A class of integral operators generated by random variables

作者：Withers Christopher S; Nadarajah Saralees^*

来源：Statistics, 2014, 48(1): 159-166.

DOI：10.1080/02331888.2012.708033

• Hermite polynomials
• integral operator
• random variable

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
K(x)g(x) = E[g(x + X)] = integral g(x + y) dP(X <= y)
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 L-2 generally do not have inverses. But K-X is not in L-2. We show that K-X has inverse K-X(-1) = K-iX, i = root-1.

• 出版日期2014-1-2