A version of Chevet's theorem for stable processes |
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Authors: | Evarist Giné Michael B Marcus Joel Zinn |
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Institution: | Department of Mathematics, Texas A & M University, College Station, Texas 77843 USA |
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Abstract: | Let be the injective tensor product of the separable Banach spaces X and Y and let SX, SY and 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 . It is shown that η1? η2 is the spectral measure of a p-stable random variable W on , 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 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 <2, xi?C(S), and yi?C(T), then the series ∑ijθijxi(s) yj(t) converges uniformly a.s. iff the series ∑iθixi(s) and ∑iθiyi(t) both converge uniformly a.s. When p = 2 this follows from Chevet's theorem on Gaussian processes. Several examples are given. One of them requires an interesting upper bound on the probability distribution of the maximum of i.i.d. p-stable random variables taking values in a general Banach space. |
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