Abstract
This paper is concerned with various aspects of the Slepian process $(B_{t+1} - B_t, t \ge 0)$ derived from a one-dimensional Brownian motion $(B_t, t \ge 0 )$. In particular, we offer an analysis of the local structure of the Slepian zero set $\{t : B_{t+1} = B_t \}$, including a path decomposition of the Slepian process for $0 \le t \le 1$. We also establish the existence of a random time $T$ such that $T$ falls in the the Slepian zero set almost surely and the process $(B_{T+u} - B_T, 0 \le u \le 1)$ is standard Brownian bridge.
Highlights
Introduction and main resultIn a recent work [66], we were interested in continuous paths of length 1 in Brownian motion (Bt; t ≥ 0)
We proved that Brownian meander m and the three-dimensional Bessel process R can be embedded into Brownian motion by a random translation of origin in spacetime, while it is not the case for either normalized Brownian excursion e or reflected Brownian bridge |b0|
Can we find a random time T ≥ 0 such that (BT +u − BT ; 0 ≤ u ≤ 1) has the same distribution as standard Brownian bridge (b0u; 0 ≤ u ≤ 1)?
Summary
In a recent work [66], we were interested in continuous paths of length 1 in Brownian motion (Bt; t ≥ 0). In terms of the moving-window process, it is equivalent to find a random time T ≥ 0 such that XT has the same distribution as Brownian bridge b0. While the paper was under review, we learned from Hermann Thorisson [79] an explicit embedding of Brownian bridge into Brownian motion by a spacetime shift His argument is mainly from Last et al [51], that is to construct allocation rules balancing stationary diffuse random measures on the real line. They were able to characterize unbiased shifts of Brownian motion, those are random times T ∈ R such that (BT +u − BT ; u ≥ 0) is a two-sided Brownian motion, independent of BT Though it appears to be easier, Thorisson’s constructive proof relies on a deep and powerful theory. The constructive proof in Subsection 5.4 is due to Hermann Thorisson
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