Convergence Rate for the Strong Law of Large Numbers: General Lattice Distributions

摘要

Let X1, X2, be a sequence of independent random variables with common lattice distribution function F having zero mean, and let (S Sub n) be the random walk of partial sums. The strong law of large numbers (SLLN) implies that for any alpha an element of IR and epsilon > 0 P sub m : = P(Sub n > alpha + (epsilon X n) for some n > or = decreases to 0 as m increases to infinity. Under conditions on the moment generating function of F, we obtain the convergence rate by determining P sub m up to asymptotic equivalence. When alpha = 0 and epsilon is a point in the lattice for F, the result is due to Siegmund; but this restriction on epsilon precludes all small values of epsilon, and these values are the most interesting vis-a-vis the SLLN. Even when alpha = 0 result handles important distributions F for which Siegmund's result is vacuous, for example, the two-point distribution F giving rise to simple symmetric random walk on the integers.