Peer-reviewedCC BY 4.0Stochastic Processes and their Applications
Gaussian heat kernel asymptotics for conditioned random walks
Abstract
Consider a random walk Sn = n i=1 Xi with independent and identically distributed real-valued increments with zero mean, finite variance and moment of order 2 + δ for some δ > 0. For any starting point x ∈ R, let τx = inf {k ⩾ 1 : x + S k < 0} denote the first time when the random walk x+Sn exits the half-line [0, ∞). We investigate the uniform asymptotic behavior over x ∈ R of the persistence probability P(τx > n) and the joint distribution P (x + Sn ⩽ u, τx > n), for u ⩾ 0, as n → ∞. New limit theorems for these probabilities are established based on the heat kernel approximations. Additionally, we evaluate the rate of convergence by proving Berry-Esseen type bounds.
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BibTeX
@article{grama2026gaussian,
title = {{Gaussian heat kernel asymptotics for conditioned random walks}},
author = {Ion Grama and Hui Xiao},
journal = {Stochastic Processes and their Applications},
year = {2026},
month = oct,
doi = {10.1016/j.spa.2026.105113},
eprint = {2412.08932},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05848310v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Gaussian heat kernel asymptotics for conditioned random walks
AU - Ion Grama
AU - Hui Xiao
PY - 2026
DA - 2026/10/08
JO - Stochastic Processes and their Applications
DO - 10.1016/j.spa.2026.105113
UR - https://hal.science/hal-05848310v1
AB - Consider a random walk Sn = n i=1 Xi with independent and identically distributed real-valued increments with zero mean, finite variance and moment of order 2 + δ for some δ > 0. For any starting point x ∈ R, let τx = inf {k ⩾ 1 : x + S k < 0} denote the first time when the random walk x+Sn exits the half-line [0, ∞). We investigate the uniform asymptotic behavior over x ∈ R of the persistence probability P(τx > n) and the joint distribution P (x + Sn ⩽ u, τx > n), for u ⩾ 0, as n → ∞. New limit theorems for these probabilities are established based on the heat kernel approximations. Additionally, we evaluate the rate of convergence by proving Berry-Esseen type bounds.
SN - 0304-4149
SN - 1879-209X
ER -