Total variation bound for Kac's random walk

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

Published in at http://dx.doi.org/10.1214/11-AAP810 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• On the convergence to equilibrium of Kac's random walk on matrices
• On the spectral gap of the Kac walk and other binary collision processes

Cited by in corpus (2)

• A Gibbs sampler on the $n$-simplex
• Cut-off phenomenon in the uniform plane Kac walk