Probability limit theorems assuming only the first moment. I. (English) Zbl 0042.37601

Chung, Kai Lai et al., Four papers on probability. Providence, RI: American Mathematical Society (AMS). Mem. Am. Math. Soc. 6, 19 p. (1951).
Let \(X_i\) \((i=1,2,\dots)\) be independent, identically distributed random variables which assume only integer values and write \(P(X=k)=p_k\). The authors consider the following assumptions: \[ \sum_{k=-\infty}^\infty | k| p_k < \infty,\ \sum_{k=-\infty}^\infty k p_k=0 \tag{1} \] \[ \sum_{k=0}^\infty k p_k=-\sum_{k=-\infty}^0 k p_k = \infty. \tag{2} \] They derive the following main results:
I. Under either (1) or 2) \[ \lim_{n=\infty } \frac{P \{ S_n=a \}}{P\{ S_n=a' \}}=1 \] where \(S_n=X_1+X_2+\cdots +X_n\) and \(a\) and \(a'\) are arbitrary integers.
II. Under (1) \[ P \left \{\lim_{n=\infty} \frac{\sum_{k=1}^n Y_k}{\sum_{k=1}^n Y'_k}=1 \right \}=1 \] where \(Y_k=1\) if \(S_k=a\) and =0 if \(S_k \neq a\), and similarly \(Y_k'\) for \(a'\).
For the entire collection see [Zbl 1415.60003].
Reviewer: Stefan Vajda


60F05 Central limit and other weak theorems