Asymptotic Expansions for a Class of Distribution Functions

Abstract

Investigations have been made in the past by several people on the possibility of extending the content of the classical central limit Theorem when the basic random variables are no longer independent. Several interesting extensions have been made so far. Hoeffding and Robbins (1948) have established asymptotic normality for the distribution of mean of a sequence of $m$-dependent random variables, where $m$ is a finite positive constant. The result has been proved by Diananda (1953) under more general conditions and has been extended to cover situations where the random variables $\{X_t\} (t = 1, 2, 3, \cdots)$ are of the type $X_t - E(X_t) = \sum^\infty_{j = 0} g_j Y_{t-j}$ where $\{Y_t\} (t = 0, \pm 1, \cdots)$ is an $m$-dependent stationary process and $E(Y_t) = 0,\quad\sum^\infty_{j = 0} g_j < \infty$. Walker (1954) has established asymptotic normality for the distributions of serial correlations based on $X_t$ of the above form. However, so far no attempt has been made to investigate whether the type of asymptotic expansions as discussed by Cramer (1937), Berry (1941) and Hsu (1945) for the distributions of means of independent random variables could also be extended to apply to situations where the random variables are not independent. Chanda (1962) has made a start in this direction, but the results are understandably incomplete. An attempt has been made in this paper to investigate this problem more systematically. The conclusion is that an extension is possible under conditions precisely similar to those under which Cramer, Berry and Hsu proved their results.