The conditional expectation of the product of the first $n-1$ Hermite polynomials in a multivariate normal distribution with respect to the $n$-th variable. A fresh perspective on the Kibble-Slepian formula

Abstract

We calculate the conditional expectation of ${\prod }_{j=1}^{n-1}{H}_{k_j}$ given $X_n=z,$ if random vector $(X_1,\dots ,X_n{)}^T$ has multivariate normal distribution and $H_n(x)$ denotes $n$-th Hermite polynomial. This expectation is a polynomial in $z$ of order ${\sum }_{j=1}$. Our formula has an iterative form with respect to $n$. We also present some auxiliary observations concerning the expansion of the density of the $n$-dimensional normal distribution in the series of the Hermite polynomials. Mostly concerning the properties of the coefficients of this expansion. To perform these calculations, we give a few auxiliary formulas concerning Hermite polynomials and multivariate normal distributions. We apply this result to obtain exact, simple forms of these expansions for $n=2$ and $3$, thus looking at known results from a different perspective.