Persistence of iterated partial sums

Dembo, Amir, Ding, Jian, and Gao, Fuchang. "Persistence of iterated partial sums." Annales de l'I.H.P. Probabilités et statistiques 49.3 (2013): 873-884. <http://eudml.org/doc/272046>.

@article{Dembo2013,
abstract = {Let $S_\{n\}^\{(2)\}$ denote the iterated partial sums. That is, $S_\{n\}^\{(2)\}=S_\{1\}+S_\{2\}+\cdots +S_\{n\}$, where $S_\{i\}=X_\{1\}+X_\{2\}+\cdots +X_\{i\}$. Assuming $X_\{1\},X_\{2\},\ldots ,X_\{n\}$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities \[p\_\{n\}^\{(2)\}:=\mathbb \{P\}\Bigl (\max \_\{1\le i\le n\}S\_\{i\}^\{(2)\}&lt;0\Bigr )\le c\sqrt\{\frac\{\mathbb \{E\}|S\_\{n+1\}|\}\{(n+1)\mathbb \{E\}|X\_\{1\}|\}\},\] 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 $0&lt;\gamma &lt;1/4$ there exist integrable, zero-mean random variables for which the rate of decay of $p_\{n\}^\{(2)\}$ is $n^\{-\gamma \}$.},
author = {Dembo, Amir, Ding, Jian, Gao, Fuchang},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {first passage time; iterated partial sums; persistence; lower tail probability; one-sided probability; random walk},
language = {eng},
number = {3},
pages = {873-884},
publisher = {Gauthier-Villars},
title = {Persistence of iterated partial sums},
url = {http://eudml.org/doc/272046},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Dembo, Amir
AU - Ding, Jian
AU - Gao, Fuchang
TI - Persistence of iterated partial sums
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 3
SP - 873
EP - 884
AB - Let $S_{n}^{(2)}$ denote the iterated partial sums. That is, $S_{n}^{(2)}=S_{1}+S_{2}+\cdots +S_{n}$, where $S_{i}=X_{1}+X_{2}+\cdots +X_{i}$. Assuming $X_{1},X_{2},\ldots ,X_{n}$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities \[p_{n}^{(2)}:=\mathbb {P}\Bigl (\max _{1\le i\le n}S_{i}^{(2)}&lt;0\Bigr )\le c\sqrt{\frac{\mathbb {E}|S_{n+1}|}{(n+1)\mathbb {E}|X_{1}|}},\] 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 $0&lt;\gamma &lt;1/4$ there exist integrable, zero-mean random variables for which the rate of decay of $p_{n}^{(2)}$ is $n^{-\gamma }$.
LA - eng
KW - first passage time; iterated partial sums; persistence; lower tail probability; one-sided probability; random walk
UR - http://eudml.org/doc/272046
ER -