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 ∏j=1n−1Hkj% (Xj) given Xn=z, if random vector (X1,…,Xn)T has multivariate normal distribution and Hn(x) denotes n-th Hermite polynomial. This expectation is a polynomial in z of order ∑j=1% n−1kj. 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. We believe that the results can be very useful in Probability Theory, in particular in the Theory of distributions, Statistics, and the Theory of Stochastic Processes, in particular Gaussian ones.
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BibTeX
@misc{szablowski2026conditional,
title = {{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}},
author = {Paweł J. Szabłowski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.22526},
eprint = {2606.22526},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.22526},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - 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
AU - Paweł J. Szabłowski
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2606.22526
UR - https://arxiv.org/abs/2606.22526
AB - We calculate the conditional expectation of ∏_j=1^n−1H_k_j% (X_j) given X_n=z, if random vector (X_1,…,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 ∑_j=1% ^n−1k_j. 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. We believe that the results can be very useful in Probability Theory, in particular in the Theory of distributions, Statistics, and the Theory of Stochastic Processes, in particular Gaussian ones.
ER -