A version of Chevet's theorem for stable processes

AbstractLet X \̌bo Y be the injective tensor product of the separable Banach spaces X and Y and let SX, SY and SX \̌bo Y be the unit spheres of these spaces. The tensor product of two symmetric finite measures η1 on SX and η2 on SY, η1⊗η2, is defined in a natural way as a measure on SX \̌bo Y. It is shown that η1⊗ η2 is the spectral measure of a p-stable random variable W on X \̌bo Y, 0 <p < 2, if and only if η1 and η2 are the spectral measures of p-stable random variables U and V on X and Y, respectively. Actually upper and lower bounds for (E∥ W∥r)1r in terms of the random variables U and V are obtained. When X = C(S), Y = C(T) with S, T compact metric spaces, and η1, and η2 are discrete, our results imply that if θi, θij are i.i.d. standard symmetric real valued p-stable random variables, 0 < p