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Practical drift conditions for subgeometric rates of convergence. (English) Zbl 1082.60062

Let $\left(X,ℬ\right)$ 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 $ℬ$, $a$ a probability on $ℕ$, then a set $C\in ℬ$ is called ${\xi }_{a}$-petite if ${\sum }_{n\ge 0}{a}_{n}{P}^{n}\left(x,B\right)\ge \xi \left(B\right)$ for all $x\in C$, $B\in ℬ$. Condition $D\left(\varphi ,V,C\right)$, where $V:X\to \left[1,\infty \right]$, $\varphi :\left[1,\infty \right]\to \left(0,\infty \right]$ is concave, nondecreasing and differentiable, $C\in ℬ$, means the existence of a constant $b$ with $PV+\varphi \circ V\le V+b{1}_{C}$. $𝒴$ is defined as the set of pairs $\left({{\Psi }}_{1},{{\Psi }}_{2}\right)$, ${{\Psi }}_{i}$ defined on $\left[1,\infty \right)$, ultimately nondecreasing, one of them tending to $\infty$ at $\infty$, such that ${{\Psi }}_{1}\left(x\right){{\Psi }}_{2}\left(y\right)\le x+y$. For a measure $\mu$ on $ℬ$ and a $f:X\to \left[1,\infty \right)$, ${\parallel \mu \parallel }_{f}$ is defined as ${sup}_{|g|\le f}\phantom{\rule{0.166667em}{0ex}}|\int gd\mu |$. The main result, proved after seven lemmas, states: if $D\left(\varphi ,V,C\right)$ holds, ${\varphi }^{\text{'}}\to 0$ at $\infty$, $C$ is petite, ${sup}_{C}V<\infty$ and if $\left({{\Psi }}_{1},{{\Psi }}_{2}\right)\in 𝒴$, then there exists an invariant probability $\pi$ and, for $x\in \left(V<\infty \right)$, ${lim}_{n}{{\Psi }}_{1}\left({r}_{\varphi }\left(n\right)\right){\parallel {P}^{n}\left(x,·\right)-\pi \parallel }_{{{\Psi }}_{2}\left(\varphi \circ V\right)}=0$, where ${r}_{\varphi }$ is $\varphi \circ {K}_{\varphi }$, ${K}_{\varphi }$ being the inverse of ${\int }_{{}_{1}}^{{}^{·}}dx/\varphi \left(x\right)$. Also that every probability $\lambda$ on $ℬ$ with $\int Vd\lambda <\infty$ is $\left({{\Psi }}_{2}\left(\varphi \circ V\right),{{\Psi }}_{1}\left({r}_{\varphi }\right)\right)$-regular, i.e. ${E}_{\lambda }\left({\sum }_{k=0}^{{}^{\tau }B}{{\Psi }}_{1}\left({r}_{\varphi }\left(k\right)\right){{\Psi }}_{2}\left(\varphi \circ V\left({{\Phi }}_{k}\right)\right)<\infty$, for all $B\in ℬ$, $\psi \left(B\right)>0$, where ${\tau }_{B}$ is the first $\ge 1$ visit in $B$ of the $P$-chain ${{\Phi }}_{k}$, and there exists a constant $c$ such that, if $\mu$ is another such $\lambda$,

$\sum _{n\ge 0}{{\Psi }}_{1}\left({r}_{\varphi }\left(n\right)\right)\int \int \lambda \left(dx\right)\mu \left(dy\right){\parallel {P}^{n}\left(x,·\right)-{P}^{n}\left(y,·\right)\parallel }_{{{\Psi }}_{2}\left(\varphi \circ V\right)}\le c\int Vd\left(\lambda +\mu \right)·$

As applications (supposing $D\left(\varphi ,V,C\right)$): $\varphi \left(t\right)=c{t}^{\alpha }$, $\alpha \in \left[0,1\right)$, $c\in \left(0,1\right]$, ${{\Psi }}_{1}\left(t\right)={\left(qt\right)}^{q}$, ${{\Psi }}_{2}\left(t\right)={\left(pt\right)}^{p}$, $p\in \left(0,1\right)$, $q=1-p$ (polynomial rates of convergence), $\varphi \left(t\right)=c{\left(1+logt\right)}^{\alpha }$, $\alpha \ge 0$, $c\in \left(0,1\right]$ and same ${\Psi }$’s (logarithmic rates), $\varphi$ concave, differentiable, $\varphi \left(t\right)=ct/{log}^{\alpha }\phantom{\rule{0.166667em}{0ex}}t$ for $t\ge {t}_{0}$, $\alpha >0$, $c>0$ (subexponential rates). The above when $X=ℕ,P\left(n,i\right)=0$ for $i\ne 0,n+1$ ($D\left(\varphi ,V,C\right)$ is shown to hold). Hastings-Metropolis algorithm: $P\left(x,A\right)={\int }_{{}_{A}}\alpha \left(x,x+y\right)q\left(y\right)d\mu \left(y\right)+c{1}_{A}\left(x\right)$, $\alpha \left(x·y\right)=min\left(1,\pi \left(y\right)/\pi \left(x\right)\right)$, $\mu$ being the Lebesgue measure on $X={ℝ}^{d}$ and $\pi$ a probability density function (for it and for the following it is proved, as theorems, that $D\left(\varphi ,V,C\right)$ is valid, in each case under some conditions, for some $\varphi ,V,C$). Nonlinear autoregressive model: ${{\Phi }}_{n+1}=g\left({{\Phi }}_{n}\right)+{\epsilon }_{n+1}$; ${\epsilon }_{n}{ℝ}^{d}$-valued, independent, identically distributed, $E\epsilon =0$, $E\left({e}^{{z|\epsilon |}^{\gamma }}\right)<\infty$, $\gamma \in \left(0,1\right]$, $g$ continuous, $|g\left(t\right)|<|t|\left(1-r|t{|}^{-\varrho }\right)$, $\varrho \in \left[0,2\right)$, for $|t|\ge {R}_{0}$. Stochastic unit root: ${{\Phi }}_{n+1}={1}_{\left({U}_{n+1}\le g\left({{\Phi }}_{n}\right)\right)}{{\Phi }}_{n}+{\epsilon }_{n+1}$, ${U}_{n}$ independent, all uniformly distributed on $\left[0,1\right]$.

##### MSC:
 60J05 Discrete-time Markov processes on general state spaces