On the probability that integrated random walks stay positive

TL;DR

This paper investigates the asymptotic probability that integrated random walks remain positive, extending known results to new classes of walks including exponential and geometric types, and establishing bounds for lattice and exponential walks.

Contribution

It extends the asymptotic analysis of positivity probabilities of integrated random walks to broader classes of walks beyond simple symmetric cases.

Findings

For certain non-symmetric walks, the probability decays as N^{-1/4}.

Established upper bounds for lattice and exponential walks.

Extended Sinai's result to exponential and geometric walks.

Abstract

Let $S_n$ be a centered random walk with a finite variance, and define the new sequence $A_n:=\sum_{i=1}^n S_i$, which we call an integrated random walk. We are interested in the asymptotics of $$p_N:=P(\min_{1 \le k \le N} A_k \ge 0)$$ as $N \to \infty$. Sinai (1992) proved that $p_N \asymp N^{-1/4}$ if $S_n$ is a simple random walk. We show that $p_N \asymp N^{-1/4}$ for some other types of random walks that include double-sided exponential and double-sided geometric walks, both not necessarily symmetric. We also prove that $p_N \le c N^{-1/4}$ for lattice walks and for upper exponential walks, that are the walks such that $Law (S_1 | S_1>0)$ is an exponential distribution.