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

Wolfgang Bischoff () and Andreas Gegg
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Wolfgang Bischoff: Catholic University of Eichstätt-Ingolstadt
Andreas Gegg: Catholic University of Eichstätt-Ingolstadt

Journal of Theoretical Probability, 2016, vol. 29, issue 2, 707-715

Abstract: 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.

Keywords: Cameron–Martin theorem; (p-)Slepian process; Radon–Nikodym derivative; 60G15; 60H99 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10959-014-0591-7

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