Positive Definiteness of Multivariate Densities Based on Hermite Polynomials

Positive Definiteness of Multivariate Densities Based on Hermite Polynomials

This paper develops both univariate and multivariate distributions based on Gram-Charlier and Edgeworth expansions, attempting to ensure non negativity by exploiting the orthogonal properties of the Hermite polynomials. The article motivates the problems underlying some specifications (in particular those involving other conditional moments beyond the variance) and provides empirical examples c...

Strict Positive Definiteness on Spheres via Disk Polynomials

Multivariate Positive Laurent Polynomials

Recently Dritschel proves that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Féjer-Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. F...

On Hermite-hermite Matrix Polynomials

In this paper the definition of Hermite-Hermite matrix polynomials is introduced starting from the Hermite matrix polynomials. An explicit representation, a matrix recurrence relation for the Hermite-Hermite matrix polynomials are given and differential equations satisfied by them is presented. A new expansion of the matrix exponential for a wide class of matrices in terms of Hermite-Hermite ma...

Factorization of multivariate positive Laurent polynomials

Recently Dritschel proves that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Féjer-Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. F...