A note on convergence in Banach spaces of cotype p

Let B be a separable Banach space. The following is one of the results proved in this paper. The Banach space B is of cotype p if and only if 1. dn, n [greater-or-equal, slanted] 1, has no subsequence converging in probability, and 2. [summation operator]n [greater-or-equal, slanted] 1anp < [infinity] whenever [summation operator]n [greater-or-equal, slanted] 1andn converges almost surely are equivalent for every sequence dn, n [greater-or-equal, slanted] 1, of symmetric independent random elements taking values in B.