An Extension of Rosen's Theorem to Non-identically Distributed Random Variables

Abstract

In [5], B. Rosen showed that if $\{X_k: k = 1,2, \cdots\}$ is an independent sequence of identically distributed random variables with $EX_k = 0$ and $\operatorname{Var} X_k = \sigma^2, 0 < \sigma^2 < \infty$ and if $S_n = X_1 + \cdots + X_n$, then the series $\sum^\infty_{n=1} n^{-1} (P(S_n < 0) - \frac{1}{2})$ is absolutely convergent. This theorem was motivated by a result of Spitzer [6] who, under the same conditions, established the convergence of this series as a corollary to a result in the theory of random walks. Rosen's theorem was generalized by Baum and Katz [1] who showed that if $EX_k = 0$ and $E|X_k|^{2+\alpha} < \infty$ for $0 \leqq \alpha < 1$ then $\sum^\infty_{n=1}n^{-(1-\alpha/2)} |P(S_n < 0) - \frac{1}{2}| < \infty.$ These results led to the study of series convergence rate criteria for the central limit theorem and a partial solution of this problem was obtained for the case of identically distributed random variables in [2]. A more complete solution has been recently obtained by Heyde [4]. The first study of series convergence rates for $P(S_n < 0)$ in the case of independent but non-identically distributed random variables was made by Heyde [3]. Based on an extension of Rosen's theorem utilizing certain uniform bounds on the characteristic function of the $X_k$'s he concluded the absolute convergence of the series $\sum^\infty_{n=1}n^{-(1-\alpha/2)} (P(S_n < n^px) - \frac{1}{2})$ for $- \infty < x < \infty$ and $0 \leqq p < \frac{1}{2}(1 - \alpha), 0 \leqq \alpha < 1,$ thus obtaining what he termed small deviation convergence rates. In the present paper two more extensions of Rosen's theorem to independent but non-identically distributed random variables are given under different hypotheses than Heyde's. The first (Theorem 1) reduces to Rosen's theorem in the case of identically distributed random variables. The second (Theorem 2) results in a theorem similar to that of Baum and Katz [1] as required in Heyde's small deviation result. This will make it possible to obtain his conclusion by simply carrying out the last step in his proof. These results are obtained in Section 3. In Section 2 some preliminary results are stated and examples are given in Section 4 to show that the first two hypotheses of Theorem 1 cannot, in general, be relaxed.