A note on convergence in Banach spaces of cotype \(p\) (Q753237)

The authors prove some theorems on convergence in Banach spaces of cotype p. The most interesting one is Theorem. The following statements are equivalent: (i) The Banach space B is of cotype p. (ii) \(\sum_{n\geq 1}| a_ n|^ p<\infty\), whenever \(\sum_{n\geq 1}a_ nd_ n\) converges almost surely for any sequence \(d_ n\), \(n\geq 1\), of symmetric independent random elements in B, which has no subsequence converging in probability.