Positivity of integrated random walks

TL;DR

This paper investigates the probability that the partial sums of a centered random walk remain positive, deriving asymptotic formulas under specific distributional assumptions and extending results to integrated bridges.

Contribution

It provides new asymptotic results for the positivity probability of integrated random walks with stable law attraction and specific tail behaviors.

Findings

Asymptotic probability p_N ~ C_α N^{1/(2α) - 1/2} for large N.

Results apply to walks with right-exponential or right-continuous increments.

Extension to positivity of integrated discrete bridges.

Abstract

Take a centered random walk S_n and consider the sequence of its partial sums A_n = S_1 + ... + S_n. Suppose S_1 is in the domain of normal attraction of an \alpha-stable law with 1 < \alpha <= 2. Assuming that S_1 is either right-exponential (that is P(S > x | S > 0)=e^{-ax} for some a > 0 and all x > 0) or right-continuous (skip free), we prove that p_N = P(A_1 > 0, ..., A_N > 0) ~ C_\alpha N^{1/(2\alpha) - 1/2} as N tends to infinity, where C_\alpha > 0 depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.