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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 $[0,1]$.

60J05Discrete-time Markov processes on general state spaces
Full Text: DOI arXiv
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