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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On a conjecture of Révész


Author: Qi Man Shao
Journal: Proc. Amer. Math. Soc. 123 (1995), 575-582
MSC: Primary 60F15; Secondary 60G17
DOI: https://doi.org/10.1090/S0002-9939-1995-1231304-9
MathSciNet review: 1231304
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Abstract: Let $\{ {X_n},n \geq 1\}$ be i.i.d. random variables with $P({X_i} = \pm 1) = \frac {1}{2}$. Révész (1990) proved \[ \begin {array}{*{20}{c}} {1 \le \lim \inf \limits _{n \to \infty } \max \limits _{0 \le j < n} \max \limits _{1 \le k \le n - j} {{(2k\log n)}^{ - 1/2}}({S_{j + k}} - {S_j})} \hfill \\ { \le \lim \sup \limits _{n \to \infty } \max \limits _{0 \le j < n} \max \limits _{1 \le k \le n - j} {{(2k\log n)}^{ - 1/2}}({S_{j + k}} - {S_j}) \le K\quad {\rm {a}}.{\rm {s}}.} \hfill \\ \end {array}\] and conjectured $K = 1$, where ${S_n} = \sum \nolimits _{i = 1}^n {{X_i}}$. In this we show that Révész’s conjecture is true but the conclusion is not valid for general i.i.d. random variables with finite moment generating function.


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Keywords: Increments, partial sums, a.s. convergence, random walk
Article copyright: © Copyright 1995 American Mathematical Society