The 1971 Rietz Lecture Sums of Independent Random Variables--Without Moment Conditions

Abstract

Analogues of classical limit laws for sums of independent random variables (central limit theorem, laws of large numbers and law of the iterated logarithm) are discussed. We stress results which go through without moment or smoothness assumptions on the underlying distributions. These include (i) estimates for the spread of the distribution of $S_n = \sum_1^nX_i$ in terms of concentration functions (Levy-Rogozin inequality), (ii) comparison of the distribution of $S_n$ on different intervals (ratio limit theorems and Spitzer's theorem for the existence of the potential kernel for recurrent random walk), (iii) study of the set of accumulation points of $S_n/\Upsilon(n)$ for suitable $\Upsilon (n) \uparrow \infty$. Only the following parallel to the law of the iterated logarithm is new: If $X_1, X_2, \cdots$ are independent random variables all with distribution $F, S_n = \sum_1^nX_1, m_n = \operatorname{med} (S_n)$, then there exists a sequence $\{\Upsilon (n)\}$ such that $\Upsilon (n) \rightarrow \infty$ and $- \infty < \lim \inf (S_n - m_n)/\Upsilon (n) < \lim \sup (S_n - m_n)/\Upsilon (n) < \infty$ w.p. 1, if and only if $F$ belongs to the domain of partial attraction of the normal law.