Bardina, Xavier; Jolis, Maria; Tudor, Ciprian A. On the convergence to the multiple Wiener-Itô integral. (English) Zbl 1163.60022 Bull. Sci. Math. 133, No. 3, 257-271 (2009). Summary: We study the convergence to the multiple Wiener-Itô integral from processes with absolutely continuous paths. More precisely, consider a family of processes, with paths in the Cameron-Martin space, that converges weakly to a standard Brownian motion in \(\mathcal C_0([0,T])\). Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-Itô integral process of a function \(f\in L^{2}([0,T]^n)\). We prove also the weak convergence in the space \(\mathcal C_0([0,T])\) to the second-order integral for two important families of processes that converge to a standard Brownian motion. Cited in 4 Documents MSC: 60H05 Stochastic integrals 60B10 Convergence of probability measures 60F05 Central limit and other weak theorems Keywords:multiple Wiener-Itô integrals; weak convergence; Donsker theorem PDFBibTeX XMLCite \textit{X. Bardina} et al., Bull. Sci. Math. 133, No. 3, 257--271 (2009; Zbl 1163.60022) Full Text: DOI arXiv References: [1] Avram, F., Weak convergence of the variations, iterated integrals and Doléans-Dade exponentials of sequences of semimartingales, Ann. Probab., 16, 1, 246-250 (1988) · Zbl 0636.60029 [2] Bardina, X.; Jolis, M., Weak convergence to the multiple Stratonovich integrals, Stochastic Process. Appl., 90, 2, 277-300 (2000) · Zbl 1047.60048 [3] Billinsgley, P., Convergence of Probability Measures (1968), John Wiley and Sons [4] Itô, K., Multiple Wiener integral, J. Math. Soc. Japan, 3, 157-169 (1951) · Zbl 0044.12202 [5] Kac, M., A stochastic model related to the telegrapher’s equation, Rocky Mountain J. Math., 4, 497-509 (1974), reprinting of an article published in 1956 · Zbl 0314.60052 [6] Solé, J. L.; Utzet, F., Stratonovich integral and trace, Stochastics Stochastics Rep., 29, 2, 203-220 (1990) · Zbl 0706.60056 [7] Stroock, D., Topics in Stochastic Differential Equations (1982), Tata Institute of Fundamental Research/Springer-Verlag: Tata Institute of Fundamental Research/Springer-Verlag Bombay · Zbl 0516.60065 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.