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Wald’s equation and asymptotic bias of randomly stopped $U$-statistics

Authors: Victor H. de la Peña and Tze Leung Lai
Journal: Proc. Amer. Math. Soc. 125 (1997), 917-925
MSC (1991): Primary 60G40, 62L12; Secondary 62L10
MathSciNet review: 1350937
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Abstract: In this paper we make use of decoupling arguments and martingale inequalities to extend Wald’s equation for sample sums to randomly stopped de-normalized $U$-statistics. We also apply this result in conjunction with nonlinear renewal theory to obtain asymptotic expansions for the means of normalized $U$-statistics from sequential samples.

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Additional Information

Victor H. de la Peña
Affiliation: Department of Statistics, Columbia University, 617 Mathematics Bldg., New York, New York 10027
MR Author ID: 268889

Tze Leung Lai
Affiliation: Department of Statistics, Stanford University, Sequoia Hall, Stanford, California 94305-4065

Keywords: Hoeffding decomposition, decoupling, martingales, Wald’s equation, stopping times
Received by editor(s): October 15, 1994
Received by editor(s) in revised form: July 28, 1995
Additional Notes: The first author’s research was supported by the National Science Foundation under DMS-9310682.
The second author’s research was supported by the National Science Foundation under DMS-9403794.
Communicated by: Wei Y. Loh
Article copyright: © Copyright 1997 American Mathematical Society