Abstract
In a recent breakthrough, Teyssier (Ann Probab 48(5):2323–2343, 2020) introduced a new method for approximating the distance from equilibrium of a random walk on a group. He used it to study the limit profile for the random transpositions card shuffle. His techniques were restricted to conjugacy-invariant random walks on groups; we derive similar approximation lemmas for random walks on homogeneous spaces and for general reversible Markov chains. We illustrate applications of these lemmas to some famous problems: the k-cycle shuffle, sharpening results of Hough (Probab Theory Relat Fields 165(1–2):447–482, 2016) and Berestycki, Schramm and Zeitouni (Ann Probab 39(5):1815–1843, 2011), the Ehrenfest urn diffusion with many urns, sharpening results of Ceccherini-Silberstein, Scarabotti and Tolli (J Math Sci 141(2):1182–1229, 2007), a Gibbs sampler, which is a fundamental tool in statistical physics, with Binomial prior and hypergeometric posterior, sharpening results of Diaconis, Khare and Saloff-Coste (Stat Sci 23(2):151–178, 2008).
Highlights
We prove the following lemma for random walks on homogeneous spaces corresponding to a Gelfand pair started from some element x ∈ K stabilised by K, ie kx = xfor all k ∈ K
We apply the approximation lemma for random walks on homogeneous spaces, ie Lemma C, using the character theory developed by Ceccherini-Silberstein, Scarabotti and Tolli [5]
We prove the our TVapproximation lemma (Lemma A) via an application of the spectral decomposition for reversible Markov chains
Summary
In this paper we present three lemmas for obtaining the TV profile for random walks; see Lemmas A–C. They work by finding a decomposition of the TV distance as a sum using either a spectral decomposition or Fourier analysis. One separates out the ‘important’ terms in the sum to give a ‘main term’ (which asymptotically captures all the TV mass) and an ‘error’ term. Lemmas A and C are original contributions; Lemma B is due to Teyssier [20]. We give an example application, establishing a limit profile of the TV convergence to equilibrium. We denote the cdf of the standard normal distribution by throughout the paper
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