Orthogonal polynomials and natural exponential families

Pommeret, Denys

doi:10.1007/bf02562683

Cited by 21 publications

(23 citation statements)

References 7 publications

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“…In contrast, natural multivariate generalizations of Proposition 2 involve more familiar polynomials in commuting variables, orthogonal with respect to n-dimensional measures. They have been investigated by a number of people; see for example [11,14,8,17,9,18]. This analysis is usually performed in the context of natural exponential families.…”

Section: Here the Conditions On U And R Are The Same As Above These mentioning

confidence: 99%

Orthogonal polynomials with a resolvent-type generating function

Anshelevich

2008

Trans. Amer. Math. Soc.

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Abstract. The subject of this paper are polynomials in multiple non-commuting variables. For polynomials of this type orthogonal with respect to a state, we prove a Favard-type recursion relation. On the other hand, free Sheffer polynomials are a polynomial family in non-commuting variables with a resolvent-type generating function. Among such families, we describe the ones that are orthogonal. Their recursion relations have a more special form; the best way to describe them is in terms of the free cumulant generating function of the state of orthogonality, which turns out to satisfy a type of secondorder difference equation. If the difference equation is in fact first order, and the state is tracial, we show that the state is necessarily a rotation of a free product state. We also describe interesting examples of non-tracial infinitely divisible states with orthogonal free Sheffer polynomials.

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Section: Here the Conditions On U And R Are The Same As Above These mentioning

confidence: 99%

Orthogonal polynomials with a resolvent-type generating function

Anshelevich

2008

Trans. Amer. Math. Soc.

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“…K Corollary 2.1 (Pommeret, 1996). With the assumptions of Theorem 2.2, if F is irreducible the two following statements are equivalent:…”

Section: Orthogonal Polynomialsmentioning

confidence: 96%

“…We may suppose A=id (the identity), Lemma 2.1 giving the general case. We content ourselves only with showing part (i) since part (ii) follows this proof (see Pommeret 1996). Let P n =P id, n .…”

Section: Orthogonal Polynomialsmentioning

confidence: 96%

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Multidimensional Bhattacharyya Matrices and Exponential Families

Pommeret

1997

Journal of Multivariate Analysis

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“…Examples of multivariate polynomials Q A,n can be found in [15] and their norms Q A,n (., m 0 ) 2 are given in [16].…”

Section: Multivariate Extensionmentioning

confidence: 99%

Polynomial expansions of density of power mixtures

Pommeret

2007

ESAIM: PS

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Abstract. For any given random variable Y with infinitely divisible distribution in a quadratic natural exponential family we obtain a polynomial expansion of the power mixture density of Y . We approach the problem generally, and then consider certain distributions in greater detail. Various applications are indicated and the results are also applied to obtain approximations and their error bounds. Estimation of density and goodness-of-fit test are derived.Mathematics Subject Classification. 60E05, 62E17.

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Orthogonal polynomials and natural exponential families

Abstract: Orthogonal Polynomials, Natural Exponential Families, (Simple) Quadratic Variance Functions,

Cited by 21 publications

References 7 publications

Orthogonal polynomials with a resolvent-type generating function

Orthogonal polynomials with a resolvent-type generating function

Multidimensional Bhattacharyya Matrices and Exponential Families

Polynomial expansions of density of power mixtures

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