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Rate of convergence in the central limit theorem and in the strong law of large numbers for von mises statistics

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Abstract

This paper provides the rate of convergence in the central limit theorem and in the strong law of large numbers forvon Mises statistics\(V_N = N^{ - m} \sum\limits_{i_1 = 1}^N \ldots \sum\limits_{i_m = 1}^N {h(X_{i_1 } , \ldots ,X_{i_m } ),N \geqslant m} \), based on i.i.d. random variablesX 1 ,..., X N .

The proofs rely on a decomposition ofvon Mises statistics into a linear combination ofU-statistics and then use (generalized) results on the convergence rates forU-statistics obtained byGrams/Serfling [1973] andCallaert/Janssen [1978].

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  1. Limburgs Universitair Centrum, Universitaire Campus, B-3610, Diepenbeek

    P. Janssen

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  1. P. Janssen
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Janssen, P. Rate of convergence in the central limit theorem and in the strong law of large numbers for von mises statistics. Metrika 28, 35–46 (1981). https://doi.org/10.1007/BF01902875

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  • DOI: https://doi.org/10.1007/BF01902875

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