A note on invariance principles for v. Mises' statistics
type: Item
instance of: scholarly article
title: A note on invariance principles for v. Mises' statistics
zbMATH Open document ID: 0609.62029
DOI: 10.1007/BF01897813
author: G. Keller
author: Manfred Denker
author: Christian Grillenberger
published in: Metrika
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/176003
review text: Let \((X_ n)_{n\geq 1}\) (resp. \((Y_ n)_{n\geq 1})\) be a process of \(E_ 1\) (resp. \(E_ 2)\)-valued i.i.d. random variables with distribution \(L(X_ 1)\) (resp. \(L(Y_ 1))\). Suppose that the two processes are independent. Using these processes and a measurable function \(h(x_ 1,...,x_{m_ 1},y_{m_ 1+1},...,y_ m)\) \((x_ i\in E_ 1\) \((i=1,...,m_ 1)\), \(y_ j\in E_ 2\) \((j=m_ 1+1,...,m))\) define the two-sample von Mises statistics \(V_ n(h)\) \((n=(n_ 1,n_ 2))\). Finally, define \(T^ h_ n\) \((n\in N^ 2)\) to be the \(D([0,1]^ 2)\)-valued random element given by \[ T^ h_ n(t_ 1,t_ 2)=(n_ 1+n_ 2)^{(m_ 1+m_ 2)/2}t_ 1^{m_ 1}t_ 2^{m_ 2}V_{([n_ 1t_ 1],[n_ 2t_ 2])}(h). \] In this paper, extending \textit{A. A. Filippova}'s result [Teor. Veroyatn. Primen. 7, 26-60 (1962; Zbl 0118.145); English translation in Theory Probab. Appl. 7, 24-57 (1962)], on weak convergence of von Mises functionals, the authors proved a weak convergence principle for \(\{T^ h_ n\}\). Applications to U- statistics and extensions to contiguity and some weakly dependent processes are also considered.
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 3986382
zbMATH Keywords: multiple stochastic integral
zbMATH Keywords: Kiefer process
zbMATH Keywords: invariance principles
zbMATH Keywords: functional limit theorems
zbMATH Keywords: Brownian bridge
zbMATH Keywords: two-sample von Mises statistics
zbMATH Keywords: weak convergence
zbMATH Keywords: von Mises functionals
zbMATH Keywords: U-statistics
zbMATH Keywords: contiguity
zbMATH Keywords: weakly dependent processes
reviewed by: Ken-ichi Yoshihara
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3986382