Consider a non-linear autoregressive Markov chain $\{\Phi_n: n\in\mathbb{Z}_+\}$ on $\mathbb{R}$ defined by

$ {{\Phi }_{n+1}}=F({{\Phi }_{n}})+{{U}_{n+1}}, \ \ \ n\in {{\mathbb{Z}}_{+}}, $

where $F$: $\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function, $\{U_n: n\in\mathbb{Z}_+\}$ is a sequence of i.i.d. random variables with distribution

$ \Gamma (-\infty, x]=\mathbb{P}\{{{U}_{n}}\le x\}, \ \ x\in \mathbb{R}, $

and $\Phi_0$ is independent of $\{U_n: n\in\mathbb{Z}_+\}$. Assume that the distribution $\Gamma$ is absolutely continuous with respect to the Lebesgue measure $\lambda $, and has a density which is positive everywhere. The non-linear autoregressive model attracted a large amount of attention in the literature. Most of the studies focued on conditions implying ergodicity, sub-geometric ergodicity and geometric ergodicity, see e.g. [1-4] and references therein. In the paper, we aim to present a sufficient condition for geometric transience for the non-linear autoregressive model.

First, let us recall some notations and definitions, see [3, 5, 6] for details. Denote by $\mathscr{B}(\mathbb{R})$ the Borel $\sigma$-field on $\mathbb{R}$, and write ${{\mathscr{B}}^{+}}(\mathbb{R})=\{A\in \mathscr{B}(\mathbb{R}):\lambda (A)>0\}$. The $n$-step transition kernel of the chain $\Phi_n$ is defined as

$ {{P}^{n}}(x, A)={{\mathbb{P}}_{x}}\{{{\Phi }_{n}}\in A\}, \ \ \ n\in {{\mathbb{Z}}_{+}}, \ x\in \mathbb{R}, \ A\in \mathscr{B}(\mathbb{R}), $

where $\mathbb{P}_x$ is the conditional distribution of the chain given $\Phi_0=x$. The corresponding expectation operator will be denoted by $\mathbb{E}_x$. The operator $P$ acts on non-negative measurable functions $f$ via

$ P f(x)=\int_{\mathbb{R}} f(y)P(x, d y), \ \ \ x\in \mathbb{R}. $

The chain $\Phi_n$ is Lebesgue-irreducible, if for every $A\in\mathscr{B}^+(\mathbb{R})$,

$ \sum\limits_{n=0}^{\infty }{{{P}^{n}}}(x, A)>0, \ \ \ x\in \mathbb{R}. $

A set $A\in\mathscr{B}(\mathbb{R})$ is called petite, if there exist a probability distribution $a=\{a_n: n\in\mathbb{Z}_+\}$ and a non-trivial measure $\nu_a$ satisfying for all $x\in A$ and $B\in \mathscr{B}(\mathbb{R})$,

$ \sum\limits_{n=0}^{\infty }{{{a}_{n}}}{{P}^{n}}(x, B)\ge {{\nu }_{a}}(B). $

Obviously, the subset of a petite set is still petite. By [7, Lemma 2.1] or [8, Theorem 1], we know that the non-linear autoregressive model is Lebesgue-irreducible, and every compact set in $\mathscr{B}^+(\mathbb{R})$ is petite.

For $A\in\mathscr{B}(\mathbb{R})$, let

$ \tau_{A}=\inf\{n\geq1: \Phi_{n}\in A\}\ \ \ \mbox{and}\ \ \ \sigma_{A}=\inf\{n\geq0: \Phi_{n}\in A\} $

be the first return and first hitting times, respectively, on $A$. It is obvious that $\tau_{A}=\sigma_{A}$ if $\Phi_0\in A^c$. Denote by $L(x, A)={{\mathbb{P}}_{x}}\{{{\tau }_{A}}<\infty \}$ the probability of the chain $\Phi_n$ ever returning to $A$.

Recall that a set $A\in \mathscr{B}^+(\mathbb{R})$ is called a uniformly geometrically transient set of the chain $\Phi_n$, if there exists a constant $\kappa>1$ such that

$ \mathop {\sup }\limits_{x \in A} {\mkern 1mu} \sum\limits_{n = 1}^\infty {{\kappa ^n}} {P^n}(x,A) < \infty . $

The chain $\Phi_n$ is called geometrically transient, if it is $\psi$-irreducible for some non-trivial measure $\psi$, and $\mathbb{R}$ can be covered $\psi$-a.e. by a countable number of uniformly geometrically transient sets. That is, there exist sets $D$ and $A_i$, $i=1, 2, \cdots$ such that $\mathbb{R}=D\cup \left( \bigcup\limits_{i=1}^{\infty }{{{A}_{i}}} \right)$, where $\psi(D)=0$ and each $A_i$ is a uniformly geometrically transient set of the chain $\Phi_n$.

To state the main result of this paper, we need the following assumptions:

(A1) $\int{{{\rm{e}}^{s|x|}}}\Gamma (dx)<\infty $ for some constant $s>0$;

(A2) $\underset{|x|\to \infty }{\mathop{\lim \inf }}\, \frac{|F(x)|}{|x|}>1$.

Theorem 1.1  Assume (A1) and (A2). Then the non-linear autoregressive model $\Phi_n$ is geometrically transient.

Remark 1.2  It is easy to see that (A2) is equivalent to the condition in [9, Theorem 3.1], where transience for the the non-linear autoregressive model $\Phi_n$ was confirmed. Here, we get a stronger result (i.e. geometric transience) in Theorem 1.1.

This section is devoted to proving Theorem 1.1 by using the Foster-Lyapunov (or drift) condition for geometric transience.

It is well known that Foster-Lyapunov conditions have been widely used to study the stochastic stability for Markov chains. For examples, Down, Meyn and Tweedie [10-13] have studied the drift conditions for recurrence, ergodicity, geometric ergodicity and uniform ergodicity. The drift conditions for sub-geometric ergodicity have been discussed in [1, 4, 14-17] and so on. In [18, 19], the drift conditions for transience have been obtained.

Recently, we have investigated the drift condition for geometric transience in [6]. One of the main results shows that the chain $\Phi_n$ is geometrically transient, if there exist some set $A\in \mathscr{B}^+(\mathbb{R})$, constants $\lambda, b\in (0, 1)$, and a function $W\geq 1_A$ (with $\mathit{W}\left( {{\mathit{x}}_{\rm{0}}} \right)<\infty $ for some $x_0\in \mathbb{R}$) satisfying the drift condition

$ PW(x)\le \lambda W(x){{1}_{{{A}^{c}}}}(x)+b{{1}_{A}}(x), \ \ \ x\in \mathbb{R}. $

As far as we know, however, this drift condition can not be applied directly for the non-linear autoregressive model considered in this paper. Alternatively, we will establish a more practical drift condition for geometric transience. First, we need the following two lemmas, which are taken from [6].

Lemma 2.1  The chain $\Phi_n$ is geometrically transient if and only if there exist some set $A\in \mathscr{B}^+(\mathbb{R})$ and a constant $\kappa >\rm{1}$ such that

$ _{x\in A}^{\ \sup }L(x, A)<1, \ \ _{x\in A}^{\ \sup }{{\mathbb{E}}_{x}}[{{\kappa }^{{{\tau }_{A}}}}{{1}_{\{{{\tau }_{A}}<\infty \}}}]<\infty . $

Lemma 2.2   $(1)$ For $A\in \mathscr{B}^{+}(\mathbb{R})$ and $\kappa {\rm{ }} \ge {\rm{1}}$,

$ {{\mathbb{E}}_{x}}[{{\kappa }^{{{\tau }_{A}}}}{{1}_{\{{{\tau }_{A}}<\infty \}}}]\rm{=}\kappa \int_{{{\mathit{A}}^{\mathit{c}}}}{{{\mathbb{E}}_{y}}[{{\kappa }^{{{\sigma }_{A}}}}{{1}_{\{{{\sigma }_{A}}<\infty \}}}]}\mathit{P}\left( \mathit{x}, \mathit{dy} \right)+\kappa \mathit{P}\left( \mathit{x}, \mathit{A} \right), \ \mathit{x}\in \mathbb{R}.\ $

(2) $\{{{\mathbb{E}}_{x}}[{{\kappa }^{{{\sigma }_{A}}}}{{1}_{\{{{\sigma }_{A}}<\infty \}}}], \ \mathit{x}\in \mathbb{R}\}$ is the minimal non-negative solution to the equations

$ \left\{ \begin{array}{*{35}{l}} g(x)=\kappa \int_{{{A}^{c}}}{g}(y)P(x, dy)+\kappa P(x, A), & x\in {{A}^{c}}, \\ g(x)=1, & x\in A. \\ \end{array}\rm{ } \right. $

Proposition 2.3  The chain $\Phi_n$ is geometrically transient, if there exist a petite set $A\in \mathscr{B}^{+}(\mathbb{R})$, constants $\lambda \in \rm{(0, 1)}$, $b\in (0, \infty )$, and a non-negative measurable function $W$ bounded on $A$ satisfying

$ PW(x)\le \lambda W(x)+b{{1}_{A}}(x), \ \ \ \mathit{x}\in \mathbb{R} $

(2.1)

and

$ D:=\{x:W(x)<_{y\in A}^{\ \ \inf }W(y)\}\in {{\mathscr{B}}^{+}}(\mathbb{R}). $

(2.2)

Proof  Since $W$ is non-negative and $D\in\mathscr{B}^{+}(\mathbb{R})$, we have $\mathop {\inf }\limits_{y \in A} W(y) > 0$. Set

$ \bar{W}(x)=\frac{W(x)}{_{y\in A}^{\ \inf }W(y)}, \ \ \ x\in \mathbb{R}. $

Then $\bar W(x)\geq1$ for $x\in D^c$, $\bar W(x)<1$ for $x\in D$, and (2.1) yields that

$ \left\{ {\begin{array}{*{20}{l}} {\bar W(x) \ge {\lambda ^{ - 1}}P\bar W(x) \ge {\lambda ^{ - 1}}\int_{{A^c}} {\bar W} (y)P(x,dy) + {\lambda ^{ - 1}}P(x,A),} & {x \in {A^c},}\\ {\bar W(x) \ge 1,} & {x \in A.} \end{array}} \right. $

(2.3)

According to Lemma 2.2 (2), $\{{{\mathbb{E}}_{x}}[{{\lambda }^{-{{\sigma }_{A}}}}{{1}_{\{{{\sigma }_{A}}<\infty \}}}], x\in \mathbb{R}\}$ is the minimal non-negative solution to the equations

$ \left\{ \begin{array}{*{35}{l}} g(x)={{\lambda }^{-1}}\int_{{{A}^{c}}}{g}(y)P(x, dy)+{{\lambda }^{-1}}P(x, A), & x\in {{A}^{c}}, \\ g(x)=1, & x\in A. \\ \end{array} \right. $

(2.4)

Hence by the comparison theorem of the minimal non-negative solution (see [20, Theorem 2.6]), we know from (2.3) and (2.4) that

$ {{\mathbb{E}}_{x}}[{{\lambda }^{-{{\sigma }_{A}}}}{{1}_{\{{{\sigma }_{A}}<\infty \}}}]\le \mathit{\bar{W}}\left( \mathit{x} \right), \ \ x\in {{\mathit{A}}^{\mathit{c}}}. $

(2.5)

By (2.5) and noting that $D\subset A^c$, we have for all $x\in\mathbb{R}$,

$ \begin{align} & L(x, A)=\int_{D}{L}(y, A)P(x, dy)+\int_{{{D}^{c}}}{L}(y, A)P(x, dy) \\ & \ \ \ \ \ \ \ \ \ \ \le \int_{D}{L}(y, A)P(x, dy)+P(x, {{D}^{c}}) \\ & \ \ \ \ \ \ \ \ \ \ \le \int_{D}{{{\mathbb{E}}_{y}}[{{\lambda }^{-{{\sigma }_{A}}}}{{1}_{\{{{\sigma }_{A}}<\infty \}}}]}P(x, dy)+P(x, {{D}^{c}}) \\ & \ \ \ \ \ \ \ \ \ \ \le \int_{D}{{\bar{W}}}(y)P(x, dy)+P(x, {{D}^{c}})P(x, D) \\ & \ \ \ \ \ \ \ \ \ \ +P(x, {{D}^{c}})=1. \\ \end{align} $

Thus there exists some set $C\subset A$ with $C\in\mathscr{B}^{+}(\mathbb{R})$ such that

$ \mathop {\sup }\limits_{x \in C} L(x, C) < 1. $

(2.6)

According to Lemma 2.1, in the following, it is enough to prove that for some $\kappa {\rm{ > 1}}$, $_{x\in C}^{\ \sup }{{\mathbb{E}}_{x}}[{{\kappa }^{{{\tau }_{C}}}}{{1}_{\{{{\tau }_{C}}<\infty \}}}]<\infty $.

Combining Lemma 2.2 (1) with (2.5) and (2.1), we get for all $x\in A$,

$ \begin{align} & {{\mathbb{E}}_{x}}[{{\lambda }^{-{{\tau }_{A}}}}{{1}_{\{{{\tau }_{A}}<\infty \}}}]={{\lambda }^{-1}}\int_{{{A}^{c}}}{{{\mathbb{E}}_{y}}[{{\lambda }^{-{{\sigma }_{A}}}}{{1}_{\{{{\sigma }_{A}}<\infty \}}}]P(x,dy)} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +{{\lambda }^{-1}}P(x,A)\ \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \le {{\lambda }^{-1}}\int_{{{A}^{c}}}{{\bar{W}}}(y)P(x,dy)+{{\lambda }^{-1}}P(x,A)\ \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \le {{\lambda }^{-1}}P\bar{W}(x)\le \bar{W}(x)+\frac{b}{\lambda _{y\in A}^{\ \inf }W(y)}. \\ \end{align} $

Since $W$ is bounded on $A$,

$ _{x\in A}^{\ \sup }{{\mathbb{E}}_{x}}[{{\lambda }^{{{\tau }_{A}}}}{{1}_{\{{{\tau }_{A}}<\infty \}}}]<\infty . $

(2.7)

Noting that $A$ is petite and $C\subset A$, according to (2.7) and the proof of [3, Theorem 15.2.1], we obtain that for all $\rm{1}<\kappa \le {{\lambda }^{-1/2}}$, $_{x\in C}^{\ \sup }{{\mathbb{E}}_{x}}[{{\kappa }^{{{\tau }_{C}}}}{{1}_{\{{{\tau }_{C}}<\infty \}}}]<\infty $. This together with (2.6) yields the desired assertion.

Now, we are ready to prove Theorem 1.1.

Proof of Theorem 1.1  By (A2), there exist constants $\theta >\rm{0}$ and $c>0$ satisfying

$ |F(x)|\ge (1+\theta )|x|, \ \ \ |x|\ge c. $

(2.8)

Choose

$ \begin{array}{l} \\ A = \left\{ {\left. {x:|x| < c \vee \frac{{1 + \log \int {{{\rm{e}}^{s|x|}}} \Gamma (\mathit{dx})}}{{s\theta }}} \right\}} \right., \;\;W(x) = {{\rm{e}}^{ - \mathit{s}|\mathit{x}|}}.\\ \end{array} $

Then $A\in\mathscr{B}^{+}(\mathbb{R})$ is petite and $D\in\mathscr{B}^{+}(\mathbb{R})$, where $D$ is defined in (2.2). From (A1) and (2.8), we have for $x\in A^{c}$,

$ \begin{align} & \frac{PW(x)-W(x)}{W(x)}=\int{\left( \frac{W(F(x)+y)}{W(x)}-1 \right)}\Gamma (dy) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\int{({{\rm{e}}^{-s|F(x)+y|+s|x|}}-1)}\Gamma (dy) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \le \int{({{\rm{e}}^{-s|F(x)|+s|y|+s|x|}}-1)\Gamma }(dy) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \le \int{({{\rm{e}}^{-s(1+\theta )|x|+s|y|+s|x|}}-1)\Gamma }(dy) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ={{\rm{e}}^{-s\theta |x|}}\int{{{\rm{e}}^{s|y|}}}\Gamma (dy)-1\le {{\rm{e}}^{-1}}-1. \\ \end{align} $

That is,

$ PW(x)\le {{\rm{e}}^{-1}}\mathit{W}(\mathit{x}), \ \ \ \mathit{x}\in {{\mathit{A}}^{\mathit{c}}}. $

(2.9)

Noting that $W$ is bounded, it is obvious that for some $b\in (0, \rm{ }\infty )$,

$ PW(x)\le {{\rm{e}}^{-1}}W(x)+b, \ \ \ x\in A. $

Combining this with (2.9), the drift condition (2.1) holds. Thus, the non-negative autoregressive model $\Phi_n$ is geometrically transient by Proposition 2.3.