A Remark on the Strong Law

Abstract

Let $X_1, X_2, \cdots$ be a sequence of random variables, each having conditional mean zero given the past. Let $V_n$ be the conditional variance of $X_n$ given the past. Les $S_n = X_1 + \cdots + X_n$ and $T_n = V_1 + \cdots + V_n$. For simplicity, suppose $\sum V_i = \infty$ a.e. For which non-decreasing function $\phi$ does $S_n/\phi(T_n)$ necessarily lead to 0 a.e. as $n$ increases? It is necessary and sufficient that $\int^\infty 1/\phi(t)^2 dt < \infty$. That is, if the integral is finite, convergence to zero a.e. is guaranteed for all $X_i$ and $V_i$ satisfying the stated condition. If the integral is infinite, there is a sequence of independent, symmetric random variables $X_i$, each having variance 1, such that $S_n/\phi(n)$ oscillates between $\pm\infty$. The sufficiency is known, but a new proof is given.