Celebratio Mathematica — Chung

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K. L. Chung and C. Derman: “Non-recurrent random walks,” Pacific J. Math. 6 : 3 (1956), pp. 441–447. MR 0081587 Zbl 0072.35301 article

Let \( \{X_i\},\ i=1,2,\dots \) be a sequence of independent and identically distributed integral valued random variables such that 1 is the absolute value of the greatest common divisor of all values of \( x \) for which \( P(X_i=x) > 0 \). Define \( S_n=\sum_{i=1}^n X_i \). Chung and Fuchs [1951] showed that if \( x \) is any integer, \( S_n=x \) infinitely often with probability 1 according as \( EX_i=0 \) or \( \neq 0 \), provided that \( E|X_i| < \infty \). Let \( 0 < EX_i < \infty \), and \( A \) denote a set of integers containing an infinite number of positive integers. It will be shown that any such set \( A \) will be visited infinitely often with probability 1 by the sequence \( \{S_n\}\ n=1,2,\dots \). Conditions are given so that similar results hold for the case where \( X_i \) has a continuous distribution and the set \( A \) is a Lebesgue measurable set whose intersection with the positivfe real numbers has infinite Lebesgue measure.