## The Annals of Applied Probability

### Total variation bound for Kac’s random walk

Yunjiang Jiang

#### 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

https://projecteuclid.org/euclid.aoap/1344614209

Digital Object Identifier
doi:10.1214/11-AAP810

Mathematical Reviews number (MathSciNet)
MR2985175

Zentralblatt MATH identifier
1248.60004

#### 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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