Abstract. We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions. Key words and phrases: Gibbs sampler, running time analyses, exponential families, conjugate priors, location families, orthogonal polynomials, singular value decomposition. 1.

We consider sequences of orthogonal polynomials and pursue the question of how (partial) knowledge of the orthogonalizing measure for the associated polynomials can lead to information about the orthogonalizing measure for the original polynomials. In particular, we relate the supports of the two measures, and their moments. As an application we analyse the relation between two decay rates connected with a birth-death process. Keywords and phrases: orthogonal polynomials, associated polynomials, numerator polynomials, birth-death process, decay rate, rate of convergence, firstentrance time 2000 Mathematics Subject Classification: Primary 42C05, Secondary 60J80 1

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Traditional linear signal processing is based on viewing signals as sequences or functions in time that flow in one direction, from past through present into future. Somewhat surprisingly, the assumption that the most basic operation that can be performed on a signal is a time shift, or “delay, ” is sufficient to derive many relevant signal processingconcepts, includingspectrum, Fourier transform, frequency response and others. This observation has led us to search for other linear, shift-invariant signal models that are based on a different definition of a basic signal shift, and hence have different notions of filtering, spectrum, and Fourier transform. Such models can serve as alternatives to the time signal model traditionally assumed in modern linear signal processing, and provide valuable insights into signal modeling in different areas of signal processing. The platform for our work is the algebraic signal processing theory, a recently developed axiomatic approach to, as well as a generalization of linear signal processing. In this thesis we present a new class of infinite and finite discrete signal models built on a new basic shift called the generic nearest-neighbor shift. We construct filter and signals spaces for these newmodels, andidentify thecorrespondingsignal processingconcepts, suchasfrequency, spectrum,

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Abstract not found

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. 1. Introduction. The