6 Spaces of type 2 and cotype 2

Remark 7.5.1. G. Pisier (1975/76) has also demonstrated that if X is a Banach space then each zero-mean random vector X ∈ L2(X) satisfies the Law of the Iterated Logarithm if, and only if, there exists a constant C such that for all n ∈ N, and every sequence (xi) ⊂X,

E ∥∥∥ n∑ i=1

rixi

∥∥∥ ≤ C(log log n)1/2( n∑ i=1

‖xi‖2 )1/2

.