ASYMPTOTIC REPRESENTATIONS OF THE PROPORTION OF THE SAMPLE BELOW THE SAMPLE MEAN FOR $\phi$-MIXING RANDOM VARIABLES

Abstract

Let $\{ \rm X_i;\, -\infty \lt i \lt \infty \}$ be a stationary sequence of random variables. Let $\rm F_n(x)$ be the corresponding empirical distribution function of $\rm X_1,\ldots,\rm X_n$, and let $\bar{\rm X} = \sum^n_{i=1} \rm X_i/n$ be the sample mean. In this paper, we derive the asymptotic almost sure representation, the central limit theorem, a law of iterated logarithm, a Wiener precess embedding and an invariant principle for $\rm F_n(\bar{\rm X})$ under different $\phi$-mixing conditions.