## The Annals of Mathematical Statistics

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

Harry Kesten

#### 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.

#### Article information

Source
Ann. Math. Statist., Volume 43, Number 3 (1972), 701-732.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177692541

Digital Object Identifier
doi:10.1214/aoms/1177692541

Zentralblatt MATH identifier
0267.60053

JSTOR
links.jstor.org

#### Citation

Kesten, Harry. The 1971 Rietz Lecture Sums of Independent Random Variables--Without Moment Conditions. Ann. Math. Statist. 43 (1972), no. 3, 701--732. doi:10.1214/aoms/1177692541. https://projecteuclid.org/euclid.aoms/1177692541