Celebratio Mathematica — Chung

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Works connected to Jacob Wolfowitz

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K. L. Chung and J. Wolfowitz: “On a limit theorem in renewal theory,” Ann. of Math. (2) 55 : 1 (January 1952), pp. 1–6. MR 0044769 Zbl 0047.12402 article

Let \( X \) be a random variable which assumes only integral values and define for all \( n \), \( p_n=P\{X=n\} \), where \( P\{\ \} \) is the probability of the event in braces. Let \( \{X_i\} \), \( i=1,2,\dots \) ad inf. be an infinite sequence of independent random variables with the same distribution as \( X \). Define \[ S_j=\sum_{i=1}^j X_i \quad\text{and}\quad u_n=\sum_{j=1}^{\infty}P\{S_j=n\} .\] Let \( m=E(X) \) be the expectation of \( X \). Let \( t \) be the absolute value of the greatest common divisor of all indices \( n \) such that \( p_n > 0 \). We shall prove the follow theorem: If \( 0 < m\leq\infty \), then \[ \lim_{n\to\infty}u_{nt}=t/m \quad\text{and}\quad \lim_{n\to -\infty} u_{nt} = 0 .\] If \( -\infty\leq m < 0 \) then the limits given above should be interchanged.