TOTAL VARIATION BOUND FOR KAC'S RANDOM WALK

TOTAL VARIATION BOUND FOR KAC'S RANDOM WALK

We show that the classical Kac's random walk on (n − 1)-sphere S n−1 starting from the point mass at e₁ mixes in O(n⁵(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 L² convergence using the spectral gap...

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Veröffentlicht in:The Annals of applied probability 2012-08, Vol.22 (4), p.1712-1727
1. Verfasser: Jiang, Yunjiang
Format: Artikel
Sprache:eng
Schlagworte:
60-XX
Convergence
Coordinate systems
Density
Entropy
interacting particle systems
Kac random walk
Markov chain mixing time
Markov chains
Mathematics
orthogonal group
Particle collisions
Polynomials
Probability distribution
Random walk
Random walk theory
Sine function
Transportation
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Zusammenfassung:We show that the classical Kac's random walk on (n − 1)-sphere S n−1 starting from the point mass at e₁ mixes in O(n⁵(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 L² convergence using the spectral gap information derived by other authors. This improves upon a previous bound by Diaconis and Saloff-Coste of order O(n² n ).
ISSN:1050-5164
2168-8737
DOI:10.1214/11-AAP810