Persistence of iterated partial sums

TL;DR

This paper investigates the probability that the iterated partial sums of i.i.d. zero-mean variables stay negative, establishing bounds and decay rates, revealing a universal decay of approximately n^{-1/4} under certain conditions.

Contribution

It provides new bounds on persistence probabilities of iterated sums and characterizes their decay rates, including a universal n^{-1/4} decay for squared integrable variables.

Findings

Persistence probability decays roughly as n^{-1/4} for certain variables.

Upper bounds on persistence probabilities involving expected absolute sums.

Existence of variables with slower decay rates of persistence probabilities.

Abstract

Let $S_n^{(2)}$ denote the iterated partial sums. That is, $S_n^{(2)}=S_1+S_2+ ... +S_n$, where $S_i=X_1+X_2+ ... s+X_i$. Assuming $X_1, X_2,....,X_n$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities $$p_n^{(2)}:=\PP(\max_{1\le i \le n}S_i^{(2)}< 0) \le c\sqrt{\frac{\EE|S_{n+1}|}{(n+1)\EE|X_1|}},$$ with $c \le 6 \sqrt{30}$ (and $c=2$ whenever $X_1$ is symmetric). The converse inequality holds whenever the non-zero $\min(-X_1,0)$ is bounded or when it has only finite third moment and in addition $X_1$ is squared integrable. Furthermore, $p_n^{(2)}\asymp n^{-1/4}$ for any non-degenerate squared integrable, i.i.d., zero-mean $X_i$. In contrast, we show that for any $0 < \gamma < 1/4$ there exist integrable, zero-mean random variables for which the rate of decay of $p_n^{(2)}$ is $n^{-\gamma}$.