Convergence rates in the law of large numbers

Introduction. Let Xk: k ^ 1 denote a sequence of random variables, a": n 2ï 1 a sequence of real numbers, bn: n ^ 1 a nondecreasing sequence of positive real numbers and let Sn = lTk = xXk. Many of the limit theorems of probability theory may then be formulated as theorems concerning the convergence of either the sequence P (| (S" -an) jb" | > e): n ^ 1 or P (sup, è" | (Sk -ak) jbk \>s): n ^ 1, fore > 0, to an appropriate limiting value. It is the purpose of this paper to study the rates of convergence of such sequences. The results of this paper will include those previously announced in 1. In the first part of the paper attention is restricted to sequences of independent and identically distributed random variables. In analogy with the Law of Large Numbers the normalizing constants b" are chosen to be n", cc> 1/2, and the centering constants a" = ESn, provided the expectation exists and is finite. Necessary and sufficient conditions are found, in terms of the order of magnitude of P (| Xk | > n), for the sequences P (| (S" -EStt) jna \ > e): n ^ 1 and P (supf. g"| (St-ES^/k"!> e): n ^ 1 to converge t0 zero at specified rates. These results extend and complete previous work on this problem. The next results, again for independent and identically distributed random variables, consider the problem of convergence rates when the Z>"'s are a sequence in the upper class cf the S"'s and again a" = ESn. Next the independence conditions on the sequeice Xk: k 5: 1 are relaxed and it is assumed only that the random variables forn a stationary sequence. It is shown here that no conditions on the size of the variables, i. e. conditions on the magnitude of £* (| Xfc | > n), can insure a prescribid rate of convergence of P (j (Sn -ES") jn \> s) to zero when the Xk's form ai ergodic stationary sequence. However, in the converse direction, a prescribed rate of convergence to zero of the above probabilities does imply conditions oi the magnitude of P (| Xk | > n). Finally in the last se<tion of the paper some propositions and examples are presented for the case oi independent but not necessarily identically distributed random variables.