Exponential Convergence Rates for the Law of Large Numbers

Leonard E. Baum, Melvin Katz, Robert R. Read
1962 Transactions of the American Mathematical Society  
Introduction. Consider a sequence of random variables {Xk:k = l,2, ■ • ■ ] obeying the law of large numbers, i.e., there exists a constant c such that for every e>0 the sequence of probabilities P{\n~12~2î-x^k-c\>e}=Pn(e) converges to zero as «-»<». The object of the present paper is to study the relationships among an exponential convergence rate (i.e., Pn(e) = 0(pn) for p=p(t) <1), the existence of the individual moment generating functions and the stochastic structure of the sequence \Xk\.
more » ... pers containing related studies (e.g., [l ; 3; 5]) have treated the case of independent random variables and demonstrated exponential convergence under the hypothesis that the moment generating functions exist. The present paper studies the extent to which an exponential convergence rate implies the existence of the moment generating function and conversely. In particular, satisfactory necessary and sufficient conditions (Theorem 2) are found for exponential convergence of sequences of independent (not necessarily identically distributed) random variables. In the first section it is proved that an exponential rate of convergence for any stationary sequence necessarily implies the existence of the moment generating function of the random variables. Conversely an example is constructed showing that restrictions on the size of the variables cannot be sufficient to insure exponential convergence for the general stationary sequence. The terms size, smallness, etc., as used throughout the paper, refer essentially to the tail probabilities. The case of independence is treated in §2 where it is shown that the existence of all the moment generating functions on the same interval is necessary and that a growth restriction on the product of the first n of these generating functions is necessary and sufficient. In §3 it is shown that the existence of the moment generating functions is necessary for exponential convergence of the averages of a function of the variables of a Markov sequence having stationary transition probabilities. If the process satisfies Doeblin's condition then this condition is proved sufficient. §4 contains an example showing that the existence of moment generating functions Received by the editors
doi:10.2307/1993673 fatcat:uks2xjf32ngwtmh42ujmndvaxu