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On a conjecture of Révész

Author: Qi Man Shao
Journal: Proc. Amer. Math. Soc. 123 (1995), 575-582
MSC: Primary 60F15; Secondary 60G17
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{displaymath}\begin{array}{*{20}{c}} {1 \le \mathop {\lim \inf }\limits_{n... ...- {S_j}) \le K\quad {\rm {a}}.{\rm {s}}.} \hfill \\ \end{array}\end{displaymath}

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