ψ, H) an element of (0, 1) × Θ 0 such that H, ζ(ψ, H) To determine the derivative of ζ we will need the infinite dimensional implicit function theorem, which we briefly repeat. Moreover, G is Fréchet differentiable with derivative We will adapt this to our setup, which corresponds to U 1 = L r w (R + ) × (0, 1), for a r > 1, E 2 = Θ 0 , E 3 = R and f = ∇ θ F . A straightforward calculation shows that the map θ → ∇ 2 θ F (ϕ, H, θ) is differentiable with derivative at (ϕ ξ 0 , ξ 0 ) represented by the Hessian The linear independence of the maps ∂ θ 1 ϕ ξ 0 and ∂ θ 2 ϕ ξ 0 immediately shows the invertibility of the Hessian. Moreover, standard theory for convergence in L r (R + ), r > 1, shows that the map (ϕ, H) → ∇ 2 θ F (ϕ, H, ·) is continuous, which is needed to assert that ∇ θ F is C 1 . The determination of the remaining derivative D 1 ϕ,ξ ∇ θ F for a point (ϕ, ξ) ∈ L r w (R + ) × (0, 1) × Θ 0 is slightly more involved. It is given by its two components D 1 = (D 1,1 , D 1,2 ) corresponding to the partial derivatives. Indeed, D 1,1 ϕ,ξ ∇ θ F is the derivative with respect to the functional coordinate ϕ ∈ L r w (R + ) and D 1,2 ϕ,ξ ∇ θ F the derivative with respect to the Hurst parameter H ∈ (0, 1), where ξ = (H, α, σ). It is easily seen that An application of Hölder's inequality proves the continuity of the linear map ξ → D 1,1 ξ ∇ θ F . The second partial derivative at (ϕ, ξ) is the linear map represented by the two dimensional vector Evaluated at the point (ϕ ξ 0 , ξ 0 ) = (ϕ ξ 0 , G(ϕ ξ 0 , H 0 )) yields the simpler expression: Suppose we are in the case k > H + 1/α then we may pick r = 2 in the discussion above. By Fréchet differentiability it follows that Recall now the derivative of G at (5.29). In order to show Theorem 3.1(ii) it suffices to prove the convergences where W = (W t ) t≥0 has been introduced in Theorem 2.1(i).
We will only consider the second convergence at (5.31) since the first is shown similarly (see also [16, page 14]). For the conditions (5.28) it suffices to find a constant C > 0 such that sup n∈N,t≥0 Var(W 2 n (t)) ≤ C < ∞. (5.32) The identity ∆ i,k X = R h k (i−s) dL s together with stationarity of the increments {∆ i,k X | i ≥ k} shows that Indeed, split the series at (5.33) into three terms. For l = 0 it follows from (2.5) that which is obviously uniformly bounded in t ≥ 0. For l = 0 with |l| ≤ k the first inequality of Lemma 5.1 implies that ).

(5.35)
Now by Cauchy-Schwarz inequality ρ l < h k α α for all l, and we obtain a uniform bound in t ≥ 0. By Lemmas 5.1 and 5.2 there exist constants C, K > 0 such that

Proof of Theorem 3.1(iii)
As in the proof of Theorem 3.1(ii) we obtain the decomposition (5.30), where the convergence rate √ n is replaced by n 1−1/β . Furthermore, as in (5.31), it suffices to prove that for some r ∈ (1, 2) then as n → ∞ As before in the Gaussian case it is enough to provide uniform bounds (in n and t) on the moments in order to use Proposition 5.6.
Recall that the dominating term in (5.23) is given by Inspired by the classical case of i.i.d. random variables, each in the domain of attraction of a stable distribution, we shall prove the following result.
For shorter notation we write We have the decomposition =: T n,t,1 + T n,t,2 .
The proposition then asserts that sup n∈N,t≥0 To prove this we observe that For the first term we obtain the inequality , which is uniformly bounded in t ≥ 0. Additionally let p α denote the density of an SαS distribution and recall that p α (x) ≤ C(1 + |x|) −1−α for all x ∈ R, cf. [28,Theorem 1.1]. We decompose E[|Y n,t,k | 2 ] into two regions corresponding to (5.37): The first term vanishes as n → ∞ since 2/β > 1 and the fact that |Φ t (x) − E t | is bounded for all t and x ∈ (x 0 , 0). The second term is further split into two terms: Using (5.37) and the boundedness of κ 2 on the first term we have that The second term contains a similar consideration, indeed In the next step we treat the term T n,t,2 . Note first that by the von Bahr-Esseen inequality we obtain E[|n −1/β T n,t,2 | r ] ≤ C r n 1−r/β E[|Z n,t,k | r ].
Decomposing as above we have that For the first term we recall that Φ t (x) is bounded uniformly in t when x lies in a bounded set, hence n 1/β > |Φ t (x) − E t | for all sufficiently large n, independent of x ∈ (0, x 0 ) and t ≥ 0, so the first term is zero for sufficiently large n.
The last term requires more computations. Due to (5.37) and the fact that κ 2 is bounded it follows that where we used that rq − α < 0 since r < β.
Combining Propositions 5.4-5.7 we finally complete the proof of Theorem 3.1(iii).