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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
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F15 Strong limit theorems
Full Text: EuDML
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