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Conditional persistence of Gaussian random walks


 
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1. Title Title of document Conditional persistence of Gaussian random walks
 
2. Creator Author's name, affiliation, country Fuchang Gao; University of Idaho; United States
 
2. Creator Author's name, affiliation, country Zhenxia Liu; Linköping University; Sweden
 
2. Creator Author's name, affiliation, country Xiangfeng Yang; Linköping University; Sweden
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Conditional persistence; random walk; integrated random walk
 
3. Subject Subject classification 60G50; 60F99
 
4. Description 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).
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Simons Foundation
 
7. Date (YYYY-MM-DD) 2014-10-10
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/3587
 
10. Identifier Digital Object Identifier 10.1214/ECP.v19-3587
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 19
 
12. Language English=en en
 
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