This chapter provides a systematic investigation of vector-valued Gaussian and Rademacher random sums, which will be a key tool for subsequent developments throughout this volume. Here we concentrate on general properties of random sums that remain valid in arbitrary Banach spaces, while more specific results connected with the geometry of the underlying space are taken up in the following chapter. In the present chapter, we establish a range of comparison results relating the L p norms of different types of random sums, as well as different L p norms of a fixed sum. We also describe the dual and bi-dual of the spaces of random sequences, and characterise the convergence of infinite random series. In the final section, we compare the L p norms of random sums and lacunary trigonometric sums.
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