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A Karhunen-Loeve expansion for a mean-centered Brownian bridge

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  • Deheuvels, Paul

Abstract

The processes of the form , where K is a constant, and B(·) a Brownian bridge, are investigated. We show that and are both Brownian bridges, and establish the independence of and , this implying that the law of coincides with the conditional law of B, given that . We provide the Karhunen-Loeve expansion on [0,1] of , making use of the Bessel functions J1/2 and J3/2. Applications and variants of these results are discussed. In particular, we establish a comparison theorem concerning the supremum distributions of and on [0,1].

Suggested Citation

  • Deheuvels, Paul, 2007. "A Karhunen-Loeve expansion for a mean-centered Brownian bridge," Statistics & Probability Letters, Elsevier, vol. 77(12), pages 1190-1200, July.
    • Handle: RePEc:eee:stapro:v:77:y:2007:i:12:p:1190-1200
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    References listed on IDEAS

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    1. Deheuvels, Paul & Peccati, Giovanni & Yor, Marc, 2006. "On quadratic functionals of the Brownian sheet and related processes," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 493-538, March.
    2. Li, Wenbo V., 1992. "Limit theorems for the square integral of Brownian motion and its increments," Stochastic Processes and their Applications, Elsevier, vol. 41(2), pages 223-239, June.
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    Cited by:

    1. Ai, Xiaohui, 2016. "A note on Karhunen–Loève expansions for the demeaned stationary Ornstein–Uhlenbeck process," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 113-117.

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