The convergence rate for the strong law of large numbers: General lattice distributions
Let \(X_ 1,X_ 2,..\). be a sequence of independent random variables with common lattice distribution function F having zero mean, and let \((S_ n)\) be the random walk of partial sums. The strong law of large numbers (SLLN) implies that for any \(\alpha\) \(\in {\mathbb{R}}\) and \(\epsilon >0\) \[ p_ m:=P\{S_ n>\alpha +\epsilon n\quad for\quad some\quad n\geq m\} \] decreases to 0 as m increases to \(\infty\). Under conditions on the moment generating function of F, we obtain the convergence rate by determining \(p_ m\) up to asymptotic equivalence. When \(\alpha =0\) and \(\epsilon\) is a point in the lattice for F, the result is due to \textit{D. Siegmund} [Z. Wahrscheinlichkeitstheor. verw. Geb. 31, 107-113 (1975; Zbl 0326.60028)]: but this restriction on \(\epsilon\) precludes all small values of \(\epsilon\), and these values are the most interesting vis-à- vis the SLLN. Even when \(\alpha =0\) our 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. We also identify for both lattice and non-lattice distributions the behavior of certain quantities in the asymptotic expression for \(p_ m\) as \(\epsilon\) decreases to 0.
scientific article; zbMATH DE number 4048761