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Non-standard Skorokhod convergence of Lévy-driven convolution integrals in Hilbert spaces. (English) Zbl 1314.60021

Summary: We study the convergence in probability in the non-standard \(\mathbf{M}_1\) Skorokhod topology of the Hilbert valued stochastic convolution integrals of the type \(\int_0^tF_\gamma(t-s)d\,L(S)\) to a process \(\int_0^tF(t-s)d\, L(s)\) driven by a Lévy process \(\mathbf{L}\). In Banach spaces, we introduce strong, weak. and product modes of \(M_1\)-convergence, prove a criterion for the \(M_1\)-convergence in probability of stochastically continuous càdlàg processes in terms of the convergence in probability of the finite dimensional marginals and a good behavior of the corresponding oscillation functions, and establish criteria for the convergence in probability of Lévy driven stochastic convolutions. The theory is applied to the infinitely dimensional integrated Ornstein-Uhlenbeck processes with diagonalizable generators.

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F17 Functional limit theorems; invariance principles
60G51 Processes with independent increments; Lévy processes
60H05 Stochastic integrals
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