The Annals of Mathematical Statistics

Some Results Relating Moment Generating Functions and Convergence Rates in the Law of Large Numbers

D. L. Hanson

Abstract

Let $X_N$ for $N = 1, 2, \cdots$ be an independent sequence of random variables with finite first absolute moments; let $a_N = \{a_{N,k} : k = 1, 2, \cdots\}$ for $N = 1, 2, \cdots$; let $A_N = 1/N \sum^N_{k = 1} (X_k - EX_k)$; and let $A_N = \sum_\infty^{k = 1} a_{N, k} (X_k - EX_k)$. Early work in probability dealt with the convergence (almost everywhere and in probability) to zero of the sequence. $A_N$. More recent work has dealt with the convergence to zero of sequences of the form $S_N$ under various assumptions on the coefficients $a_{N,k}$ and the distributions of the $X_N's$. In most cases the assumptions made about the $X_N's$ have been not much stronger or weaker than the assumption of a finite upper bound on their $\gamma$th absolute moments for some $\gamma \geqq 1$. The classical result giving exponential convergence rates in the law of large numbers was established by Cramer  (see also ) and states that if the $X_N's$ are identically distributed, and if their common moment generating function is finite in some interval about the origin, then for each $\epsilon > 0$ there exists $0 \leqq \rho < 1$ such that $P\{| A_N| \geqq \epsilon\} \leqq 2\rho^N$. Baum, Katz, and Read  investigated this exponential convergence further. However, their investigation was restricted to sequences of the form $A_N$. Koopmans  dealt with averages of the form $1/N \sum^N_{k = 1} \sum^\infty_{j=-\infty} a_jX_{k-j}$. In  the exponential rate was obtained for sequences $S_N$ provided $\sum_k \mid a_{N, k}\mid \leqq M < \infty$ for all $N$ and $\max_k |a_{N,k}| \leqq O(1/N)$. A corresponding result was obtained for continuous time stochastic processes in . More recently Chow [5, Section 2] obtained similar results under stronger assumptions on the moment generating functions involved. The results obtained here generalize and unify the results of , , and [5, Section 2]. The results are stated in Section 2 and proved in Section 2 and proved in Section 3. Corollaries and details of the relationships between these results and previous results are contained in Section 4.

Article information

Source
Ann. Math. Statist., Volume 38, Number 3 (1967), 742-750.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177698867

Digital Object Identifier
doi:10.1214/aoms/1177698867

Mathematical Reviews number (MathSciNet)
MR215342

Zentralblatt MATH identifier
0153.19603

JSTOR