Berry-Esseen inequality for random walks conditioned to stay positive

Denis Denisov; Alexander Tarasov; Vitali Wachtel

Berry-Esseen inequality for random walks conditioned to stay positive

Denis Denisov
, Alexander Tarasov
, Vitali Wachtel

Probability

Universitat Bielefeld

Research output: Contribution to journal › Article › peer-review

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Abstract

We consider random walks conditioned to stay positive. When the mean of increments is zero and variance is finite it is known that their distribution converges to the Rayleigh law. In the present paper we derive a Berry--Esseen type estimate and show that if the third absolute moment is finite then the rate of convergence is of order $n^{-1/2}$.

Original language	English
Number of pages	22
Journal	Annals of Applied Probability
Publication status	Accepted/In press - 13 Oct 2025

Keywords

random walk
exit time
Rayleigh distribution
Berry-Esseen inequality
conditioned process

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title = "Berry-Esseen inequality for random walks conditioned to stay positive",

abstract = "We consider random walks conditioned to stay positive. When the mean of increments is zero and variance is finite it is known that their distribution converges to the Rayleigh law. In the present paper we derive a Berry--Esseen type estimate and show that if the third absolute moment is finite then the rate of convergence is of order \$n\textasciicircum{}\{-1/2\}\$.",

keywords = "random walk, exit time, Rayleigh distribution, Berry-Esseen inequality, conditioned process",

author = "Denis Denisov and Alexander Tarasov and Vitali Wachtel",

note = "This is the accepted version of the paper accepted by The Annals of Applied Probability (Institute of Mathematical Statistics). The final published version will be available at: [TBA]. ",

year = "2025",

month = oct,

day = "13",

language = "English",

journal = "Annals of Applied Probability",

issn = "1050-5164",

publisher = "Institute of Mathematical Statistics",

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TY - JOUR

T1 - Berry-Esseen inequality for random walks conditioned to stay positive

AU - Denisov, Denis

AU - Tarasov, Alexander

AU - Wachtel, Vitali

N1 - This is the accepted version of the paper accepted by The Annals of Applied Probability (Institute of Mathematical Statistics). The final published version will be available at: [TBA].

PY - 2025/10/13

Y1 - 2025/10/13

N2 - We consider random walks conditioned to stay positive. When the mean of increments is zero and variance is finite it is known that their distribution converges to the Rayleigh law. In the present paper we derive a Berry--Esseen type estimate and show that if the third absolute moment is finite then the rate of convergence is of order $n^{-1/2}$.

AB - We consider random walks conditioned to stay positive. When the mean of increments is zero and variance is finite it is known that their distribution converges to the Rayleigh law. In the present paper we derive a Berry--Esseen type estimate and show that if the third absolute moment is finite then the rate of convergence is of order $n^{-1/2}$.

KW - random walk

KW - exit time

KW - Rayleigh distribution

KW - Berry-Esseen inequality

KW - conditioned process

UR - https://arxiv.org/abs/2412.08502

M3 - Article

SN - 1050-5164

JO - Annals of Applied Probability

JF - Annals of Applied Probability

ER -

Berry-Esseen inequality for random walks conditioned to stay positive

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