On a necessary conditon for the Erdős-Rényi law of large numbers

Abstract:

For a sequence ${\{ {X_i}\} _{i = 1,2, \ldots }}$ of independent, identically distributed random variables with existing moment-generating function $\varphi (t) = E\;\exp (t{X_i})$ in some nondegenerate interval, Erdös and Rényi (1970) studied the maximum $D(N,K)$ of the $N - K + 1$ sample means ${K^{ - 1}}({S_{n + K}} - {S_n}),\;0 \leqslant n \leqslant N - K$, where ${S_0} = 0,\;{S_n} = {X_1} + \cdots + {X_n}$. They showed that for a certain range of numbers a there exist positive constants $C({\mathbf {a}})$ such that ${\lim _{N \to \infty }}D(N,[C({\mathbf {a}})\log N]) = {\mathbf {a}}$ with probability one. In the present paper it is shown that the existence of the moment-generating function is also a necessary condition, i.e. that $\lim {\sup _{N \to \infty }}D(N,[C\log N]) = \infty$ for every positive constant C, if the moment-generating function does not exist for any positive number t.