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The tail of the maximum of Brownian motion minus a parabola

We analyze the tail behavior of the maximum N of Brownian motion minus a parabola and give an asymptotic expansion for P(N>x) as x tends to infinity. This extends a first order result on the tail behavior, which can be deduced from Huesler and Piterbarg (1999). We also point out the relation between certain results in Groeneboom (2010) and Janson, Louchard and Martin-L\"of (2010).


Introduction
The distribution function of the maximum of Brownian motion minus a parabola was studied in the two recent papers Janson, Louchard and Martin-Löf (2010) and Groeneboom (2010), both for one-sided and two-sided Brownian motion. The characterization of the distribution function is somewhat different in the two papers, but both characterizations (unavoidably) involve Airy functions. In this note we address the tail behavior of the distribution, a topic that was not addressed in these papers.
The tail behavior of the maximum plays an important role in certain recent studies on the asymptotic distribution of tests for monotone hazards, based on integral-type statistics measuring the distance between the empirical cumulative hazard function and its greatest convex minorant, for example in Groeneboom and Jongbloed (2010).
Let N be defined by where W is standard Brownian motion on [0, ∞). It can be deduced from Theorem 2.1 in Hüsler and Piterbarg (1999) that the distribution function F N of N satisfies: In section 2 we will give an asymptotic expansion of the left-hand side, which extends this result. The proof is based on an integral expression for the density, derived from Groeneboom (2010) (which in turn relies on Groeneboom (1989)), and uses a saddle point method for the integral over a shifted path in the complex plane. As a side effect, it also leads to a clarification of the relation between the representations of the distribution, given in Janson, Louchard and Martin-Löf (2010) and Groeneboom (2010).

Main results
In the following, we will use Corollary 2.1 of Groeneboom (2010), which is stated below for ease of reference, specialized to the density of the maximum of W (t) − t 2 (instead of the more general W (t) − ct 2 ).
Lemma 2.1 (Corollary 2.1 in Groeneboom (2010)) The density f of N is given by: where Ai is the Airy function Ai, as defined in, e.g., Olver (2010) 1 .
We deduce from this the following representation which is better suited for our purposes.
Lemma 2.2 The density f N of N is given by: Proof. Integration by parts of the second term of (2.2) yields: Let the function h be defined by Using Olver (2010) and using the Wronskian Ai(z)Bi ′ (z) − Ai ′ (z)Bi(z) = 1/π we conclude that This gives the desired result. ✷ Remark 2.1 Lemma 2.2 is in fact equivalent to relation (5.10) in Janson, Louchard and Martin-Löf (2010). The difference in the scaling constants is caused by the fact that they consider the maximum of W (t) − 1 2 t 2 instead of the maximum of W (t) − t 2 (see also section 3) and the fact that they integrate from −∞ to ∞ (in that way also the imaginary part drops out). However, they arrive at this relation in a completely different way. So in this case we can go from Corollary 2.1 in Groeneboom (2010) to the result in Janson, Louchard and Martin-Löf (2010), just by using integration by parts. This might serve as a first step in establishing the relation between the representations in the two papers.
We are now ready to prove our main result. We will give two proofs, one based on the first equality in (2.3) and the other one based on the second equality.
Theorem 2.1 Let N be defined by (1.1), and let f N and F N be the density and the distribution function of N , respectively. Then: and a k is given by: a 5 = 1601600, a 6 = −127568000, a 7 = 12287436800.
(ii) More explicitly: where the first coefficients are where the first coefficients are: Proof. Here we only derive the leading terms. Further terms in the asymptotic expansion are computed in the appendix. Let g(t) be defined by (2.6) By using the representation and this integral will be expanded for large values of x. For this we need an expansion of g(t) for large values of t.
For large values of the argument |u|, and |arg(iu)| < π, we have (see Olver (2010) 3 for the asymptotic behavior of the Airy functions): The derivative of this function vanishes at u s = −it 2 /4, and we have at u s the expansion This suggests to take u s as a saddle point (for a changed integration road) for the integral in (2.6).
We consider the following integration path: first the path P 1 , going from 0 to u s , and next the path P 2 , from u s to +∞, into the valley of exp( 4 3 (iu) 3/2 ). That is, at +∞ the phase of u is 1 6 π. See Figure 1, where we have shown the paths P 1 and P 2 for t = 1.
We write g(t) = g 1 (t) + g 2 (t), where g j (t) is the contribution from the path P j , j = 1, 2. Then: This shows that g 1 (t) is purely imaginary for t > 0. When we replace in (2.7) g(t) with g 1 (t), we see that this contribution to f N (x) becomes where G(t) is an analytic function which is real for positive values of t. The behavior of G(t) at infinity allows to take the contour along the imaginary axis. Integrating in this way, we obtain and since G(iv) + G(−iv) is real for real v (because G(t) is a real function), this integral is purely real, and, hence, we can ignore this contribution. Next we consider the integral over P 2 . A parametrization of this path follows from the equation Im{φ(u)} = Im{φ(u s )}, and we see from (2.10) that Im{φ(u s )} = 0. By writing u = r exp(iθ), it follows that the path P 2 can be described in terms of polar coordinates by r = 3t 2 cos 2 θ 16 sin 2 3 2 θ + 3 4 π , − 1 2 π ≤ θ ≤ 1 6 π, where for θ → − 1 2 π we have to apply l'Hôpital's rule to obtain r = 1 4 t 2 . There is no need to follow this path for obtaining the asymptotic expansion of g 2 (t) and for the sake of convenience we use the path from u s to u s + ∞ parallel to the positive real axis. In this way we find the contribution from the path P 2 , that is (2.11) At the lower limit u s = −it 2 /4 of the integration, we have by (2.8), us+∞ us e φ(u) du.
Next we neglect the O−term in the expansion in (2.10), and substitute u = u s + t 2 w, w ∈ R. This yields Re(g(t)) = Re(g 2 (t)) ∼ 2t 3 e − 1 12 t 3 ∞ 0 e −t 3 w 2 dw = t 3/2 e − 1 12 t 3 √ π, t → ∞. (2.12) ✷ Plugging the result in (2.12) into (2.7), and expanding at t = 2 4/3 x/3 yields: This gives the leading term of the expansion; the further coefficients are computed in the appendix. Part (iii) follows by integrating this relation. (Second proof.) Again we only derive the leading term. We start with the second representation in (2.3).
and shift the integration path in the last integral to the path P along the line, parallel to the imaginary axis, and running from c − i∞ to c + i∞, where c = 1 3 2 2/3 x. A similar path was used in the proof of (ii) of Corollary 3.4 in Groeneboom (1989). Asymptotically, as x → ∞, the exponent is now given by 4 3 (c + u) 3/2 − 2 3 (c + 2 2/3 x + u) 3/2 , which equals −8x 3/2 / √ 27 at u = 0. We get: ✷ Remark 2.2 In the first proof, the integral in (2.12) does not have a meaning for all complex values t. For example, we use values of t with phase π/3 for which t 3 is negative. However, for all complex values of t with phases in [−π/3, π/3] we can give the integral in (2.12) a meaning by turning the path of integration in the w−plane such that the integral remains convergent. For example when arg(t) = π/3 we can take arg(w) = −π/2.
For two-sided Brownian motion we get similarly: where W is standard two-sided Brownian motion, originating from zero. and let f M and F M be the density and the distribution function of M , respectively. Then: Proof. This follows from Corollary 2.2 of Groeneboom (2010), which gives the representation:

Concluding remarks
As pointed out to us by Svante Janson, the result implies certain facts for the moments of the distribution. For example, applying Theorem 4.5 in Janson and Chassaing (2004) together with Theorem 2.1 of the present paper gives: Also, by Lemma 2.2, the density of M c is given by: (3.14) 4 Appendix. Computing more coefficients of the asymptotic expansions Recall, see (2.11), where we assume for the time being t > 0. We substitute u = t 2 v and write φ(u) = −t 3 ψ(v), where φ(u) is given in (2.9). This gives where the modified path P 2 runs from −i/4 to −i/4 + ∞, and ψ(v) = − 4 3 (iv) 3/2 + iv.
We have ψ(−i/4) = 1 12 and substitute Locally, we prescribe for this mapping Upon inverting we have v = ∞ k=0 d k w k , and the first coefficients are The transformation (4.15) gives Because the argument of the Airy function is large for large values of t (for all values of w ≥ 0) we expand the function h(w) first for large values of t. For this we need the well-known expansion (see Olver (2010) (4.17) where ζ = 2 3 z 3/2 , and The first coefficients are u 0 = 1, u 1 = 5 72 , u 2 = 385 10368 , u 3 = 85085 2239488 , u 4 = 37182145 644972544 .
Remark 4.1 The Gauss hypergeometric function in (4.24) has a simple finite representation because k is an integer. Also, for α = 1, 2, 3, . . ., giving results for the derivatives of the Airy function, we have simple relations. For example, when α = 1 we have u (1) k = u k (1 + 6k)/(1 − 6k) = −v k , and the v k are used in the expansion of Ai ′ (z), see Olver (2010) Remark 4.2 Similar expansion as in (4.23) can be found in Drazin & Reid (1981) (pp. 465-478) for slightly different functions.