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Contents of PMS, Vol. 22, Fasc. 1,
pages 1 - 11
 

WEAK CONVERGENCE OF RANDOM VECTORS AND DISTRIBUTIONS IN BANACH SPACES

Ryszard Jajte
Adam Paszkiewicz

Abstract: Let (q) n be a sequence of random vectors with values in a Banach space X with distributions p qn weakly converging to a given distribution p. We characterize a general form of a distribution of a weak limit of q n in Banach space L (X) 1 of Bochner integrable vectors. We show that the weak convergence of random vectors (q) n in L (X) 1 implies that ||q (w)- q(w)||--> 0 n stochastically. Moreover, the conditions ||q (w) - q(w)|| --> 0 n stochastically and <q (w)- q(w),x*> --> 0 n stochastically for any x* (- X* are equivalent.

1991 AMS Mathematics Subject Classification: 60B11, 46B09.

Key words and phrases: Distributions in Banach spaces, weak limits of random vectors, weak limits of distributions.

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