Isotropic probability measures in infinite-dimensional spaces.

Abstract

Every isotropic probability measure on the space R(infinity) of real sequences x = (x(1), x(2),...) is a convex combination of the measure concentrated at 0 and a member of I(0)(R(infinity)), the set of all isotropic probability measures p(infinity) on R(infinity) with p(infinity)({0}) = 0. Each p(infinity) [unk] I(0)(R(infinity)) is completely determined by any one of its finite-dimensional marginal distributions p(n). Each p(n) has a density function f(n) with dp(n)(x(1),..., x(n)) = dx(1)... dx(n)f(n)(x(1) (2) +... + x(n) (2)). Each f(n) is completely monotone in 0 < xi < infinity (hence analytic in the right complex xi half-plane), and pi(n/2)Gamma(n/2)(-1) (0) (infinity)dxi xi(n/2-1)f(n)(xi) = 1. Every f that satisfies these two conditions is f(n) for a unique p(infinity) [unk] I(0)(R(infinity)). Hence the equation pi(xi) (infinity)dzeta f(2)(zeta) = (0) (infinity)dmu (t)e(-txi) defines a bijection between I(0)(R(infinity)) and the set of all probability measures mu on 0 </= t < infinity. If p(infinity) [unk] I(0)(R(infinity)) then p(infinity)({x: Sigma(i=1) (infinity)x(i) (2) < infinity}) = 0, so p(infinity) is not a "softened" or "fuzzy" version of the inequality Sigma(i=1) (infinity)x(i) (2) </= 1. If the prior information in a linear inverse problem consists of this inequality and nothing else, stochastic inversion and Bayesian inference are both unsuitable inversion techniques.