The Erdős-Rényi new law of large numbers for weighted sums

Abstract:

From the partial sums ${S_n}$ of the first $N$ random variables of a sequence of independent, identically distributed random variables, $N - K + 1$ averages of the form ${K^{ - 1}}({S_{n + K}} - {S_n})$ can be constructed, one such average for each $n$ between 0 and $N - K$, inclusive. If we denote by $\Sigma (N,K)$ the maximum of those $N - K + 1$ averages, then for a wide range of numbers $\lambda$, Erdös and Rényi (1970) proved that, as $N \to \infty ,\Sigma (N,K) \to \lambda$ a.e. for $K = [C(\lambda )\log N]$, where $C(\lambda )$ is a constant depending only on $\lambda$, not on $N$. The objective of the present article is to extend the Erdös-Rényi theorem to the case of weighted sums. The main theorem bears a relation to the law of large numbers for weighted sums of Jamison, Orey, and Pruitt (1965) similar to the one borne by the Erdös-Rényi theorem to the ordinary strong law of large numbers.