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Practical drift conditions for subgeometric rates of convergence. (English) Zbl 1082.60062
Let $(X,{\cal B})$ be a measurable space, $P$ a transition kernel on it, supposed $\psi$-irreducible (with a maximal $\psi$) and aperiodic. If $\xi$ is a measure on ${\cal B}$, $a$ a probability on ${\Bbb N}$, then a set $C\in {\cal B}$ is called $\xi_{a}$-petite if $\sum_{n\geq 0}a_{n}P^{n}(x,B)\geq\xi (B)$ for all $x\in C$, $B\in {\cal B}$. Condition $D(\varphi , V,C)$, where $V:X\rightarrow [1,\infty ]$, $\varphi :[1,\infty ]\rightarrow (0,\infty ]$ is concave, nondecreasing and differentiable, $C\in {\cal B}$, means the existence of a constant $b$ with $PV+\varphi\circ V\leq V+b1_{C}$. ${\cal Y}$ is defined as the set of pairs $(\Psi_{1},\Psi_{2})$, $\Psi_{i}$ defined on $[1,\infty )$, ultimately nondecreasing, one of them tending to $\infty$ at $\infty$, such that $\Psi_{1}(x)\Psi_{2}(y)\leq x+y$. For a measure $\mu$ on ${\cal B}$ and a $f:X\rightarrow [1,\infty )$, $\Vert\mu\Vert_{f}$ is defined as $\sup_{\vert g\vert \leq f}\,\vert \int gd\mu \vert $. The main result, proved after seven lemmas, states: if $D(\varphi ,V,C)$ holds, $\varphi '\rightarrow 0$ at $\infty$, $C$ is petite, $\sup_{C} V <\infty$ and if $(\Psi_{1},\Psi_{2})\in {\cal Y}$, then there exists an invariant probability $\pi$ and, for $x\in (V<\infty )$, $\lim_{n}\Psi_{1}(r_{\varphi }(n))\Vert P^{n}(x,\cdot )-\pi\Vert_{\Psi_{2}(\varphi\circ V)}=0$, where $r_{\varphi }$ is $\varphi\circ K_{\varphi }$, $K_{\varphi }$ being the inverse of $\int_{_{1}}^{^{\cdot }}dx/\varphi (x)$. Also that every probability $\lambda$ on ${\cal B}$ with $\int Vd\lambda <\infty$ is $(\Psi_{2}(\varphi\circ V),\Psi_{1}(r_{\varphi }))$-regular, i.e. $E_{\lambda }(\sum_{k=0}^{^{\tau }B}\Psi_{1}(r_{\varphi }(k))\Psi_{2}(\varphi\circ V(\Phi_{k}))<\infty$, for all $B\in {\cal B}$, $\psi (B) > 0$, where $\tau_{B}$ is the first $\geq 1$ visit in $B$ of the $P$-chain $\Phi_{k}$, and there exists a constant $c$ such that, if $\mu$ is another such $\lambda$, $$\sum_{n\geq 0}\Psi_{1}(r_{\varphi }(n))\int\int\lambda (dx)\mu (dy)\Vert P^{n}(x,\cdot )- P^{n}(y,\cdot )\Vert_{\Psi_{2}(\varphi\circ V)}\leq c\int Vd(\lambda +\mu ).$$ As applications (supposing $D(\varphi ,V,C)$): $\varphi (t)=ct^{\alpha }$, $\alpha\in [0,1)$, $c\in (0,1]$, $\Psi_{1}(t)=(qt)^{q}$, $\Psi_{2}(t)= (pt)^{p}$, $p\in (0,1)$, $q=1-p$ (polynomial rates of convergence), $\varphi (t)=c(1+\log t)^{\alpha }$, $\alpha\geq 0$, $c\in (0,1]$ and same $\Psi$’s (logarithmic rates), $\varphi$ concave, differentiable, $\varphi (t)=ct/\log^{\alpha }\,t$ for $t\geq t_{0}$, $\alpha >0$, $c>0$ (subexponential rates). The above when $X= {\Bbb N},P(n,i)=0$ for $i\neq 0,n+1$ ($D(\varphi ,V,C)$ is shown to hold). Hastings-Metropolis algorithm: $P(x,A) =\int_{_{A}}\alpha (x,x+y) q(y)d\mu (y)+c 1_{A}(x)$, $\alpha (x . y)=\min(1,\pi (y)/\pi (x))$, $\mu$ being the Lebesgue measure on $X={\Bbb R}^{d}$ and $\pi$ a probability density function (for it and for the following it is proved, as theorems, that $D(\varphi ,V,C)$ is valid, in each case under some conditions, for some $\varphi ,V,C$). Nonlinear autoregressive model: $\Phi_{n+1}=g(\Phi_{n})+\varepsilon_{n+1}$; $\varepsilon_{n} {\Bbb R}^{d}$-valued, independent, identically distributed, $E\varepsilon =0$, $E(e^{z\vert \varepsilon \vert ^{\gamma }}) <\infty$, $\gamma\in (0,1]$, $g$ continuous, $\vert g(t)\vert <\vert t\vert (1-r\vert t\vert ^{-\varrho })$, $\varrho\in [0,2)$, for $\vert t\vert \geq R_{0}$. Stochastic unit root: $\Phi_{n+1}= 1_{(U_{n+1}\leq g(\Phi_{n}))}\Phi_{n} +\varepsilon_{n+1}$, $U_{n}$ independent, all uniformly distributed on $[0,1]$.

MSC:
60J05Discrete-time Markov processes on general state spaces
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References:
[1] AngoNze, P. (1994). Critères d’ergodicité de modèles markoviens. Estimation non-paramétrique sous des hypothèses de dépendance. Ph.D. dissertation, Univ. Paris 9, Dauphine.
[2] AngoNze, P. (2000). Geometric and subgeometric rates for Markovian processes: A robust approach. Technical report, Univ. de Lille III.
[3] Duflo, M. (1997). Random Iterative Systems . Springer, Berlin. · Zbl 0868.62069
[4] Fort, G. and Moulines, E. (2000). V-subgeometric ergodicity for a Hastings--Metropolis algorithm. Statist. Probab. Lett. 49 401--410. · Zbl 0981.60032 · doi:10.1016/S0167-7152(00)00074-2
[5] Fort, G. and Moulines, E. (2003). Polynomial ergodicity of Markov transition kernels. Stochastic Process. Appl. 103 57--99. · Zbl 1075.60547 · doi:10.1016/S0304-4149(02)00182-5
[6] Gourieroux, C. and Robert, C. (2001). Tails and extremal behaviour of stochastic unit root models. Technical report, Centre de Recherche en Economie et Statistique du Travail.
[7] Granger, C. and Sawnson, N. (1997). An introduction to stochastic unit-root processes. J. Econometrics 80 35--62. · Zbl 0885.62100 · doi:10.1016/S0304-4076(96)00016-4
[8] Grunwald, G., Hyndman, R., Tedesco, L. and Tweedie, R. (2000). Non-Gaussian conditional linear AR(1) models. Aust. N. Z. J. Stat. 42 479--495. · Zbl 1018.62065 · doi:10.1111/1467-842X.00143
[9] Jarner, S. and Hansen, E. (2000). Geometric ergodicity of Metropolis algorithms. Stochastic Process. Appl. 85 341--361. · Zbl 0997.60070 · doi:10.1016/S0304-4149(99)00082-4
[10] Jarner, S. and Roberts, G. (2002). Polynomial convergence rates of Markov chains. Ann. Appl. Probab. 12 224--247. · Zbl 1012.60062 · doi:10.1214/aoap/1015961162
[11] Klokov, S. and Veretennikov, A. (2002). Sub-exponential mixing rate for a class of Markov processes. Technical Report 1, School of Mathematics, Univ. Leeds. · Zbl 1054.60073
[12] Malyshkin, M. (2001). Subexponential estimates of the rate of convergence to the invariant measure for stochastic differential equations. Theory Probab. Appl. 45 466--479. · Zbl 0994.60062 · doi:10.1137/S0040585X97978403
[13] Mengersen, K. and Tweedie, R. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 101--121. · Zbl 0854.60065 · doi:10.1214/aos/1033066201
[14] Meyn, S. and Tweedie, R. (1993). Markov Chains and Stochastic Stability. Springer, London. · Zbl 0925.60001
[15] Nummelin, E. and Tuominen, P. (1983). The rate of convergence in Orey’s theorem for Harris recurrent Markov chains with applications to renewal theory. Stochastic Process. Appl. 15 295--311. · Zbl 0532.60060 · doi:10.1016/0304-4149(83)90037-6
[16] Roberts, G. and Tweedie, R. (1996). Geometric convergence and central limit theorem for multidimensional Hastings and Metropolis algorithms. Biometrika 83 95--110. · Zbl 0888.60064 · doi:10.1093/biomet/83.1.95 · http://www3.oup.co.uk/biomet/hdb/Volume_83/Issue_01/
[17] Tanikawa, A. (2001). Markov chains satisfying simple drift conditions for subgeometric ergodicity. Stoch. Model. 17 109--120. · Zbl 0983.60063 · doi:10.1081/STM-100002059
[18] Tuominen, P. and Tweedie, R. (1994). Subgeometric rates of convergence of $f$-ergodic Markov chains. Adv. in Appl. Probab. 26 775--798. · Zbl 0803.60061 · doi:10.2307/1427820
[19] Veretennikov, A. (1997). On polynomial mixing bounds for stochastic differential equations. Stochastic Process. Appl. 70 115--127. · Zbl 0911.60042 · doi:10.1016/S0304-4149(97)00056-2
[20] Veretennikov, A. (2000). On polynomial mixing and convergence rate for stochastic differential and difference equations. Theory Probab. Appl. 44 361--374. · Zbl 0969.60070 · doi:10.1137/S0040585X97977550