LIMIT-THEOREMS FOR RANDOM-VARIABLES WITH VALUES IN SPACES LP (2LESS-THAN-OR-EQUAL-TO P LESS-THAN INFINITY)

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 lp or Lp with 2<p<∞ which satisfy the CLT. As an application we show that in lp (2<p<∞) the stochastic boundedness of the weighed partial sums does not imply the CLT. © 1978 Springer-Verlag.