Recursive Computation of Multivariate Hermite–Gaussian Integrals with Applications

E. Taufer${}^a$ and Gy. H. Terdik${}^b$ and M. Bee${}^a$

${}^a$University of Trento, ${}^b$University of Debrecen

An exact recursive algorithm for evaluating integrals of $d$-variate Hermite polynomials weighted by the Gaussian density over orthants is introduced. The proposed approach provides closed-form expressions that can be computed efficiently for any fixed polynomial order $k$ and threshold vector. This result addresses a fundamental computational issue and enables the systematic use of higher-order expansions in practical multivariate problems.

The algorithm has a computational complexity of order $O(k{d}^k)$ and memory requirements of order $O(d^k)$, which may become prohibitive as either the dimension $d$ or the Hermite order $k$ increases. To mitigate this issue, a streaming formulation is developed that avoids storing large intermediate arrays arising from Kronecker products. This approach computes contributions sequentially, significantly reducing memory usage and improving numerical stability.

Applications are developed through integrated versions of Gram–Charlier and Edgeworth expansions for approximating multivariate cumulative distribution functions, as well as multivariate tail conditional expectations and related risk measures, including multivariate Value at Risk. The expansions are presented in compact and intuitive forms of arbitrary order using multivariate Bell polynomials.

Simulation studies and real data applications illustrate the performance and practical relevance of the proposed method. All results can be readily implemented using the R package MultiStatM. [1].

Keywords: Multivariate cumulants; multivariate density approximation; multivariate tail conditional expectation.