Probability Theory and Related Fields ( IF 2 ) Pub Date : 2021-07-14 , DOI: 10.1007/s00440-021-01061-5 Evita Nestoridi 1 , Sam Olesker-Taylor 2
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).
中文翻译:
可逆马尔可夫链的极限曲线
在最近的一项突破中,Teyssier (Ann Probab 48(5):2323–2343, 2020) 引入了一种新方法,用于估算一组随机游走与平衡的距离。他用它来研究随机换位洗牌的极限曲线。他的技术仅限于对群体进行共轭不变的随机游走;我们为齐次空间上的随机游走和一般可逆马尔可夫链推导出类似的逼近引理。我们说明了这些引理在一些著名问题上的应用:k-Hough 的循环洗牌、锐化结果(Probab Theory Relat Fields 165(1–2):447–482, 2016)和 Berestycki、Schramm 和 Zeitouni(Ann Probab 39(5):1815–1843, 2011),Ehrenfest urn许多骨灰盒的扩散,Ceccherini-Silberstein、Scarabotti 和 Tolli (J Math Sci 141(2):1182–1229, 2007) 的锐化结果,Gibbs 采样器,它是统计物理学中的基本工具,具有二项式先验和超几何后验, Diaconis、Khare 和 Saloff-Coste 的锐化结果 (Stat Sci 23(2):151–178, 2008)。