On the maximum of a~simple random walk
Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 2, pp. 412-417
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Let $S_0=0$, $S_n=\xi_1+\xi_2+\dots+\xi_n$, $n\ge 1$, be the simple random walk generated by a sequence of independent random variables $\xi_i $, $i=1,2,\dots$, such that $\mathbf{P}\{\xi_i=1\}=1-\mathbf{P}\{\xi_i=-1\}=\frac12$, and let $T$ be the moment of the first return of $S_n$ to the state 0. We find an asymptotic representation for the probability $\mathbf{P}\{\max_{0$ which is exact (in order), assuming that $n^2 N^{-1}\to\infty$, and $nN^{-1}\le a1$. The results obtained are used to study the asymptotics of moderate and large deviations of the height of a planted plane tree with $N$ vertices.
Keywords:
random walk, return to zero, moderate and large deviations, the height of a planted plane tree.
@article{TVP_1995_40_2_a12,
author = {V. A. Vatutin},
title = {On the maximum of a~simple random walk},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {412--417},
publisher = {mathdoc},
volume = {40},
number = {2},
year = {1995},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1995_40_2_a12/}
}
V. A. Vatutin. On the maximum of a~simple random walk. Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 2, pp. 412-417. http://geodesic.mathdoc.fr/item/TVP_1995_40_2_a12/