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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