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Donsker’s theorem and Dirichlet forms. (Théorème de Donsker et formes de Dirichlet.) (French) Zbl 1093.60018

Let us consider a sequence of independent and identically distributed random variables and associate the corresponding random walk with them. We then define a sequence of stochastic processes by taking the piecewise linear interpolation of the random walk. Essentially, the classical Donsker’s theorem asserts that these processes converge weakly to a Brownian motion. The author shows that if the considered random variables are in addition erroneous, the convergence occurs in the sense of Dirichlet forms and induces the Ornstein-Uhlenbeck structure on the Wiener space. The proof requires an extension of the Donsker theorem to functions with quadratic growth. This extension is more delicate than in the case of the central limit theorem and is the main difficulty of the proof. As a consequence to his result, the author proves an invariance principle for the gradient of the maximum of the Brownian path calculated by Nualart and Vives.

MSC:

60G50 Sums of independent random variables; random walks
60J65 Brownian motion
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References:

[1] Billingsley, P., Convergence of Probability Measures (1968), Wiley · Zbl 0172.21201
[2] Bouleau, N., Error Calculus for Finance and Physics, the Language of Dirichlet Forms (2003), De Gruyter
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[9] Feyel, D.; de la Pradelle, A., Espaces de Sobolev Gaussiens, Ann. Inst. Fourier, 39, 4, 875-908 (1989) · Zbl 0664.46028
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[11] Nualart, D.; Vives, J., Continuité de la loi du maximum d’un processus continu, C. R. Acad. Sci. Paris Sér. I, 307, 349-354 (1988) · Zbl 0651.60066
[12] Nualart, N., The Malliavin Calculus and Related Topics (1995), Springer · Zbl 0837.60050
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