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Some comments on a result of Hanš on strong convergence of sequences of random elements in separable Banach spaces. (English) Zbl 0669.60015
Let ($$\Omega$$,A,P) be a complete probability space and let $$\{X_ n\}$$ be a sequence of random elements on $$\Omega$$ taking values in a separable Banach space B. Let $A=\{\omega:\quad \{X_ n(\omega)\}\quad is\quad relatively\quad strongly\quad compact\quad in\quad B\}.$ O. Hanš [Trans. 1st Prague Conf. Inf. Theor., Statist. Decis. Funct., Random Processes, Liblice 1956, 61-103 (1957; Zbl 0089.339)], proved that $$\{X_ n\}$$ converges a.e. [P] if and only if the following two conditions hold:
($$\alpha)$$ $$P(A)=1;$$
($$\beta)$$ $$\{f(X_ n)\}$$ converges a.e. [P] for every f in a total subset of $$B^*.$$
The authors of the present paper examine conditions ($$\alpha)$$ and ($$\beta)$$ in relation to
($$\gamma)$$ $$\{X_ n\}$$ converges in probability.
They also provide counter examples to show that some of Hanš’ characterizations of almost sure convergence of sequences of random variables taking values in some specific Banach spaces, are wrong. Finally they provide a result, in the spirit of Hanš’ theorem, for convergence in probability.
Reviewer: A.Dale
##### MSC:
 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60F15 Strong limit theorems
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##### References:
  BANACH S.: Theorie des Opérations Linéaires. Warszawa 1932. · Zbl 0005.20901  BECK A., WARREN P.: Strong laws of large numbers for weaklu orthogonal sequences of Banach space-valued random variables. Ann. Probabilitu, Vol. 2, 1974, 918-925. · Zbl 0293.60026  CHUNG K. L.: A Course in Probabilitu Theoru. Second edition, Academic Press, London 1974.  HANŠ O.: Generalized random variables. Trans. First Prague Conf. on Information Theoru, Statistics, Decision Functions, and Random Processes, 1957, 61-103. · Zbl 0089.33901  WANG X. C., RAO M. B.: Convergence in the pth - mean and some weak laws of large numbers for weighted sums of random elements in separable normed linear spaces. J. Multivariate Analusis, Vol. 15, No. V 124-134. · Zbl 0546.60006
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