Bilinear decompositions of two-dimensional probability densities with respect to orthogonal polynomials. (Russian) Zbl 0669.60022

Vetrov, L. G.

Teor. Veroyatn. Mat. Stat., Kiev 39, 24-29 (1988).

Let f(x) be the probability density function of an arbitrary random variable and \(\{\) F(x), \(k=0,1,...\}^ a \)sequence of orthonormal polynomials with weighting function f(x). Consider a function \[ (1)\quad g(x,y)=f(x)f(y)(1+\sum^{\infty}_{k=1}c_ kF_ k(x)F_ k(y)). \] The problem of determining necessary conditions for the right hand side of (1) to be a bivariate probability density function is considered. The answer is given in terms of the moment problem solution. When f(x) is Poisson density this condition is sufficient.

Reviewer: L.G.Vetrov

MSC:

60E05	Probability distributions: general theory
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

orthonormal polynomials; bivariate probability density function; moment problem solution