Persistence of iterated partial sums

TL;DR

This paper investigates the decay rate of the persistence probability of iterated partial sums of i.i.d. random variables, establishing bounds and decay rates under various tail conditions.

Contribution

It provides universal bounds on persistence probability and characterizes its decay rate for different classes of random variables.

Findings

p_n is bounded above by the square root of the expected empirical mean absolute value

p_n decays as n^{-1/4} for variables with finite second moment

Existence of variables with decay rates of n^{-c} for any 0 < c < 1/4

Abstract

Let p_n denote the persistence probability that the first n iterated partial sums of integrable, zero-mean, i.i.d. random variables X_k, are negative. We show that p_n is bounded above up to universal constant by the square root of the expected absolute value of the empirical average of {X_k}. A converse bound holds whenever P(-X_1>t) is up to constant exp(-b t) for some b>0 or when P(-X_1>t) decays super-exponentially in t. Consequently, for such random variables we have that p_n decays as n^{-1/4} if X_1 has finite second moment. In contrast, we show that for any 0 < c < 1/4 there exist integrable, zero-mean random variables for which the rate of decay of p_n is n^{-c}.