@article{ECP3587,
author = {Fuchang Gao and Zhenxia Liu and Xiangfeng Yang},
title = {Conditional persistence of Gaussian random walks},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {19},
year = {2014},
keywords = {Conditional persistence; random walk; integrated random walk},
abstract = {Let $\{X_n\}_{n\geq1}$ be a sequence of i.i.d. standard Gaussian random variables, let $S_n=\sum_{i=1}^nX_i$ be the Gaussian random walk, and let $T_n=\sum_{i=1}^nS_i$ be the integrated (or iterated) Gaussian random walk. In this paper we derive the following upper and lower bounds for the conditional persistence:$$\mathbb{P}\left\{\max_{1\leq k \leq n}T_{k} \leq 0\,\,\Big|\,\,T_n=0,S_n=0\right\}\lesssim n^{-1/2},\ \mathbb{P}\left\{\max_{1\leq k \leq 2n}T_{k} \leq 0\,\,\Big|\,\,T_{2n}=0,S_{2n}=0\right\}\gtrsim\frac{n^{-1/2}}{\log n},$$for $n\rightarrow\infty,$ which partially proves a conjecture by Caravenna and Deuschel (2008).},
pages = {no. 70, 1-9},
issn = {1083-589X},
doi = {10.1214/ECP.v19-3587},
url = {http://ecp.ejpecp.org/article/view/3587}}