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Conditional persistence of Gaussian random walks
University of Idaho. (Department of Mathematics)
Blåeldsvägen 12B, Sturefors, Sweden.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
2014 (English)In: Electronic Communications in Probability, ISSN 1083-589X, E-ISSN 1083-589X, Vol. 19, no 70, 1-9 p.Article in journal (Refereed) Published
Abstract [en]

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:\begin{align*}\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},\end{align*}for $n\rightarrow\infty,$ which partially proves a conjecture by Caravenna and Deuschel (2008).

Place, publisher, year, edition, pages
2014. Vol. 19, no 70, 1-9 p.
Keyword [en]
conditional persistence; random walk; integrated random walk
National Category
Probability Theory and Statistics
Identifiers
ISI: 000346594300001OAI: oai:DiVA.org:liu-112753DiVA: diva2:771412
Available from: 2014-12-13 Created: 2014-12-13 Last updated: 2017-12-05Bibliographically approved

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Cite
Citation style
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