Summary
We prove that whenever B is an infinite dimensional Banach space, there exists a B-valued random variable X failing the Central Limit Theorem (in short the CLT) and such that IE∥X∥2=∞ but yet satisfying the Law of the Iterated Logarithm (in short the LIL). We obtain a new sufficient condition for the LIL in Hilbert space and we characterize the random variables with values in l p or L p with 2<p<∞ which satisfy the CLT. As an application we show that in l p (2<p<∞) the stochastic boundedness of the weighed partial sums does not imply the CLT.
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Research partially supported by NSF Grant MCS 75-07605 A01
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Pisier, G., Zinn, J. On the limit theorems for random variables with values in the spaces L p (2≦p<∞). Z. Wahrscheinlichkeitstheorie verw Gebiete 41, 289–304 (1978). https://doi.org/10.1007/BF00533600
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DOI: https://doi.org/10.1007/BF00533600