Multivariate Krawtchouk polynomials and composition birth and death processes

This paper defines the multivariate Krawtchouk polynomials, orthogonal on the multinomial distribution, and summarizes their properties as a review. The multivariate Krawtchouk polynomials are symmetric functions of orthogonal sets of functions defined on each of N multinomial trials. The dual multivariate Krawtchouk polynomials, which also have a polynomial structure, are seen to occur naturally as spectral orthogonal polynomials in a Karlin and McGregor spectral representation of transition functions in a composition birth and death process. In this Markov composition process in continuous time there are N independent and identically distributed birth and death processes each with support 0,1, .... The state space in the composition process is the number of processes in the different states 0,1,... Dealing with the spectral representation requires new extensions of the multivariate Krawtchouk polynomials to orthogonal polynomials on a multinomial distribution with a countable infinity of states

Multivariate Kawtchouk polynomials as Birth and Death polynomials

Multivariate Krawtchouk polynomials are constructed explicitly as Birth and Death polynomials, which have the nearest neighbour interactions. They form the complete set of eigenpolynomials of a birth and death process with the birth and death rates at population $x=(x_1,\ldots,x_n)$ are $B_j(x)=\bigl(N-\sum_{i=1}^nx_i\bigr)$ and $D_j(x)=p_i^{-1}x_j$, $0<p_j$, $j=1,\ldots,n$. The corresponding stationary distribution is the multinomial distribution with the probabilities $\{\eta_i\}$, $\eta_i= p_i/(1+\sum_{j=1}^np_j)$. The polynomials, depending on $n+1$ parameters ($\{p_i\}$ and $N$), satisfy the difference equation with the coefficients $B_j(x)$ and $D_j(x)$ $j=1,\ldots,n$, which is the straightforward generalisation of the difference equation governing the single variable Krawtchouk polynomials. The polynomials are truncated $(n+1,2n+2)$ hypergeometric functions of Aomoto-Gelfand. The divariate Rahman polynomials are identified as the dual polynomials with a special parametrisation.Comment: LaTeX 28 pages, no figure, 2n parameter -> n parameter, one reference and one comment added, typo correcte

A Lie theoretic interpretation of multivariate hypergeometric polynomials

In 1971 Griffiths used a generating function to define polynomials in d variables orthogonal with respect to the multinomial distribution. The polynomials possess a duality between the discrete variables and the degree indices. In 2004 Mizukawa and Tanaka related these polynomials to character algebras and the Gelfand hypergeometric series. Using this approach they clarified the duality and obtained a new proof of the orthogonality. In the present paper, we interpret these polynomials within the context of the Lie algebra sl_{d+1}. Our approach yields yet another proof of the orthogonality. It also shows that the polynomials satisfy d independent recurrence relations each involving d^2+d+1 terms. This combined with the duality establishes their bispectrality. We illustrate our results with several explicit examples.Comment: minor change

Rates of convergence of some multivariate Markov chains with polynomial eigenfunctions

We provide a sharp nonasymptotic analysis of the rates of convergence for some standard multivariate Markov chains using spectral techniques. All chains under consideration have multivariate orthogonal polynomial as eigenfunctions. Our examples include the Moran model in population genetics and its variants in community ecology, the Dirichlet-multinomial Gibbs sampler, a class of generalized Bernoulli--Laplace processes, a generalized Ehrenfest urn model and the multivariate normal autoregressive process.Comment: Published in at http://dx.doi.org/10.1214/08-AAP562 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the hypergroup property

International audienceThe hypergroup property satisfied by certain reversible Markov chains can be seen as a generalization of the convolution related features of class random walks on groups.Carlen, Geronimo and Loss~\cite{MR2764893} developed a method for checking this property in the context of Jacobi eigen-polynomials.A probabilistic extension of their approach is proposed here, enabling to recover the discrete example of the biased Ehrenfest model due to Eagleson \cite{MR0328162}.Next a spectral characterization is provided for finite birth and death chains enjoying the hypergroup property with respect to one of the boundary points