1 99 9 the Distribution of Vector - Valued Rademacher Series

Let X = ε n x n be a Rademacher series with vector-valued coefficients. We obtain an approximate formula for the distribution of the random variable ||X|| in terms of its mean and a certain quantity derived from the K-functional of interpolation theory. Several applications of the formula are given. In [6] the second-named author calculated the distribution of a scalar Rademacher series ε n a n. The principal result of the present paper extends the results of [6] to the case of a Rademacher series ε n x n with coefficients (x n) belonging to an arbitrary Banach space E. Its proof relies on a deviation inequality for Rademacher series obtained by Talagrand [9]. A somewhat curious feature of the proof is that it appears to exploit in a non-trivial way (see Lemma 2) the platitude that every separable Banach space is isometric to a closed subspace of ℓ ∞. The principal result is applied to yield a precise form of the Kahane-Khintchine inequalities and to compute certain Orlicz norms for Rademacher series. First we recall some notation and terminology from interpolation theory (see e.g.