ON m-DEPENDENCE AND EDGEWORTH EXPANSIONS

WEI-LIEM LOH

Department of Statistics, Purdue University, West Lafayette, IN 47907-1399, U.S.A.

(Received September 25, 1992; revised April 27, 1993)

Abstract. This paper contains two results. The first establishes, under mild assumptions, the validity of an Edgeworth expansion with remainder o(N^{−1/2}) for a U-statistic with a kernel of degree two using observations from an mdependent shift. The second result gives a necessary and sufficient condition for the distribution of a sum of m-dependent random variables to possess an Edgeworth expansion. This generalizes a result of Bickel and Robinson from the i.i.d. case to the m-dependent case.

Key words and phrases: Edgeworth expansions, U-statistics, m-dependence.