Sankhya: The Indian Journal of Statistics

1996, Volume 58, Series A, Pt. 2, pp. 206--224

ASYMPTOTIC EXPANSIONS FOR SUMS OF RANDOM VECTORS UNDER POLYNOMIAL MIXING RATES

By

SOUMENDRA NATH LAHIRI, Iowa State University

SUMMARY.  Let ${X_n}_{n \geq 1}$  be a sequence of $I \hskip-3pt R^d$-valued random vectors with strong mixing coeficient $\alpha (.)$. Let EXj=0 for all $j \geq 1$ and $S_n=($X_1+...+X_n)/ \surd, n \geq 1$. This paper establishes asymptotic expansions for Ef(Sn) for a large class of Borel-measurable functions f when Sup ${E \| X_j \|_{s+\delta} :j \geq 1}< \infty$ for some integers $s \geq 3$ and some $\delta \in (0, 1]$, and the mixing coefficient $\alpha(n)$ decays at the rate n-C for some constant C 0, depending on $\delta, d$, the order of the expansion and the growth rate of the function f. Under similar conditions on $\alpha(.)$, expansions for Ef(Sn) are obtained in the cases when (i) Xj's are (uniformly bounded, or when (ii) f has (s-1)-th order partial derivatives, but Xj's do not satisfy a conditional Cramer condition.

AMS (1980) subject classification.  60F05, 60G60.

Key words and phrases. Asympotic expansion, Edgeworth expansion, strong mixing.

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This article in Mathematical Reviews.