## 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 [6] (see also [4]) 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 [3] investigated this exponential convergence further. However, their investigation was restricted to sequences of the form $A_N$. Koopmans [17] dealt with averages of the form $1/N \sum^N_{k = 1} \sum^\infty_{j=-\infty} a_jX_{k-j}$. In [11] 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 [12]. 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 [11], [12], 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

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