Statistics Papers

Document Type

Journal Article

Date of this Version

1982

Publication Source

The Annals of Probability

Volume

10

Issue

1

Start Page

234

Last Page

239

DOI

10.1214/aop/1176993926

Abstract

Let W~=W~t,0≤t≤1, be the pinned Wiener process and let ξ = ∫10|W~|. We show that the Laplace transform of ξ,ϕ(s)=Eeξs satisfies

∫∞0 eus ϕ(2√s3/2)s−1/2 ds = −√π Ai(u)/Ai′(u)

where Ai is Airy's function. Using (∗), we find a simple recurrence for the moments, n (which seem to be difficult to calculate by direct or by other techniques) namely n = enπ(36√2)n/Γ(3n+1/2) where e0 = 1,gk = Γ(3k+1/2)/Γ(k+1/2) and for n ≥ 1,

en=gn+∑nk=1enk(n k) (6k+1)/(6k−1) gk.

Keywords

airy, moments, Kac's method, Karhunen-Loeve

Included in

Probability Commons

Share

COinS
 

Date Posted: 27 November 2017

This document has been peer reviewed.