On the distribution of the square integral of the Brownian bridge

• 成果类型: Article
• 署名作者: Tolmatz, L
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2002
• 页码: 253-269
• 关键词: pinned wiener process; absolute value

摘要

Smirnov obtained the distribution F for his omega(2)-test in the form of a certain series. F is identical to the distribution of the the Brownian bridge in the L-2 norm. Smirnov, Kac and Shepp determined the Laplace-Stieltjes transform of F. Anderson and Darling expressed F in terms of Bessel functions. In the present paper we compute the moments of F and their asymptotics, obtain expansions of F and its density f in terms of the parabolic cylinder functions and Laguerre functions, and determine their asymptotics for the small and large values of the argument. A novel derivation of expansions of Smirnov and of Anderson and Darling is obtained.