The Slepian zero set, and Brownian bridge embedded in Brownian motion by a spacetime shift

Abstract

E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. AbstractThis paper is concerned with various aspects of the Slepian process (Bt+1 − Bt, t ≥ 0) derived from one-dimensional standard Brownian motion (Bt, t ≥ 0). In particular, we offer an analysis of the local structure of the Slepian zero set {t : Bt+1 = Bt}, including a path decomposition of the Slepian process for 0 ≤ t ≤ 1. We also establish the existence of a random time T ≥ 0 such that the processAs a natural candidate, the bridge-like process as below was considered:This bridge-like process bears some resemblance to Brownian bridge. At least, it starts and ends at 0, and is some part of a Brownian path in between. This leads us to the following question:The Slepian zeros and Brownian bridge embedded in Brownian motion Question 1.2.The study of the bridge-like process is challenging, because the random time F as in (1.2) does not fit into any of the above classes. We even do not know whether this bridge-like process is Markov, or whether it enjoys the semi-martingale property. Note that if the answer to (2) of Question 1.2 is positive, then we can apply Rost's filling scheme [17,72] as in Pitman and Tang [66, Section 3.5] to sample Brownian bridge from a sequence of i.i.d. bridge-like processes in Brownian motion by iteration of the construction (1.1). While we are unable to answer either of the above questions about the bridge-like process, we are able to settle Question 1.1.EJP 20 (2015), paper 61.

Patterns in Random Walks and Brownian Motion

“…The problem of embedding Brownian bridge b 0 into Brownian motion is treated in a subsequent work of Pitman and Tang [70]. Since the proof relies heavily on Palm theory of stationary random measures, we prefer not to include it in the current work.…”

“…The process (S t := B t+1 − B t ; t ≥ 0) is a stationary Gaussian process, first studied by Slepian [79] and Shepp [77]. The following result, which can be found in Pitman and Tang [70,Lemma 2.3], is needed for the proof of Proposition 2.4.…”

“…Lemma 2.5. [76,70] For each fixed t ≥ 0, the distribution of (S u ; t ≤ u ≤ t + 1) is mutually absolutely continuous with respect to the distribution of…”

Approximating Shepp’s constants for the Slepian process

“…In Table 3 we use Approximation 7 (our most accurate approximation) to approximate Λ(h) over increments 0.1 for h. Bold font indicates the decimal places which we claim accurate. Note that h = 0 has been treated as a special case, see for example [4] and [17]. For h = 0, instead of Approximation 7, we have used the approximation Λ (8) (h) = − log(F 5 (h)/F 4 (h)); we do not recommend using this approximation in general because of its high complexity.…”

The spans in Brownian motion

“…Pitman and Tang [112,Proposition 2.4] established the following invariance principle for the first hitting bridge {RW Fn+j − RW Fn ; 0 ≤ j ≤ n}.…”