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Rate of convergence for degenerate von-Mises functionals. (Russian) Zbl 0634.60030
The authors obtain the estimate of the rate of convergence for degenerate von-Mises functionals. They prove the following result:
Let \(\{X_ n,n\geq 1\}\) be a sequence of i.i.d. random variables on a measurable space (X,\({\mathcal A})\) with distribution P. Let \(Q_ n=n^{- 1}\sum^{n}_{i=1}\sum^{n}_{j=1}\phi (X_ i,X_ j)\), where \(\phi\) : (x,y)\(\to \phi (x,y)\) is symmetric and degenerate, i.e. \(\int \phi (x,y)P(dy)=0\) (P-a.s.), \(F_ n(x)=P(Q_ n<x)\), \(F_{\infty}(x)=\lim_{n\to \infty}F_ n(x).\)
If \(E| \phi (X_ 1,X_ 2)|\) \(3<\infty\) and \(E| \phi (X_ 1,X_ 1)|^{3/2}<\infty\), then \[ \sup_{x}| F_ n(x)- F_{\infty}(x)| =o(n^{-1/2}),\quad n\to \infty. \]
Reviewer: V.Sakalauskas

MSC:
60F17 Functional limit theorems; invariance principles
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