Type, cotype, and related properties
In this chapter we connect some of the deeper properties of Rademacher sums and Gaussian sums to the geometry of the Banach space in which they live. We begin with a study of the notions of type and cotype, defined through nontrivial upper and lower bounds for random sums. Under the assumption of finite cotype, we prove a refined version of the contraction principle involving function (instead of constant) coefficients; for this we also develop a necessary minimum of the theory of summing operators. In the third section we prove geometric characterisations of type and cotype 2, due to Kwapień, and of non-trivial type and cotype, due to Maurey and Pisier. The fourth section is devoted to the notion of K-convexity and its connections with the duality of the random sequence spaces; this section culminates in Pisier's characterisation of K-convexity in terms of non-trivial type. The final section investigates the properties of multiple random sums involving products of Gaussian or Rademacher variables.
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