2. Preliminaries

To formalize the discussion in the Introduction, consider $X_{t}$ ( $t=0,1,2,\ldots$ ), a time-homogeneous discrete-time Markov chain on a general measurable state space $(\textsf{X},\mathcal{B}(\textsf{X}))$ . Comprehensive treatments of the relevant Markov chain theory can be found in [Reference Meyn and Tweedie19] or [Reference Douc, Moulines, Priouret and Soulier7]. Let $\mu$ be any initial measure on $\mathcal{B}(\textsf{X})$ , and suppose that $X_{0}$ has distribution $\mu$ . Denote the transition probabilities by $P(x\,;\,A)$ ( $x\in\textsf{X}$ , $A\in\mathcal{B}(\textsf{X})$ ) and let $(\Omega,\mathcal{F},\mathbb{P}_{\mu})$ denote the probability space of the Markov process $\{X_{0},X_{1},\ldots\}$ . As usual, $\mathbb{P}_{x}$ denotes the probability measure corresponding to a fixed initial value $X_{0}=x$ and $P^{n}(x\,;\,A)=\mathbb{P}_{x}(X_{n}\in A)$ ( $x\in\textsf{X}$ , $A\in\mathcal{B}(\textsf{X})$ ) signifies the n-step transition probability measure.

Next consider the rate of convergence of the n-step probability measures $P^{n}$ to the stationary probability measure $\pi$ . To this end, for any two probability measures $\lambda_{1}$ and $\lambda_{2}$ on $(\textsf{X},\mathcal{B}(\textsf{X}))$ , the total variation distance is defined as

$$\lVert\lambda_{1}-\lambda_{2}\rVert=2\sup_{B\in\mathcal{B}(\textsf{X})}\lvert\lambda_{1}(B)-\lambda_{2}(B)\rvert=\sup_{\lvert h\rvert\leq1}\lvert\lambda_{1}(h)-\lambda_{2}(h)\rvert,$$

where the last supremum runs over all $\mathcal{B}(\textsf{X})$ -measurable functions $h\colon \textsf{X}\to\mathbb{R}$ bounded in absolute value by 1 and $\lambda_{i}(h)=\int_{\textsf{X}}\lambda_{i}({\textrm{d}} x)h(x)<\infty$ . The n-step probability measures $P^{n}$ converge in total variation norm to the stationary probability measure $\pi$ at rate r(n), $n\geq 0$ , if

(3) \begin{equation}\lim_{n\to\infty}r(n)\lVert P^{n}(x\,;\,\cdot)-\pi(\cdot)\rVert=0,\quad\text{$\pi$-a.e.}\end{equation}

If (3) holds, we say that the Markov chain $X_{t}$ is ergodic with rate r(n); geometric ergodicity obtains when $r(n)=r^{n}$ for some $r>1$ .

To define the $\beta$ -mixing coefficients, let $\mathcal{F}_{k}^{l}$ , $0\leq k\leq l\leq\infty$ , signify the $\sigma$ -algebra generated by $\{X_{k},\ldots,X_{l}\}$ . For the stochastic process $\{X_{0},X_{1},\ldots\}$ , the $\beta$ -mixing coefficients $\beta(n)$ , $n=1,2,\ldots,$ are defined by ([Reference Doukhan8, Section 1.1], [Reference Bradley3, Chapter 3])

\begin{align*}\beta(n) & =\dfrac{1}{2}\sup_{m\in\mathbb{N}}\sup\sum_{i=1}^{I}\sum_{j=1}^{J}|\mathbb{P}_{\mu}(A_{i}\cap B_{j})-\mathbb{P}_{\mu}(A_{i})\mathbb{P}_{\mu}(B_{j})|\\& =\sup_{m\in\mathbb{N}}\mathbb{E}_{\mu}\biggl[\sup_{B\in\mathcal{F}_{n+m}^{\infty}}|\mathbb{P}_{\mu}(B\mid\mathcal{F}_{0}^{m})-\mathbb{P}_{\mu}(B)|\biggr],\end{align*}

where $\mathbb{N}=\{0,1,2,\ldots\}$ , and in the first expression for $\beta(n)$ the second supremum is taken over all pairs of (finite) partitions $\{A_{1},A_{2},\ldots,A_{I}\}$ and $\{B_{1},B_{2},\ldots,B_{J}\}$ of $\Omega$ such that $A_{i}\in\mathcal{F}_{0}^{m}$ for each i and $B_{j}\in\mathcal{F}_{n+m}^{\infty}$ for each j. For our purposes it is convenient to use the following alternative expression obtained by Davydov ([Reference Davydov5, Proposition 1; note that his definition of $\beta(n)$ includes an additional factor of 2]):

(4) \begin{equation}\beta(n)=\dfrac{1}{2}\sup_{m\in\mathbb{N}}\int_{\textsf{X}}\mu P^{m}({\textrm{d}} x)\Vert P^{n}(x\,;\,\cdot)-\mu P^{n+m}(\cdot)\Vert ,\quad n=1,2,\ldots,\end{equation}

where $\mu P^{m}(\cdot)=\int_{\textsf{X}}\mu({\textrm{d}} x)P^{m}(x\,;\,\cdot)$ denotes the distribution of $X_{m}$ ( $m=1,2,\ldots$ ; $\mu P^{0}=\mu$ ). For a stationary Markov chain (i.e. one with initial distribution $\pi$ ), the $\beta$ -mixing coefficients can be expressed simply as

(5) \begin{equation}\beta(n)=\dfrac{1}{2}\int_{\textsf{X}}\pi({\textrm{d}} x)\Vert P^{n}(x\,;\,\cdot)-\pi(\cdot)\Vert ,\quad n=1,2,\ldots.\end{equation}

Process $X_{t}$ is said to be $\beta$ -mixing (or sometimes absolutely regular) if $\lim_{n\rightarrow\infty}\beta(n)=0$ . As with the convergence in (3), the rate of this convergence is of interest, and in what follows we seek for results of the form $\lim_{n\rightarrow\infty}r(n)\beta(n)=0$ with some rate function r(n).

3. The geometric case

We start by briefly discussing the relation of geometric ergodicity and $\beta$ -mixing; although these results are well known, comparing them with the subgeometric case will be illuminating. For a stationary Markov chain (i.e. one with initial distribution $\pi$ ), this relation is particularly simple. As was first shown by Nummelin and Tuominen [Reference Nummelin and Tuominen20, Theorem 2.1], a geometrically ergodic Markov chain satisfies, for some $r>1$ , $\lim_{n\rightarrow\infty}r^{n}\int\pi({\textrm{d}} x)\Vert P^{n}(x\,;\,\cdot)-\pi(\cdot)\Vert =0$ ; given expression (5), the $\beta$ -mixing property immediately follows and the mixing coefficients satisfy $\lim_{n\rightarrow\infty}r^{n}\beta(n)=0$ . Statements of this result can be found for instance in [Reference Doukhan8, page 89], [Reference Bradley2, Theorem 3.7], and [Reference Bradley3, Theorem 21.19]. For initial distributions other than the stationary one, a corresponding result seems to have first appeared in [Reference Liebscher16, Proposition 4].

To facilitate comparison with the subgeometric case, we present the ergodicity and mixing results as consequences of a particular drift criterion; as is discussed in [Reference Meyn and Tweedie19], this is how geometric ergodicity is often established. We use the following traditional Foster–Lyapunov-type geometric drift condition (see [Reference Meyn and Tweedie19, Theorem 15.0.1]). Here $\boldsymbol{1}_{C}(\cdot)$ signifies the indicator function of a set C. As a technical remark, note that in this condition we assume the function V to be everywhere finite (i.e. $V \colon \textsf{X}\rightarrow[1,\infty)$ ) and such that $\sup_{x\in C}V(x)<\infty$ . In contrast, in [Reference Meyn and Tweedie19, Theorem 15.0.1] it is only assumed that V is extended-real-valued (i.e. $V \colon \textsf{X}\rightarrow[1,\infty]$ ) and finite at some point $x_{0}\in\textsf{X}$ . Our stronger requirements hold in most practical applications and lead to more transparent exposition and proofs.

Condition Drift–G. Suppose there exist a petite set C, constants $b<\infty$ , $\beta>0$ , and a measurable function $V \colon \textsf{X}\rightarrow[1,\infty)$ such that $\sup_{x\in C}V(x)<\infty$ , satisfying

$$\mathbb{E}[V(X_{1})\mid X{}_{0}=x ]\leq V(x)-\beta V(x)+b\boldsymbol{1}_{C}(x),\quad x\in\textsf{X}.$$

For the definition of a ‘petite set’ appearing in this condition, and for the concepts of irreducibility and aperiodicity in the theorem below, we refer the reader to [Reference Meyn and Tweedie19]. Theorem 1 summarizes the relation between geometric ergodicity and $\beta$ -mixing.

Parts (a) and (b) are very well known (see e.g. [Reference Meyn and Tweedie19, Theorem 15.0.1] for part (a) and [Reference Nummelin and Tuominen20, Theorem 2.3] for part (b)) and so is also the mixing result in the stationary case (see the references given earlier). Part (c) for general initial distributions was obtained by Liebscher [Reference Liebscher16, Proposition 4], although our formulation is somewhat different from his (our formulation and proof avoid the use of so-called ‘Q-geometric ergodicity’ employed by Liebscher; for completeness, our proof of Theorem 1, which may be of independent interest, is provided in the supplementary material). Part (d) elaborates parts (b) and (c) as well as their relation in the stationary case.

4. The subgeometric case

We seek a counterpart of Theorem 1 in which the geometric rate $r^{n}$ is replaced by some slower rate function; such rate functions were already exemplified in the Introduction. More formally, the subgeometric rate functions we consider are defined as follows (see e.g. [Reference Nummelin and Tuominen21] and [Reference Douc, Fort, Moulines and Soulier6]). Let $\Lambda_{0}$ be the set of positive non-decreasing functions $r_{0} \colon \mathbb{N}\rightarrow[1,\infty)$ such that $\ln[r_{0}(n)]/n$ decreases to zero as $n\rightarrow\infty$ . The class of subgeometric rate functions, denoted by $\Lambda$ , consists of positive functions $r \colon \mathbb{N}\rightarrow(0,\infty)$ for which there exists some $r_{0}\in\Lambda_{0}$ such that

(6) \begin{equation}0<\liminf_{n\rightarrow\infty}\dfrac{r(n)}{r_{0}(n)}\leq\limsup_{n\rightarrow\infty}\dfrac{r(n)}{r_{0}(n)}<\infty.\end{equation}

Typical examples are obtained of rate functions r for which these inequalities hold with (for notational convenience, we set $\ln(0)=0$ )

$$r_{0}(n)=(1+\ln(n))^{\alpha}\,\cdot\,(1+n)^{\beta}\,\cdot\,{\textrm{e}}^{cn^{\gamma}},\quad\alpha,\beta,c\geq 0,\,\gamma\in(0,1).$$

The rate function $r_{0}(n)$ is called subexponential when $c>0$ , polynomial when $c=0$ and $\beta>0$ , and logarithmic when $\beta=c=0$ and $\alpha>0$ .

In analogy with the geometric case, subgeometric ergodicity and mixing results are most conveniently obtained by verifying an appropriate drift condition. The following drift condition for subgeometric ergodicity is adapted from [Reference Douc, Moulines, Priouret and Soulier7, Definition 16.1.7]. A somewhat more general drift condition, for instance allowing for V to be extended-real-valued, is given in [Reference Douc, Fort, Moulines and Soulier6].

Condition Drift–SubG. Suppose there exist a petite set C, a constant $b<\infty$ , a concave increasing continuously differentiable function $\phi \colon [1,\infty)\rightarrow(0,\infty)$ satisfying $\lim_{v\rightarrow\infty}\phi'(v)=0$ , and a measurable function $V \colon \textsf{X}\rightarrow[1,\infty)$ such that $\sup_{x\in C}V(x)<\infty$ and

$$\mathbb{E}[V(X_{1})\mid X{}_{0}=x ]\leq V(x)-\phi(V(x))+b\boldsymbol{1}_{C}(x),\quad x\in\textsf{X}. $$

Note that if $\phi(v)=\eta v$ ( $\eta>0$ ), then one obtains Condition Drift–G (but assumption $\lim_{v\rightarrow\infty}\phi'(v)=0$ rules this out; as we are interested in subgeometric rates of ergodicity, assuming this means no loss of generality; see [Reference Douc, Moulines, Priouret and Soulier7, Remark 16.1.8]).

Following Douc et al. [Reference Douc, Fort, Moulines and Soulier6] we next introduce a rate function, denoted by $r_{\phi}$ . First define the function

$$H_{\phi}(v)=\int_{1}^{v}\dfrac{{\textrm{d}} x}{\phi(x)},$$

where $\phi$ is as in Condition Drift–SubG. The definition implies that $H_{\phi}$ is a non-decreasing, concave, and differentiable function on $[1,\infty)$ , and it has an inverse $H_{\phi}^{-1} \colon [0,\infty)\rightarrow[1,\infty)$ which is increasing and differentiable (see [Reference Douc, Fort, Moulines and Soulier6, Section 2.1]). Thus we can define the rate function

$$r_{\phi}(z)=(H_{\phi}^{-1})'(z)=\phi\circ H_{\phi}^{-1}(z).$$

Douc et al. [Reference Douc, Fort, Moulines and Soulier6, Lemma 2.3 and Proposition 2.5] showed that this rate function is subgeometric and that Condition Drift–SubG implies the convergence (3) at rate $r_{\phi}(n)$ .

Theorem 2 summarizes the relation between subgeometric ergodicity and $\beta$ -mixing. Here $\lfloor k\rfloor$ denotes the integer part of the real number k.

Of the results in Theorem 2, part (a) is given in Proposition 2.5 of [Reference Douc, Fort, Moulines and Soulier6]. Part (b) can be obtained by combining Theorem 4.1 of [Reference Tuominen and Tweedie23] and Proposition 2.5 of [Reference Douc, Fort, Moulines and Soulier6], but in the proof we make use of [Reference Nummelin and Tuominen21]. Part (c) is new and illuminates the relation between subgeometrically ergodic Markov chains and their $\beta$ -mixing properties, thereby providing a counterpart of a result obtained by Liebscher [Reference Liebscher16, Proposition 4] in the case of geometric ergodicity. Part (d) is analogous to its counterpart in Theorem 1 and provides further insight into parts (b) and (c), whereas part (e) makes part (c) more concrete in the case of the most common rate functions. For completeness, we give a detailed proof in the Appendix.

As discussed in [Reference Douc, Fort, Moulines and Soulier6, Section 2.3] and [Reference Meitz and Saikkonen17, Theorem 1], there is a connection between the function $\phi$ and the rate function $r_{\phi}$ , which can be used to find out the latter in particular cases. For instance, polynomial rate functions are associated with cases where the function $\phi$ is of the form $\phi(v)=cv^{\alpha}$ with $\alpha\in(0,1)$ and $c\in(0,1]$ , and then the rate obtained is $r_{\phi}(n)=n^{\alpha/(1-\alpha)}$ (an alternative form is $r_{\phi}(n)=n^{\kappa-1}$ with $\kappa=1+\alpha/(1-\alpha)$ already given by Jarner and Roberts [Reference Jarner and Roberts11]). In the subexponential case the function $\phi$ is such that $v/\phi(v)$ goes to infinity slower than polynomially so that a possibility, given in [Reference Meitz and Saikkonen17, Theorem 1], is $\phi(v)=c(v+v_{0})/[\ln(v+v_{0})]^{\alpha}$ for some $\alpha,c,v_{0}>0$ . This results in the rate $r_{\phi}(n)=({\textrm{e}}^{d})^{n^{1/(1+\alpha)}}$ for some $d>0$ , which is faster than polynomial. A logarithmic rate is an example of a rate slower than polynomial. Then the function $\phi$ is of the form $\phi(v)=c[1+\ln(v)]^{\alpha}$ for some $\alpha>0$ and $c\in(0,1]$ , and the resulting rate is $r_{\phi}(n)=[\ln(v)]^{\alpha}$ (see [Reference Douc, Fort, Moulines and Soulier6, Section 2.3]).

Theorem 2 (or 1) also provides information about the moments of the stationary distribution of $X_{t}$ . Specifically, once part (a) of Theorem 2 (or 1) has been established, one can deduce from Condition Drift–SubG (or Drift–G) and Theorem 14.3.7 of [Reference Meyn and Tweedie19] that $\int_{\textsf{X}}\pi({\textrm{d}} x)\phi(V(x))<\infty$ (or $\int_{\textsf{X}}\pi({\textrm{d}} x)V(x)<\infty$ ). This can be very useful when one aims to apply limit theorems developed for $\beta$ -mixing processes, where moment conditions are typically assumed.

We close this section by noting that Condition Drift–SubG can also be used to obtain more general ergodicity results than provided in Theorem 2. Without going into details, we only mention that Theorem 2.8 of [Reference Douc, Fort, Moulines and Soulier6] and Theorem 1 of [Reference Meitz and Saikkonen17] show how a stronger form of ergodicity, called (f,r)-ergodicity, can be established.

5. Example

To illustrate our results we consider the self-exciting threshold autoregressive (SETAR) model studied by Chan et al. [Reference Chan, Petruccelli, Tong and Woolford4]. These authors analyzed the model

(7) \begin{equation}X_{t}=\varphi(\,j)+\theta(\,j)X_{t-1}+W_{t}(\,j),\quad X_{t-1}\in(r_{j-1},r_{j}],\end{equation}

where $-\infty=r_{0}<\cdots<r_{M}=\infty$ , and for each $j=1,\ldots,M$ , $\{W_{t}(\,j)\}$ is an independent and identically distributed mean zero sequence independent of $\{W_{t}(i)\}$ , $i\neq j$ , and with $W_{t}(\,j)$ having a density that is positive on the whole real line. They considered the following conditions:

(8a) \begin{align} \theta(1)<1,\quad\theta(M)<1,\quad\theta(1)\theta(M)<1, \end{align}

(8b) \begin{align} \theta(1)=1,\quad\theta(M)<1,\quad0<\varphi(1), \end{align}

(8c) \begin{align} \theta(1)<1,\quad\theta(M)=1,\quad\varphi(M)<0, \end{align}

(8d) \begin{align}\theta(1)=1,\quad\theta(M)=1,\quad\varphi(M)<0<\varphi(1), \end{align}

(8e) \begin{align}\theta(1)<0,\quad\theta(1)\theta(M)=1,\quad\varphi(M)+\varphi(1)\theta(M)>0, \end{align}

and showed that the SETAR model is ergodic if and only if one of the conditions (8a)–(8e) holds [Reference Chan, Petruccelli, Tong and Woolford4, Theorem 2.1]. Moreover, if $\mathbb{E}[\vert W_{t}(\,j)\vert]<\infty$ for each j, they showed that condition (8a) ensures geometric ergodicity [Reference Chan, Petruccelli, Tong and Woolford4, Theorem 2.3]. To our knowledge, in the cases (8b)–(8e) no results regarding the rate of ergodicity have as yet appeared in the literature and our Theorem 4(b) below indicates that geometric ergodicity may not always hold without stronger assumptions. Related to this, we note that Meyn and Tweedie [Reference Meyn and Tweedie19, Section 11.4.3 and Section B.2] discussed the (geometric) ergodicity of the SETAR model (7), reproducing the ergodicity result of [Reference Chan, Petruccelli, Tong and Woolford4, Theorem 2.1] as their Proposition 11.4.5. On their page 541, Meyn and Tweedie [Reference Meyn and Tweedie19] also stated that (our additions in brackets) ‘in the interior of the parameter space [the union of (8a)–(8e)] we are able to identify geometric ergodicity in Proposition 11.4.5 … the stronger form [geometric ergodicity] is actually proved in that result’ but no formal proof is given for this statement.

We consider rates of ergodicity and $\beta$ -mixing in case (8d) when the autoregressive coefficients $\theta(1)$ and $\theta(M)$ equal unity. For intuition, note that due to non-zero intercept terms $\varphi(1)$ and $\varphi(M)$ , both the first and the last regimes exhibit non-stationary random-walk-type behavior with a drift. As the intercept terms satisfy $\varphi(M)<0<\varphi(1)$ , the drift is increasing in the first regime and decreasing in the last regime. This feature prevents the process $y_{t}$ from exploding to (plus or minus) infinity, thereby providing intuition why ergodicity can hold true. It is noteworthy that ergodicity is in no way dependent on the behavior of the process in the middle regimes ( $2,\ldots,M-1$ ), which can exhibit stationary, random-walk-type (with or without drift), or even explosive behavior.

In their results, Chan et al. [Reference Chan, Petruccelli, Tong and Woolford4] allowed for regime-dependent distributions for the error term $W_{t}(\,j)$ . To obtain our results for the case (8d), we strengthen the assumptions on the error term and, in particular, assume that the error distribution is the same in each regime (this stronger assumption is needed to apply the results mentioned in the proof of Theorem 3 below, and relaxing it appears less than straightforward). To compensate, we obtain results for a model more general than the SETAR model (7) with (8d). Specifically, we formulate our results in terms of the general non-linear autoregressive model

(9) \begin{equation}X_{t}=g(X_{t-1})+\varepsilon_{t},\quad t=1,2,\ldots,\end{equation}

where the function $g \colon \mathbb{R}\rightarrow\mathbb{R}$ and the error term $\varepsilon_{t}$ satisfy the following conditions:

(A1) g is a measurable function with the property $|g(x)|\rightarrow\infty$ as $|x|\rightarrow\infty$ and such that there exist positive constants r and $M_{0}$ such that

$$|g(x)| \le (1 - r/|x|)|x|\quad {\rm{for}}|x| \ge {M_0}\;{\rm{and}}\;\mathop {\sup }\limits_{|x| \le {M_0}} |g(x)|<\infty;} $$

(A2) $\{\varepsilon_{t},\,t=1,2,\ldots\}$ is a sequence of independent and identically distributed mean zero random variables that is independent of $X_{0}$ and the distribution of $\varepsilon_{1}$ has a (Lebesgue) density that is bounded away from zero on compact subsets of $\mathbb{R}$ .

Model (9) with conditions (A1) and (A2) is a special case of models considered by Fort and Moulines [Reference Fort and Moulines10, Section 2.2], Douc et al. [Reference Douc, Fort, Moulines and Soulier6, Section 3.3], and Meitz and Saikkonen [Reference Meitz and Saikkonen17, Sections 3-4]. These authors considered much more general models, but for clarity of presentation we have simplified the model as much as possible while still being able to obtain results for the SETAR model (7) with (8d) (papers [Reference Fort and Moulines10] and [Reference Douc, Fort, Moulines and Soulier6] considered a multivariate version of (9) whereas [Reference Meitz and Saikkonen17] considered a higher-order generalization of (9); the inequality constraint for the function g in condition (A1) is also more general in these papers, where it is only required that $|g(x)| \leq (1-r|x|^{-\rho})|x|$ for some $0<\rho\leq2$ ).

The following theorem establishes ergodicity and $\beta$ -mixing results for model (9) with varying rates of convergence. The proof (in the Appendix) makes use of results in [Reference Fort and Moulines10], [Reference Douc, Fort, Moulines and Soulier6], and [Reference Meitz and Saikkonen17] to obtain rates of ergodicity, as well as Theorems 1 and 2 above to obtain rates of $\beta$ -mixing (only the subgeometric mixing results in parts (b) and (c) are new).

Theorem 3 shows that there is a trade-off between rates of ergodicity and $\beta$ -mixing and finiteness of moments of the error term. The fastest geometric rate is obtained when $\mathbb{E}[{\textrm{e}}^{z_{0}|\varepsilon_{1}|}]<\infty$ ( $z_{0}>0$ ), so that $\varepsilon_{1}$ has finite moments of all orders and the slowest polynomial rate is obtained when only $\mathbb{E}[\varepsilon_{1}^{2}]<\infty$ . As discussed after Theorem 2, we also have $\int_{\textsf{X}}\pi({\textrm{d}} x)\phi(V(x))<\infty$ so that there is a similar trade-off between these convergence rates and finiteness of moments of the stationary distribution (expressions of V and $\phi$ are available in the proof of Theorem 3).

Above it was mentioned that [Reference Fort and Moulines10], [Reference Douc, Fort, Moulines and Soulier6], and [Reference Meitz and Saikkonen17] considered (subgeometric) ergodicity of models more general than (9) with conditions (A1) and (A2). Making use of our Theorems 1 and 2, subgeometric rates of $\beta$ -mixing can easily be obtained for these more general models too. We omit the details for brevity.

In a series of papers, Veretennikov and co-authors also considered the model (9) with function g satisfying $|g(x)|\leq(1-r|x|^{-\rho})|x|$ for some $1\leq\rho\leq2$ . Using methods very different from ours, they obtained results on subgeometric ergodicity and subgeometric rates for $\beta$ -mixing coefficients. The cases $1<\rho<2$ and $\rho=2$ are considered in [Reference Veretennikov27], [Reference Klokov and Veretennikov14, Reference Klokov and Veretennikov15], and [Reference Klokov13] and are shown to lead to subgeometric rates. For the case $\rho=1$ relevant for the SETAR example, these papers refer to [Reference Veretennikov25, Reference Veretennikov26] and [Reference Veretennikov and Gulinskii28]. A result corresponding to our Theorem 3(a) can be found in [Reference Veretennikov and Gulinskii28, Theorem 1], but subgeometric rates, such as those in our Theorem 3 parts (b) and (c), do not seem to be established in the case $\rho=1$ .

We now specialize the results above to the SETAR model (7) with (8d). It is easy to see that this model, with the function g in (9) defined as

$$g(x)=\sum_{j=1}^{M}[\varphi(\,j)+\theta(\,j)x]\boldsymbol{1}\{x\in(r_{j-1},r_{j}]\}$$

(with $\boldsymbol{1}\{\cdot\}$ denoting the indicator function), satisfies the condition in (A1). Namely, for x large enough and positive we have $\lvert g(x)\rvert=g(x)=x+\varphi(M)=\lvert x\rvert-(-\varphi(M))$ , whereas for x small enough and negative we have $\lvert g(x)\rvert=-g(x)=-x-\varphi(1)=\lvert x\rvert-\varphi(1)$ , so that the inequality in (A1) holds for $M_{0}>\max\{|r_{1}|,|r_{M-1}|\}$ and $r=\min\{\varphi(1),-\varphi(M)\}$ (and the supremum condition is obviously satisfied).

Part (a) of the next theorem simply restates the result of Theorem 3 for the SETAR model (7) with (8d), whereas part (b) establishes that geometric ergodicity cannot hold under the weaker moment assumptions of Theorem 3 parts (b) and (c).

Theorem 4(b) shows that for the SETAR model (7) with (8d), the subgeometric rates of Theorem 3 parts (b) and (c) cannot be improved to a geometric rate unless stronger moment assumptions are made regarding the error term. This result is obtained by making use of a necessary condition for geometric ergodicity of certain specific type of Markov chains in [Reference Jarner and Tweedie12] (using their necessary condition to obtain this result appears possible only in case (8d) out of (8a)–(8e)).