Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series A, Pt. 1, 88-101

ASYMPTOTICS OF SOME EMPERICAL FUNCTIONS UNDER LONG RANGE DEPENDENCE

By

KANCHAN MUKHERJEE, National University of Singapore, Singapore

And

SUMAN MAJUMDAR, University of Connecticut, Stanford

SUMMARY. This paper obtains the asymptotic representation of the supremum of a class of functionals of the empirical distribution, with application to estimating the strength of a bundle of parallel filaments, when the observations are strongly dependent. Unlike the case of weakly dependent observations discussed by Sen and Bhattacharya (1976), the limiting distributions of these functionals are not always normal. The nature of the limiting distribution depends heavily on the Hermite rank of a class of indicator functions, and the rate of convergence is much slower in this case (compared to the case of weak dependence). A law of iterated logarithm (LIL) for these functionals is also derived.

AMS (1991) subject classification. 62G20, 60F17

Key words and phrases. Long range dependence, Hermite rank, empirical distribution, LIL, strength of a bundle.

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