A class of integral operators generated by random variables

Abstract

Let X be a random variable in ℝ^p 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 K_X defined by acting on the space of functions g: ℂ^p→ℂ^q with Taylor series expansions about zero. By Fredholm theory, non-degenerate integral operators in L₂ generally do not have inverses. But K_X is not in L₂. We show that K_X has inverse , i=√−1.

Keywords:

• Hermite polynomials
• integral operator
• random variable

AMS 2000 subject classifications:

• Primary: 45P05
• Secondary: 62E15