An extension of the Erdős-Rényi new law of large numbers

Abstract:

If ${S_n}$ is the $n$th partial sum of a sequence of independent, identically distributed random variables ${X_1},{X_2} \cdots$ such that $E({X_1}) = 0$ and $E(\exp (t{X_1})) < \infty$ for some nonempty interval of $t$’s, then, for a wide range of positive numbers $\lambda$, Erdös and Rényi (1970) showed that $\Sigma (N,[C(\lambda )\log N])$ converges with probability one to $\lambda$ as $N \to \infty$, where $\Sigma (N,K)$ is the maximum of the $N - K + 1$ averages of the form ${K^{ - 1}}({S_{n + K}} - {S_n})$ for $0 \leq n \leq N - K$, and $C(\lambda )$ is a known constant depending on $\lambda$ and the distribution of ${X_1}$. The objective of the present article is to state and prove the Erdös-Rényi theorem for the $N - K + 1$ “averages” of the form ${K^{ - 1/r}}({S_{n + K}} - {S_n})$, where $1 < r < 2$. This form of the Erdös-Rényi theorem arises from the extended form of the strong law of large numbers which asserts that, if $E(|{X_1}{|^r}) < \infty$ for some $r,1 \leq r < 2$, and $E({X_1}) = 0$, then ${n^{ - 1/r}}{S_n}$ converges with probability one to 0 as $n \to \infty$.