Spectral analysis of Markov kernels and application to the convergence rate of discrete random walks. (English) Zbl 1305.60060
Summary: Let \(\{X_n\}_{n\in \mathbb{N}}\) be a Markov chain on a measurable space \(\mathbb{X}\) with transition kernel \(P\) and let \(V:\mathbb{X} \to [1, +\infty)\). The Markov kernel \(P\) is here considered as a linear bounded operator on the weighted-supremum space \(\mathcal{B}_V\) associated with \(V\). Then the combination of quasicompactness arguments with precise analysis of the eigenelements of \(P\) allows us to estimate the geometric rate of convergence \(\rho_V(P)\) of \(\{X_n\}_{n\in \mathbb{N}}\) to its invariant probability measure in the operator norm on \(\mathcal{B}_V\). A general procedure to compute \(\rho_V(P)\) for discrete Markov random walks with identically distributed bounded increments is specified.
MSC:
| 60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |
| 47B07 | Linear operators defined by compactness properties |
References:
| [1] | Baxendale, P. H. (2005). Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Prob. 15 , 700-738. · Zbl 1070.60061 · doi:10.1214/105051604000000710 |
| [2] | Guibourg, D., Hervé, L. and Ledoux, J. (2011). Quasi-compactness of Markov kernels on weighted-supremum spaces and geometrical ergodicity. Preprint. Available at http://uk.arxiv.org/abs/1110.3240. |
| [3] | Hennion, H. (1993). Sur un théorème spectral et son application aux noyaux lipchitziens. Proc. Amer. Math. Soc. 118 , 627-634. · Zbl 0772.60049 · doi:10.2307/2160348 |
| [4] | Hervé, L. and Ledoux, J. (2014). Approximating Markov chains and \({V}\)-geometric ergodicity via weak perturbation theory. Stoch. Process. Appl. 124 , 613-638. · Zbl 1305.60059 · doi:10.1016/j.spa.2013.09.003 |
| [5] | Hordijk, A. and Spieksma, F. (1992). On ergodicity and recurrence properties of a Markov chain with an application to an open Jackson network. Adv. Appl. Prob. 24 , 343-376. · Zbl 0766.60085 · doi:10.2307/1427696 |
| [6] | Klimenok, V. and Dudin, A. (2006). Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory. Queueing Systems. 54 , 245-259. · Zbl 1107.60060 · doi:10.1007/s11134-006-0300-z |
| [7] | Kontoyiannis, I. and Meyn, S. P. (2003). Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Prob. 13 , 304-362. · Zbl 1016.60066 · doi:10.1214/aoap/1042765670 |
| [8] | Kontoyiannis, I. and Meyn, S. P. (2012). Geometric ergodicity and the spectral gap of non-reversible Markov chains. Prob. Theory Relat. Fields 154 , 327-339. · Zbl 1263.60064 · doi:10.1007/s00440-011-0373-4 |
| [9] | Kovchegov, Y. (2009). Orthogonality and probability: beyond nearest neighbor transitions. Electron. Commun. Prob. 14 , 90-103. · Zbl 1189.60075 · doi:10.1214/ECP.v14-1447 |
| [10] | Lund, R. B. and Tweedie, R. L. (1996). Geometric convergence rates for stochastically ordered Markov chains. Math. Operat. Res. 21 , 182-194. · Zbl 0847.60053 · doi:10.1287/moor.21.1.182 |
| [11] | Malyshev, V. A. and Spieksma, F. M. (1995). Intrinsic convergence rate of countable Markov chains. Markov Process. Relat. Fields 1 , 203-266. · Zbl 0901.60037 |
| [12] | Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability . Springer, London. · Zbl 0925.60001 |
| [13] | Meyn, S. P. and Tweedie, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Prob. 4 , 981-1011. · Zbl 0812.60059 · doi:10.1214/aoap/1177004900 |
| [14] | Roberts, G. O. and Tweedie, R. L. (1999). Bounds on regeneration times and convergence rates for Markov chains. Stoch. Process. Appl. 80 , 211-229. (Corrigendum: 91 (2001), 337-338.) · Zbl 0961.60066 · doi:10.1016/S0304-4149(98)00085-4 |
| [15] | Rosenthal, J. S. (1996). Markov chain convergence: from finite to infinite. Stoch. Process. Appl. 62 , 55-72. · Zbl 0849.60071 · doi:10.1016/0304-4149(95)00082-8 |
| [16] | Van Doorn, E. A. and Schrijner, P. (1995). Geometric ergodicity and quasi-stationarity in discrete-time birth-death processes. J. Austral. Math. Soc. B 37 , 121-144. · Zbl 0856.60080 · doi:10.1017/S0334270000007621 |
| [17] | Wu, L. (2004). Essential spectral radius for Markov semigroups. I. Discrete time case. Prob. Theory Relat. Fields 128 , 255-321. · Zbl 1056.60068 · doi:10.1007/s00440-003-0304-0 |
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