The Annals of Applied Probability

Total variation bound for Kac’s random walk

Yunjiang Jiang

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Abstract

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 $\mathcal{O}(n^{2n})$.

Article information

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

Dates
First available in Project Euclid: 10 August 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1344614209

Digital Object Identifier
doi:10.1214/11-AAP810

Mathematical Reviews number (MathSciNet)
MR2985175

Zentralblatt MATH identifier
1248.60004

Subjects
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}

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

Citation

Jiang, Yunjiang. Total variation bound for Kac’s random walk. Ann. Appl. Probab. 22 (2012), no. 4, 1712--1727. doi:10.1214/11-AAP810. https://projecteuclid.org/euclid.aoap/1344614209


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References

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