A class of integral operators generated by random variables

Abstract

Let X be a random variable in ℝp distributed symmetrically about zero with cumulants of order 4, 8, 12, … equal to zero. This class of random variables includes the multivariate normal. Consider the linear integral operator KX defined byacting on the space of functions g: ℂp→ℂq with Taylor series expansions about zero. By Fredholm theory, non-degenerate integral operators in L2 generally do not have inverses. But KX is not in L2. We show that KX has inverse , i=√−1.