On the probability that integrated random walks stay positive

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 $$p_N:=P(\underset{1\le k\le N}{\min }{A}_k\ge 0)$$ 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.

Theorems 2 and 3 were restated and merged into one theorem; a new lemma (Lemma 1) added; Lemma 3 and Remark 1 were restated and merged into Proposition 1; the proof of Lemma 3 is reworked. The paper is accepted to SPA.

References in corpus (3)

• Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas
• Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction
• Clustering in a stochastic model of one-dimensional gas

Cited by in corpus (8)

• Persistence probabilities \& exponents
• Persistence of iterated partial sums
• Survival probabilities of weighted random walks
• Persistence probabilities for an integrated random walk bridge
• Exit times for integrated random walks
• Stationary entrance Markov chains, inducing, and level-crossings of random walks
• Persistence problems for additive functionals of one-dimensional Markov processes
• Stationary entrance chains and applications to random walks