ON THE DISTRIBUTION OF BROWNIAN AREAS
成果类型：
Article
署名作者：
Perman, Mihael; Wellner, Jon A.
署名单位：
University of Ljubljana; University of Washington; University of Washington Seattle
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
发表日期：
1996
页码：
1091-1111
关键词：
摘要：
We study the distributions of the areas under the positive parts of a Brownian motion process B and a Brownian bridge process U: with A(+) = integral B-1(0)+ (t) dt and A(0)(+) = integral U-1(0)+ (t) dt, we use excursion theory to show that the Laplace transforms psi(1) (s) = E exp(-sA(1)) and psi(1)(0) (s) = E exp(-sA(0)(+)) of A(+) and A(0)(+) satisfy integral(infinity)(0)e(-lambda s)psi(+)(root 2s(3/2)) ds = lambda(-1/2)Ai(lambda) + (1/3-integral(lambda)(0)Ai(t)dt)/root lambda Ai(lambda)-Ai'(lambda) and integral(infinity)(0)e(lambda s)/root s psi(+)(0)(root 2s(s/2)) ds = 2 root pi Ai(lambda)/root lambda Ai(lambda)-Ai'(lambda), where Ai is Airy's function. At the same time, our approach via excursion theory unifies previous calculations of this type due to Kac, Groeneboom, Louchard, Shepp and Takacs for other Brownian areas. Similarly, we use excursion theory to obtain recursion formulas for the moments of the positive part'' areas. We have not yet succeeded in inverting the double Laplace transforms because of the structure of the function appearing in the denominators, namely, root lambda Ai(lambda) - Ai'(lambda)