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Denisov, Denis; Sakhanenko, Alexander; Wachtel, Vitali

First-passage times for random walks with nonidentically distributed increments. (English) Zbl 1434.60126

Ann. Probab. 46, No. 6, 3313-3350 (2018).
Let \((S_n)_{n\in\mathbb{N}}\) be a random walk with independent, not necessarily identically distributed steps taking values on the real line. Assume that the random walk, properly normalized and rescaled, converges in distribution to a Brownian motion. For a deterministic sequence \((g_n)_{n\in\mathbb{N}}\), put \(T_g:=\inf\{n\in\mathbb{N}:S_n\leq g_n\}\).
The main result of the paper gives the asymptotics of \(\mathbb{P}\{T_g>n\}\) as \(n\to\infty\) under a couple of additional assumptions. The asymptotic equivalent of \(\mathbb{P}\{T_g>n\}\) involves a slowly varying function denoted by \(U_g\). The authors find conditions under which \(\lim_{t\to\infty}U_g(t)\) exists and is finite. To prove the main result the authors work out a modification of the universality approach which is, roughly speaking, based on replacing random walks attracted to a Brownian motion with the Brownian motion itself. The aforementioned modification is achieved by exploiting an argument appealing directly to the functional central limit theorem for \((S_n)\) rather than the Komlós-Major-Tusnády coupling which is commonly used in this context. As a consequence of the result on the tail behavior the authors derive a functional limit theorem for \((S_n)_{n\in\mathbb{N}}\), again properly normalized and rescaled, conditionally on \(\{T_g>n\}\) as \(n\to\infty\). The assertions proved in the paper improve upon several earlier results available in the literature.
Reviewer: Alexander Iksanov (Kiev)

Cited in 3 Reviews
Cited in 17 Documents

MSC:

60G50 Sums of independent random variables; random walks
60G40 Stopping times; optimal stopping problems; gambling theory
60F17 Functional limit theorems; invariance principles

Keywords:

random walk; Brownian motion; first-passage time; overshoot; moving boundary
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Full Text: DOI arXiv Euclid

References:

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