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The LIL when X is in the domain of attraction of a Gaussian law. (English) Zbl 0572.60010

A mean zero random variable X with values in a Banach space B in the domain of normal attraction of a Gaussian law satisfies the compact law of the iterated logarithm if and only if \(E(\| X\|^ 2/L_ 2\| X\|)<\infty\). In the paper under review, the author mainly shows that an analogous result can be obtained when X is merely in the domain of attraction of a Gaussian law Z, i.e. when there exist a sequence \(d_ n\nearrow \infty\) and \(\{\delta_ n\}\) in B such that \((S_ n-\delta_ n)/d_ n\) converges in distribution to Z.
Under this assumption, the author actually proves the existence of smooth coefficients \(d_ n\) which leads to a law of the iterated logarithm with regularity properties of the normalizing sequence. The corresponding functional law of the iterated logarithm is obtained in this setting and the clustering phenomena are examined with great care: in contrast with the situation of normal attraction, when X is in the domain of attraction of Z, the unit ball of the R.K.H.S. of Z may be a proper subset of the cluster set of the law of the iterated logarithm.
Reviewer: M.Ledoux

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60B11 Probability theory on linear topological spaces
60F15 Strong limit theorems
60F10 Large deviations
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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