Weak convergence of random vectors and distributions in Banach spaces

• Autorzy: Jajte, R. , Paszkiewicz, A.
• Języki publikacji: EN

Abstrakty

EN

Let (ξ_n) be a sequence of random vectors with values in a Banach space X with distributions p_ξn weakly converging to a given distribution p. We characterize a general form of a distribution of a weak limit of ξ _n in Banach space L₁(X) of Bochner integrable vectors. We show that the weak convergence of random vectors (ξ_n) in L₁(X) implies that ‖ξ_n(ω)−ξ(ω)‖ →0 stochastically. Moreover, the conditions ‖ξ_n(ω)−ξ(ω)‖ →0 stochastically and〈ξ_n(ω)−ξ(ω), x*〉→0 stochastically for any x*∈X*are

Słowa kluczowe

EN

distributions in Banach spaces; limits of random vectors; limits of distributions

• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2002
• Tom: Vol. 22, Fasc. 1
• Strony: 1--11
• Opis fizyczny: Bibliogr. 6 poz.

Twórcy

autor

Jajte, R.

• Faculty of Mathematics, Institute of Mathematics, University of Łódź , Banacha 22, PL-90-238 Łódź, Poland ,

autor

Paszkiewicz, A.

• Faculty of Mathematics, Institute of Mathematics, University of Łódź , Banacha 22, PL-90-238 Łódź, Poland ,
• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, PL-00-950 Warszawa, P.O. Box 17, Poland

Bibliografia

• [1] P. Billingsley, Convergence of Probability Measures, Willey, New York-Toronto 1968.
• [2] R. Jajte and A. Paszkiewicz, Conditioning and weak convergence, Probab. Math. Statist. 19 (1999), pp. 453-461.
• [3] K. Kuratowski, Topologie. I, Monografie Matematyczne, Vol. 3, Warsaw 1933.
• [4] P. A. Meyer, Probability and Potentials, Blaisdell Publish. Comp., Toronto-London 1966.
• [5] L. Pratelli, Un caractérisation de la convergence dans L1. Application aux quasimartingales, Sémin. Probab. XXVI, Lecture Notes in Math. 1526, Springer, 1992, pp. 61-69.
• [6] L. Pratelli, Sur la convergence en moyenne pour des vecteurs aléatoires intégrables au sens de Bochner, Rend. Mat. Accad. Lincei 3 (1992), pp. 299-306.