The Annals of Applied Probability

Time inhomogeneous Markov chains with wave-like behavior

L. Saloff-Coste and J. Zúñiga

Full-text: Open access

Abstract

Starting from a given Markov kernel on a finite set V and a bijection g of V, we construct and study a time inhomogeneous Markov chain whose kernel at time n is obtained from K by transport of gn−1. We show that this construction leads to interesting examples, and we obtain quantitative results for some of these examples.

Article information

Source
Ann. Appl. Probab. Volume 20, Number 5 (2010), 1831-1853.

Dates
First available in Project Euclid: 25 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1282747402

Digital Object Identifier
doi:10.1214/09-AAP661

Mathematical Reviews number (MathSciNet)
MR2724422

Zentralblatt MATH identifier
1206.60067

Subjects
Primary: 60J05: Discrete-time Markov processes on general state spaces 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Time inhomogeneous Markov chains wave like behavior singular values

Citation

Saloff-Coste, L.; Zúñiga, J. Time inhomogeneous Markov chains with wave-like behavior. Ann. Appl. Probab. 20 (2010), no. 5, 1831--1853. doi:10.1214/09-AAP661. https://projecteuclid.org/euclid.aoap/1282747402


Export citation

References

  • [1] Del Moral, P., Ledoux, M. and Miclo, L. (2003). On contraction properties of Markov kernels. Probab. Theory Related Fields 126 395–420.
  • [2] Diaconis, P. (1991). Finite Fourier methods: Access to tools. In Probabilistic Combinatorics and Its Applications (San Francisco, CA, 1991). Proc. Sympos. Appl. Math. 44 171–194. Amer. Math. Soc., Providence, RI.
  • [3] Diaconis, P. and Ram, A. (2000). Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques. Michigan Math. J. 48 157–190.
  • [4] Diaconis, P. and Saloff-Coste, L. (1993). Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 696–730.
  • [5] Diaconis, P. and Saloff-Coste, L. (1996). Nash inequalities for finite Markov chains. J. Theoret. Probab. 9 459–510.
  • [6] Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695–750.
  • [7] Flatto, L., Odlyzko, A. M. and Wales, D. B. (1985). Random shuffles and group representations. Ann. Probab. 13 154–178.
  • [8] Ganapathy, M. (2007). Robust mixing time. Electron. J. Probab. 12 262–299.
  • [9] Goel, S. (2004). Modified logarithmic Sobolev inequalities for some models of random walk. Stochastic Process. Appl. 114 51–79.
  • [10] Iosifescu, M. (1980). Finite Markov Processes and Their Applications. Wiley, Chichester.
  • [11] Lubotzky, A. (1994). Discrete Groups, Expanding Graphs and Invariant Measures. Progress in Mathematics 125. Birkhäuser, Basel.
  • [12] Lubotzky, A. (1995). Cayley graphs: Eigenvalues, expanders and random walks. In Surveys in Combinatorics. London Mathematical Society Lecture Note Series 218 155–189. Cambridge Univ. Press, Cambridge.
  • [13] Mossel, E., Peres, Y. and Sinclair, A. (2004). Shuffling by semi-random transpositions. In 45th Symposium on Foundations of Comp. Sci. Available at arXiv:math.PR/0404438.
  • [14] Păun, U. (2001). Ergodic theorems for finite Markov chains. Math. Rep. (Bucur.) 3 383–390.
  • [15] Saloff-Coste, L. and Zúñiga, J. (2007). Convergence of some time inhomogeneous Markov chains via spectral techniques. Stochastic Process. Appl. 117 961–979.
  • [16] Saloff-Coste, L. and Zúñiga, J. (2008). Refined estimates for some basic random walks on the symmetric and alternating groups. ALEA Lat. Am. J. Probab. Math. Stat. 4 359–392.
  • [17] Saloff-Coste, L. and Zúñiga, J. (2009). Merging of time inhomogeneous Markov chains, part I: Singular values and stability. Electron. J. Probab. 14 1456–1494.
  • [18] Saloff-Coste, L. and Zúñiga, J. (2010). Merging of time inhomogeneous Markov chains, part II: Nash and log-Sobolev inequalities. To appear.
  • [19] Seneta, E. (1973). On strong ergodicity of inhomogeneous products of finite stochastic matrices. Studia Math. 46 241–247.