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The Cameron–Martin Theorem for (p-)Slepian Processes

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  • Wolfgang Bischoff

    (Catholic University of Eichstätt-Ingolstadt)

  • Andreas Gegg

    (Catholic University of Eichstätt-Ingolstadt)

Abstract

We show a Cameron–Martin theorem for Slepian processes $$W_t:=\frac{1}{\sqrt{p}}(B_t-B_{t-p}), t\in [p,1]$$ W t : = 1 p ( B t - B t - p ) , t ∈ [ p , 1 ] , where $$p\ge \frac{1}{2}$$ p ≥ 1 2 and $$B_s$$ B s is Brownian motion. More exactly, we determine the class of functions $$F$$ F for which a density of $$F(t)+W_t$$ F ( t ) + W t with respect to $$W_t$$ W t exists. Moreover, we prove an explicit formula for this density. p-Slepian processes are closely related to Slepian processes. p-Slepian processes play a prominent role among others in scan statistics and in testing for parameter constancy when data are taken from a moving window.

Suggested Citation

  • Wolfgang Bischoff & Andreas Gegg, 2016. "The Cameron–Martin Theorem for (p-)Slepian Processes," Journal of Theoretical Probability, Springer, vol. 29(2), pages 707-715, June.
    • Handle: RePEc:spr:jotpro:v:29:y:2016:i:2:d:10.1007_s10959-014-0591-7
    DOI: 10.1007/s10959-014-0591-7
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    References listed on IDEAS

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    1. Wolfgang Bischoff & Frank Miller, 2000. "Asymptotically Optimal Tests and Optimal Designs for Testing the Mean in Regression Models with Applications to Change-Point Problems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(4), pages 658-679, December.
    2. Liu, Jin V. & Huang, Zongfu & Mao, Hongjun, 2014. "Karhunen–Loève expansion for additive Slepian processes," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 93-99.
    3. Bischoff, Wolfgang & Hashorva, Enkelejd, 2005. "A lower bound for boundary crossing probabilities of Brownian bridge/motion with trend," Statistics & Probability Letters, Elsevier, vol. 74(3), pages 265-271, October.
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