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.
Citation
Cheun-Der Lea. "ASYMPTOTIC REPRESENTATIONS OF THE PROPORTION OF THE SAMPLE BELOW THE SAMPLE MEAN FOR $\phi$-MIXING RANDOM VARIABLES." Taiwanese J. Math. 10 (5) 1379 - 1390, 2006. https://doi.org/10.11650/twjm/1500557308
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