The Annals of Applied Probability

Total variation bound for Kac’s random walk

Yunjiang Jiang

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We show that the classical Kac’s random walk on $(n-1)$-sphere $S^{n-1}$ starting from the point mass at $e_{1}$ mixes in $\mathcal{O}(n^{5}(\log n)^{3})$ steps in total variation distance. The main argument uses a truncation of the running density after a burn-in period, followed by $\mathcal{L}^{2}$ convergence using the spectral gap information derived by other authors. This improves upon a previous bound by Diaconis and Saloff-Coste of order $\mathcal{O}(n^{2n})$.

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Ann. Appl. Probab., Volume 22, Number 4 (2012), 1712-1727.

First available in Project Euclid: 10 August 2012

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Zentralblatt MATH identifier

Primary: 60-XX: PROBABILITY THEORY AND STOCHASTIC PROCESSES {For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX}

Markov chain mixing time orthogonal group Kac random walk interacting particle systems


Jiang, Yunjiang. Total variation bound for Kac’s random walk. Ann. Appl. Probab. 22 (2012), no. 4, 1712--1727. doi:10.1214/11-AAP810.

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