## 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

### MSC:

 60F05 Central limit and other weak theorems

### Keywords:

probability theory