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Characterization of the law of the iterated logarithm in Banach spaces. (English) Zbl 0662.60008
Using a Gaussian randomization technique, we prove that a random variable X with values in a Banach space B satisfies the (compact) law of the iterated logarithm if and only if (i) $$E(\| X\|^ 2/L L \| X\|)<\infty$$, (ii) $$\{| <x^*,X>|^ 2$$; $$x^*\in B^*$$, $$\| x^*\| \leq 1\}$$ is uniformly integrable and (iii) $$S(x)/a_ n\to 0$$ in probability.
In particular, if B is of type 2, in order that X satisfies the law of the iterated logarithm it is necessary and sufficient that X has mean zero and satisfies (i) and (ii).
The proof uses tools of the theory of Gaussian random vectors as well as by now classical arguments of probability in Banach spaces. It also sheds some light on the usual law of the iterated logarithm on the line.

##### MSC:
 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60B11 Probability theory on linear topological spaces 60G15 Gaussian processes 46B20 Geometry and structure of normed linear spaces
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