Persistence of iterated partial sums

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 $n^{-\gamma }$.

overlaps and improves upon an earlier version by Dembo and Gao at arXiv:1101.5743

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