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On the convergence rate in the multi-dimensional invariance principle for functionals of integral form. (Russian) Zbl 0621.60037

Let \(\xi_{j,n}\), \(n=1,2,...\), \(1\leq j\leq n\), be a triangular array of independent q-dimensional random vectors with expectation 0 and covariance \(\sigma_{j,n}I_ q\), where \(\sum^{n}_{j=1}\sigma^ 2_{j,n}=1\) and \(I_ q\) denotes the unit matrix. Defining the summation process \(S_ n(t)\), \(0\leq t\leq 1\), in a natural way, the authors consider an integral-type functional of the form \[ F(S_ n)=\int^{1}_{0}f(S_ n(t),t)dt, \] where \(\frac{\partial}{\partial x}f(x,t)\) fulfils a uniform Hölder condition in both arguments. Estimations are obtained for the error made when approximating \(F(S_ n)\) by F(W) with an appropriately given q-dimensional standard Wiener process W, in terms of \[ L_{3,n}=\sum^{n}_{j=1}E\| \xi_{j,n}\|^ 3. \] Similar results for \(q=1\) were obtained by I. S. Borisov [Teor. Veroyatn. Primen. 21, 293-308 (1976; Zbl 0366.60036)] via Skorohod embedding. Here the authors apply a modification of the method of I. Berkes and W. Philipp [Ann. Probab. 7, 29-54 (1979; Zbl 0392.60024)].
Reviewer: T.F.Mori

MSC:

60F17 Functional limit theorems; invariance principles
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