Reproducing kernel orthogonal polynomials on the multinomial distribution

Diaconis, Persi; Griffiths, Robert

doi:10.1016/j.jat.2019.01.007

Cited by 6 publications

(6 citation statements)

References 38 publications

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“…See also the study of similar models arried out by Diaconis et al (2008) using spectral theory. Theorems 1.12 and 1.13 sharpen the results in (Khare and Zhou, 2009) and (Diaconis and Griffiths, 2019), since they provide the explicit limit profiles for the chi-square and the total variation distances, for p ≥ 0 and p = 0, respectively.…”

mentioning

confidence: 82%

“…(Khare and Zhou, 2009, Prop. 2.8) and (Diaconis and Griffiths, 2019). Also, for more details on the kernel polynomials for the Dirichlet multinomial distribution see e.g.…”

Section: Neutral Multi-allelic Moran Type Process With Parent Indepen...mentioning

confidence: 99%

“…Khare and Zhou (2009) proved bounds for the chi-square distance in a discrete-time multi-allelic Moran process that implies the existence of a cutoff. Diaconis and Griffiths (2019) studied the existence of a chi-square and total variation cutoffs for a discrete-time analogous of the mutation process generated by L N , i.e. when p = 0.…”

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confidence: 99%

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On the spectrum and ergodicity of a neutral multi-allelic Moran model

Corujo¹

2023

ALEA

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The purpose of this paper is to provide a complete description of the eigenvalues of the generator of a neutral multi-type Moran model, and the applications to the study of the speed of convergence to stationarity. The Moran model we consider is a non-reversible in general, continuoustime Markov chain with an unknown stationary distribution. Specifically, we consider N individuals such that each one of them is of one type among K possible allelic types. The individuals interact in two ways: by an independent irreducible mutation process and by a reproduction process, where a pair of individuals is randomly chosen, one of them dies and the other reproduces. Our main result provides explicit expressions for the eigenvalues of the infinitesimal generator matrix of the Moran process, in terms of the eigenvalues of the jump rate matrix. As consequences of this result, we study the convergence in total variation of the process to stationarity and show a lower bound for the mixing time of the Moran process. Furthermore, we study in detail the spectral decomposition of the neutral multi-allelic Moran model with parent independent mutation scheme, which is the unique mutation scheme that makes the neutral Moran process reversible. Under the parent independent mutation, we also prove the existence of a cutoff phenomenon in the chi-square and the total variation distances when initially all the individuals are of the same type and the number of individuals tends to infinity. Additionally, in the absence of reproduction, we prove that the total variation distance to stationarity of the parent independent mutation process when initially all the individuals are of the same type has a Gaussian profile.

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confidence: 82%

“…(Khare and Zhou, 2009, Prop. 2.8) and (Diaconis and Griffiths, 2019). Also, for more details on the kernel polynomials for the Dirichlet multinomial distribution see e.g.…”

Section: Neutral Multi-allelic Moran Type Process With Parent Indepen...mentioning

confidence: 99%

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On the spectrum and ergodicity of a neutral multi-allelic Moran model

Corujo¹

2023

ALEA

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“…This shows that (9.17) is a cutoff time because if C is large and positive both the upper and lower bounds are small, and if C is large and negative both bounds are large. Calculations here are related to chi-squared cutoff calculations for a multinomial model in Diaconis and Griffiths (2019), Section 4.1.…”

Section: Spectral Representation Via Tensor Productsmentioning

confidence: 99%

Three steps mixing for general random walks on the hypercube at criticality

Collevecchio,

Griffiths

2020

Preprint

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We introduce a general class of random walks on the N -hypercube, study cut-off for the mixing time, and provide several types of representation for the transition probabilities. We observe that for a sub-class of these processes with long range (i.e. non-local) there exists a critical value of the range that allows an "almost-perfect" mixing in at most three steps. In other words, the total variation distance between the three steps transition and the stationary distribution decreases geometrically in N , which is the dimension of the hypercube. In some cases, the walk mixes almost-perfectly in exactly two steps. Notice that a well-known result (Theorem 1 in Diaconis and Shahshahani ( 1986)) shows that there exist no random walk on Abelian groups (such as the hypercube) which mixes perfectly in exactly two steps.

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“…Khare and Zhou [KZ09] proved bounds for the chi-square distance in a discrete-time multi-allelic Moran process that implies the existence of a cutoff. Diaconis and Griffiths [DG19] studied the existence of a chi-square and total variation cutoffs for a discrete-time analogous of the mutation process generated by L N . Our methods for proving Theorem 1.7 could be used to improve the results related to the existence of cutoffs in [KZ09] and [DG19], in order to obtain the explicit profiles for the limits of the chi-square distance, for p ≥ 0, and the total variation distance, for p = 0, when N → ∞.…”

mentioning

confidence: 99%

On the spectrum and ergodicity of a neutral multi-allelic Moran model

Corujo¹

2020

Preprint

View full text Add to dashboard Cite

The purpose of this paper is to provide a complete description of the eigenvalues of the generator of a neutral multi-type Moran model, which is non-reversible in general, continuous-time Markov chain with unknown stationary distribution. Specifically, we consider N individuals, where each one of them is of one type among K possible allelic types. The individuals interact in two ways: by an independent irreducible mutation process and by a reproduction process, where a pair of individuals is randomly chosen, one of them dies and the other reproduces. Our main result provides explicit expressions for the eigenvalues of the infinitesimal generator matrix of the Moran process, in terms of the eigenvalues of the jump rate matrix. We also study in detail the spectral decomposition of the neutral multi-allelic Moran model with parent independent mutation scheme, which turns to be the unique mutation scheme that makes the neutral Moran process reversible. Under the parent independent mutation assumption we provide a complete description of the left and right eigenfunctions of the infinitesimal generator of the Markov process. We use spectral techniques to provide non-asymptotic bounds for the rate of convergence of the neutral multi-allelic Moran process with parent independent mutation to stationarity in the total variation and chi-square distances. Moreover, we prove the existence of a strongly optimal cutoff phenomenon in the chi-square distance when initially all the individuals are of the same type and the number of individuals tends to infinity. Additionally, in the absence of reproduction, we prove that the the mutation process presents a strongly optimal total variation cutoff with Gaussian profile when initially all the individuals are of the same type. Contents 1. Introduction and main results Applications to the neutral multi-allelic Moran model with parent independent mutation Cutoff phenomenon Links with other models Structure of the article 2. State spaces for distinguishable and indistinguishable particle processes 3. Spectrum of the neutral multi-allelic Moran process 3.1. Proof of Theorem 1.1 3.2. Proof of Theorem 1.2 3.3. Proof of Theorem 1.3 3.4. Applications to the exponential ergodicity of neutral multi-allelic Moran processes 4. Spectral decomposition of the neutral multi-allelic Moran type process with parent independent mutation 4.1. Background on reversible finite continuous-time Markov chains 4.2. Proof of Theorems 1.6 and 1.7 Multivariate orthogonal Hahn and Krawtchouk polynomials Kernel polynomials for Dirichlet multinomial and multinomial distributions 5. Discussion and open problems Appendix A.

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Reproducing kernel orthogonal polynomials on the multinomial distribution

Cited by 6 publications

References 38 publications

On the spectrum and ergodicity of a neutral multi-allelic Moran model

On the spectrum and ergodicity of a neutral multi-allelic Moran model

Three steps mixing for general random walks on the hypercube at criticality

On the spectrum and ergodicity of a neutral multi-allelic Moran model

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