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

Let XN for N = 1, 2, ⋯ be an independent sequence of random variables with finite first absolute moments; let aN = ,k : k = 1, 2, ⋯\ for N = 1, 2, ⋯; let AN = 1/N ∑Nk = 1 (Xₖ - EXₖ); and let AN = ∑_∞k = 1 aN, k (Xₖ - EXₖ). Early work in probability dealt with the convergence (almost everywhere and in probability) to zero of the sequence. AN. More recent work has dealt with the convergence to zero of sequences of the form SN under various assumptions on the coefficients aN,k and the distributions of the XN's. In most cases the assumptions made about the XN's have been not much stronger or weaker than the assumption of a finite upper bound on their γth absolute moments for some γ 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 XN's are identically distributed, and if their common moment generating function is finite in some interval about the origin, then for each ε > 0 there exists 0 ρ < 1 such that P\| AN| ε\ 2ρN. Baum, Katz, and Read [3] investigated this exponential convergence further. However, their investigation was restricted to sequences of the form AN. Koopmans [17] dealt with averages of the form 1/N ∑Nk = 1 ∑^∞j=-∞ aⱼXₖ₋ⱼ. In [11] the exponential rate was obtained for sequences SN provided ∑ₖ aN, k M < ∞ for all N and maxₖ |aN,k| 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.