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The invariance principle for stochastic integrals in Hilbert spaces. (English) Zbl 0756.60035

Summary: Let \(H\) be a separable Hilbert space and let \(\{X_ n\), \(n\geq 1\}\) be a sequence of i.i.d. \(H\)-valued random variables. We investigate limit behaviour, in the sense of \(D([0,1];H)\)-weak convergence, of the sequence \[ \left\{\left(n^{-1/2}\sum_{i=0}^{[nt]-1}f_ n\left(i/n, n^{- 1/2}\sum_{k=1}^ i X_ k\right)X_{i+1}:\;0\leq t\leq 1\right),\;n\geq 1\right\}. \]

MSC:

60F17 Functional limit theorems; invariance principles
60H05 Stochastic integrals
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