2.1. Environment

Throughout this article, $\mathbb X$ is a Polish space, $\mathcal B$ is its Borel sets, and $\preceq$ is a closed partial order on $\mathbb X$ . The last statement means that the graph of $\preceq$ , denoted by

(1) \begin{equation} \mathbb G = \{ (x, x^{\prime}) \in \mathbb X \times \mathbb X \,:\, x \preceq x^{\prime} \} ,\end{equation}

is closed under the product topology on $\mathbb X \times \mathbb X$ . A map $h \colon \mathbb X\to \mathbb R$ is called increasing if $x \preceq x^{\prime}$ implies $h(x) \leqslant h(x^{\prime})$ . We take $p\mathcal B$ to be the set of all probability measures on $\mathcal B$ and let $b\mathcal B$ be the bounded Borel measurable functions sending $\mathbb X$ into $\mathbb R$ . Given $h \in b\mathcal B$ and $\mu \in p\mathcal B$ we set $\mu(h) \,:\!=\, \int h \mathop{}\!\mathrm{d} \mu$ . The symbol $ib\mathcal B$ represents all increasing $h \in b\mathcal B$ .

For $\mu, \nu$ in $p\mathcal B$ , we say that $\mu$ is stochastically dominated by $\nu$ and write $\mu \preceq_{s} \nu$ if $\mu(h) \leqslant \nu(h)$ for all $h \in ib\mathcal B$ . In addition, we set

(2) \begin{equation} \rho(\mu, \nu) \,:\!=\, \sup\nolimits_{I \in i\mathcal B} (\mu(I) - \nu(I)) + \sup\nolimits_{I \in i\mathcal B} (\nu(I) - \mu(I)).\end{equation}

This is the total ordered variation metric on $p\mathcal B$ . A proof that $\rho$ is indeed a metric can be found in Lemma 4.1 in [Reference Kamihigashi and Stachurski16]. Positive definiteness follows from the fact that $\rho(\mu, \nu)=0$ implies $\mu \preceq_{s} \nu$ and $\nu \preceq_{s} \mu$ . Since $\preceq_{s} $ is antisymmetric on $p\mathcal B$ [Reference Kamae and Krengel14, Lemma 1], we then have $\mu=\nu$ . Connections between $\rho$ and the total variation and Wasserstein metrics are discussed in Section 5.

A function $Q \colon (\mathbb X, \mathcal B) \to \mathbb R$ is called a transition kernel on $\mathbb X$ if Q is a map from $\mathbb X \times \mathcal B$ to [0,1] such that $x\mapsto Q(x, A)$ is measurable for each $A \in \mathcal B$ and $A \mapsto Q(x, A)$ is a probability measure on $\mathcal B$ for each $x \in \mathbb X$ . At times we use the symbol $Q_x$ to represent the distribution $Q(x, \cdot)$ at given x. A transition kernel Q on $\mathbb X$ is called increasing if $Qh \in ib\mathcal B$ whenever $h \in ib\mathcal B$ . Equivalently,

\begin{equation*} Q_x \preceq_{s} Q_{x^{\prime}} \quad \text{whenever} \quad x \preceq x^{\prime}.\end{equation*}

For a transition kernel Q on $\mathbb X$ , we define the left and right Markov operators generated by Q via

\begin{equation*} \mu Q(A) = \int Q(x, A) \mu (\mathop{}\!\mathrm{d} x) \quad \text{and} \quad Qf(x) = \int f(y) Q(x, \mathop{}\!\mathrm{d} y).\end{equation*}

(The left Markov operator $\mu \mapsto \mu Q$ maps $p\mathcal B$ to itself, while the right Markov operator $f \mapsto Qf$ acts on $f \in b\mathcal B$ .) A discrete-time $\mathbb X$ -valued stochastic process $(X_t)_{t \geqslant 0}$ on a filtered probability space $(\Omega, \mathscr F, \mathbb P, (\mathscr F_t)_{t \geqslant 0})$ is called Markov- $(Q,\mu)$ if $X_0 \stackrel {\scriptsize{d}} {=} \mu$ and $\mathbb E [ h(X_{t+1}) \,|\, \mathscr F_t ] = Qh(X_t)$ with probability 1 for all $t \geqslant 0$ and $h \in b \mathcal B$ .

2.2. Couplings

A coupling of $(\mu, \nu) \in p\mathcal B \times p\mathcal B$ is a probability measure $\pi$ on $\mathcal B \otimes \mathcal B$ satisfying $\pi(A \times \mathbb X) = \mu(A) $ and $\pi(\mathbb X \times A) = \nu(A)$ for all $A \in \mathcal B$ . Let $\mathscr {C} \, (\mu, \nu)$ denote the set of all couplings of $(\mu, \nu)$ and let

(3) \begin{equation} \alpha(\mu, \nu) = \sup\nolimits_{\pi \in \mathscr {C} \, (\mu, \nu)} \pi(\mathbb G).\end{equation}

The value $\alpha(\mu, \nu)$ can be understood as a measure of ‘partial stochastic dominance’ of $\nu$ over $\mu$ [Reference Kamihigashi and Stachurski17]. In line with this interpretation, and applying Strassen’s theorem [Reference Lindvall18, Reference Strassen26], we have

(4) \begin{equation} \alpha(\mu, \nu) = 1 \quad \text{whenever} \quad \mu \preceq_{s} \nu.\end{equation}

Let Q be a transition kernel on $\mathbb X$ . A Markov coupling of Q is a real-valued function $\hat Q$ on $(\mathbb X \times \mathbb X) \times (\mathcal B \otimes \mathcal B)$ such that

1. (i) $(x,x^{\prime}) \mapsto \hat Q((x, x^{\prime}), E)$ is measurable for each $E \in \mathcal B \otimes \mathcal B$ and

2. (ii) $\hat Q_{(x, x^{\prime})}$ is a coupling of $Q_x$ and $Q_{x^{\prime}}$ for all $x, x^{\prime} \in \mathbb X$ .

In other words, $\hat Q$ is a transition kernel on $\mathbb X \times \mathbb X$ that couples the distributions $Q_x$ and $Q_{x^{\prime}}$ at every pair of points in the state space.

We call $\hat Q$ a $\preceq$ -maximal Markov coupling of Q if $\hat Q$ is a Markov coupling of Q and, in addition,

(5) \begin{equation} \hat Q((x, x^{\prime}), \mathbb G) = \alpha(Q_x, Q_{x^{\prime}}) \quad \text{for all } (x,x^{\prime}) \in \mathbb X \times \mathbb X.\end{equation}

Informally, $\hat Q$ serves as a transition kernel of a ‘joint’ chain $((X_t,X_t'))_{t \geqslant 0}$ that maximizes the probability of attaining $X_t \preceq X_t'$ at each step. Below we use the following existence result.

Proof. Let Q be a transition kernel on $\mathbb X$ . By Theorem 1.1 in [Reference Zhang27], given lower semicontinuous $g \colon \mathbb X \times \mathbb X \to \mathbb R$ , there exists a transition kernel $\hat Q$ on $\mathbb X \times \mathbb X$ such that $\hat Q$ is a Markov coupling of Q and, in addition

\begin{equation*} (\hat Q g)(x, x^{\prime}) = \inf \left\{ \int g \mathop{}\!\mathrm{d} \pi \, : \, \pi \in \mathscr {C} \, (Q_x, Q_{x^{\prime}}) \right\} \! . \end{equation*}

As $\mathbb G$ is closed, this equality is attained when $g = 1 - \unicode{x1D7D9}_{\mathbb G}$ . Since $\hat Q_{(x, x^{\prime})}$ and $\pi$ are probability measures, we then have

\begin{equation*} \hat Q((x, x^{\prime}), \mathbb G) = \sup \left\{ \pi(\mathbb G) \, : \, \pi \in \mathscr {C} \, (Q_x, Q_{x^{\prime}}) \right\} \! . \end{equation*}

This shows that $\hat Q$ is a $\preceq$ -maximal Markov coupling of Q.

The following simple lemma will be important for our bounds.

3.1. A quantitative bound

Let Q be an increasing transition kernel on $\mathbb X$ and let $\hat Q$ be a $\preceq$ -maximal Markov coupling of Q. Let W be a measurable function from $\mathbb X \times \mathbb X$ to $[1,\infty)$ . Suppose that, for some measurable set C in $\mathbb X$ and some strictly increasing convex function $\delta \colon \mathbb R_+ \to\mathbb R_+$ with $\delta(0)=0$ , the kernel $\hat Q$ obeys the drift condition

(6) \begin{equation} \hat Q W(x, x^{\prime}) \leqslant \delta( W(x, x^{\prime}) ) \quad \text{for all } (x,x^{\prime}) \notin C \times C.\end{equation}

We set

(7) \begin{equation} B = \max \big\{1, \;\delta^{-1} \left(B_0\right)\big\} \! , \quad \text{where} \quad B_0 \,:\!=\, \sup\nolimits_{(x, x^{\prime}) \in C \times C} \hat R W(x,x^{\prime}),\end{equation}

and

(8) \begin{equation} \hat R W (x,x^{\prime}) \,:\!=\, \int \int W(y,y') \unicode{x1D7D9}\{y \npreceq y'\} \hat Q((x,x^{\prime}), \mathop{}\!\mathrm{d} (y,y')).\end{equation}

In addition, let

(9) \begin{equation} \varepsilon = \inf \left\{ \alpha(Q_x, Q_{x^{\prime}}) \, : \, (x, x^{\prime}) \in C \times C \right\} \! . \end{equation}

Given $\mu, \mu'$ in $p\mathcal B$ , we set

\begin{equation*} (\mu \times \mu')(W) = \int \int W(x, x^{\prime}) \mu(\! \mathop{\mathrm{d}}x) \mu'(\! \mathop{\mathrm{d}} x^{\prime}).\end{equation*}

In (7), $\delta^{-1}$ is the inverse of $\delta$ . Below, $\delta^{-t}$ indicates t compositions of $\delta^{-1}$ with itself. We are now ready to state the main result on quantitative bounds.

Theorem 1. For all $j, t \in \mathbb N$ with $j \leqslant t$ , we have

\begin{equation*} \rho( \mu Q^t, \mu' Q^t) \leqslant 2(1-\varepsilon)^j + \frac{B^{\, j-1}}{\delta^{-t}(1)} [(\mu \times \mu')(W) + (\mu' \times \mu)(W)]. \end{equation*}

Theorem 1 provides a total ordered variation bound on the deviation between the time-t distributions $\mu Q^t$ and $\mu' Q^t$ generated by iteration with the Markov operator Q, taking as given an arbitrary pair of initial distributions $\mu, \mu'$ in $p\mathcal B$ . Convergence of the bound in Theorem 1 to zero as $t \to \infty$ requires that $\varepsilon > 0$ and $\delta^{-1}(1) \to \infty$ .

When $\delta$ is linear we obtain the geometric case. Since this case is important we state it as a corollary.

Corollary 1. If the conditions above hold with (6) replaced by

(10) \begin{equation} \hat Q W(x, x^{\prime}) \leqslant \gamma W(x, x^{\prime}) \quad \text{for all } (x,x^{\prime}) \notin C \times C \end{equation}

for some positive constant $\gamma$ , then, for all $j, t \in \mathbb N$ with $j \leqslant t$ ,

\begin{equation*} \rho( \mu Q^t, \mu' Q^t) \leqslant 2(1-\varepsilon)^j + \gamma^t B^{j-1} [(\mu \times \mu')(W) + (\mu' \times \mu)(W)]. \end{equation*}

Convergence of the bound in Corollary 1 to zero as $t \to \infty$ requires that the constants $\varepsilon$ and $\gamma$ obey $\varepsilon > 0$ and $\gamma \in [0, 1)$ . The bound can be viewed as an order-theoretic version of the geometric total variation bound in Theorem 1 in [Reference Rosenthal24]. Further comparisons are given in Section 5.1.

3.2. Sketch of the proof

The bound in Theorem 1 is obtained by tracking a joint chain $((X_t, X_t'))_{t \geqslant 0}$ generated by a $\preceq$ -maximal Markov coupling $\hat Q$ and started from initial condition $\mu \times \mu'$ . Because $\hat Q$ is a Markov coupling of Q, the individual chains $(X_t)_{t \geqslant 0}$ and $(X'_t)_{t \geqslant 0}$ are Markov- $(Q, \mu)$ and Markov- $(Q, \mu')$ , respectively. Taking $\tau$ to be the first time that $X_t \preceq X_t'$ occurs, we can use the fact that $\mathbb G$ is absorbing for $\hat Q$ (Lemma 2) to obtain

(11) \begin{equation} \tau \leqslant t \quad \text{if and only if} \quad X_t \preceq X_t'.\end{equation}

Next, an order-theoretic version of a standard total variation coupling argument is used to generate the bound $(\mu Q^t)(I) - (\mu' Q^t)(I) \leqslant \mathbb P \{X_t \npreceq X'_t\}$ for all $I \in i\mathcal B$ . In view of (11), the left-hand side is also bounded above by $\mathbb P \{\tau > t\}$ . This in turn is bounded with the use of the drift to $C \times C$ implied by (6), and the $\varepsilon$ -probability of the joint chain $((X_t, X_t'))_{t \geqslant 0}$ entering $\mathbb G$ after $C \times C$ implied by (9). Reversing the roles of $\mu$ and $\mu'$ and then adding the two inequalities leads to the bound in Theorem 1. Details are given in Section 4.

3.3. Univariate drift

In applications, drift conditions on the underlying kernel Q are usually easier to test and interpret than drift conditions on a joint kernel such as (6). Fortunately, there are relatively straightforward ways to map the former (let us call them ‘univariate’ drift conditions) to the latter (‘joint’ drift conditions). For example, suppose that Q is a transition kernel on $\mathbb X$ and V is a measurable function from $\mathbb X$ to $\mathbb R_+$ . Suppose there exist $\lambda, \beta \in \mathbb R_+$ such that $\lambda < 1$ and

(12) \begin{equation} QV(x) \leqslant \lambda V(x) + \beta \qquad \text{for all } x \in \mathbb X.\end{equation}

In this setting, we can attain (10) by setting $C = \{ x \in\mathbb X \, : \, V(x) \geqslant d \} $ for some fixed $d \geqslant 1$ , and then

\begin{equation*} W(x,x^{\prime}) = 1 + V(x) + V(x^{\prime}) \quad \text{ and } \quad \gamma = \frac{1 + \lambda d + 2\beta}{1 + d}.\end{equation*}

A proof that (10) holds with these definitions can be found in Theorem 12 in [Reference Rosenthal23].

Alternatively, if $V \geqslant 1$ then we can choose C in the same way and then set

\begin{equation*} W(x,x^{\prime}) = \frac{V(x) + V(x^{\prime})}{2} \quad \text{ and } \quad \gamma = \lambda + \frac{2\beta}{d}.\end{equation*}

Indeed, since $\hat Q$ is a Markov coupling of Q, an application of (12) yields

\begin{equation*} \hat Q W(x, x^{\prime}) = \frac{Q V(x) + QV(x^{\prime})}{2} \leqslant \lambda W(x, x^{\prime}) + \beta = \left( \lambda + \frac{\beta}{W(x, x^{\prime})}\right) W(x, x^{\prime}).\end{equation*}

The drift condition (10) now follows from $d/2 \leqslant W$ on the complement of $C \times C$ .

5.1. Connection to total variation results

One interesting special case of Theorem 1 is obtained by setting $\preceq$ to the identity order, so that $x \preceq y$ if and only if $x = y$ . For this order we have $ib\mathcal B = b\mathcal B$ , so every transition kernel is increasing, and, moreover, the total ordered variation distance becomes the total variation distance. In this setting, Theorem 1 becomes a version of well-known geometric bounds for total variation distance, such as Theorem 1 in [Reference Rosenthal24].

In the total variation setting, $\varepsilon$ in (9) is at least as large as the analogous term $\varepsilon$ in Theorem 1 in [Reference Rosenthal24]. Indeed, in [Reference Rosenthal24], the value $\varepsilon$ , which we now write as $\hat \varepsilon$ to avoid confusion, comes from an assumed minorization condition: there exists a $\nu \in p\mathcal B$ such that

(21) \begin{equation} \hat \varepsilon \nu(B) \leqslant Q(x, B) \quad \text{for all } B \in \mathcal B \text{ and } x \in C.\end{equation}

To compare $\hat \varepsilon$ with $\varepsilon$ defined in (9), suppose that this minorization condition holds and set $R(x, B) = (Q(x, B) - \hat \varepsilon \nu(B))/(1-\hat \varepsilon)$ . Fixing $(x, x^{\prime}) \in C \times C$ , we draw $(X, X^{\prime})$ as follows: With probability $\hat \varepsilon$ , we draw $X \sim \nu$ and set $X' = X$ . With probability $1-\hat \varepsilon$ , we independently draw $X \sim R(x, \cdot)$ and $X' \sim R(x^{\prime}, \cdot)$ . Simple arguments confirm that X is a draw from $Q(x, \cdot)$ and $ X^{\prime}$ is a draw from $Q(x^{\prime}, \cdot)$ . Recalling that $\preceq$ is the identity order, we obtain $\hat \varepsilon \leqslant \mathbb P\{X = X'\} = \mathbb P\{X \preceq X'\} \leqslant \alpha(Q(x, \cdot), Q(x^{\prime}, \cdot))$ . Since, in this discussion, the point (x,x’) was arbitrarily chosen from $C\times C$ , we conclude that $\hat \varepsilon \leqslant \varepsilon$ , where $\varepsilon$ is as defined in (9).

5.2. Connection to Wasserstein bounds

Theorem 1 is also connected to research on convergence rates for distributions in Wasserstein distance. To see this, recall that if d is a metric on $\mathbb X$ then the induced Wasserstein distance between probability measures $\mu$ and $\nu$ is expressed as

(22) \begin{equation} W_d(\mu, \nu) \,:\!=\, \inf_{\pi \in \mathscr {C} \, (\mu, \nu)} \int d(x, x^{\prime}) \pi(\mathop{}\!\mathrm{d} x, \mathop{}\!\mathrm{d} x^{\prime}).\end{equation}

To connect this distance to the total ordered variation metric, we use Theorem 3.1 in [Reference Kamihigashi and Stachurski16] to write

\begin{equation*} \sup\nolimits_{I \in i\mathcal B} (\mu(I) - \nu(I)) = \inf_{\pi \in {\mathscr{C}} \, (\mu, \nu)} \int \unicode{x1D7D9}\{ x \npreceq x^{\prime} \} \pi(\mathop{}\!\mathrm{d} x, \mathop{}\!\mathrm{d} x^{\prime}).\end{equation*}

Thus, if

(23) \begin{equation} s(x, x^{\prime}) = \unicode{x1D7D9}\big\{x \npreceq x^{\prime}\big\} \quad \text{and} \quad W_s(\lambda, \kappa) = \inf_{\pi \in {\mathscr {C}} \, (\lambda, \kappa)} \int s(x, x^{\prime}) \pi(\mathop{}\!\mathrm{d} x, \mathop{}\!\mathrm{d} x^{\prime}),\end{equation}

then, by the definition of $\rho$ in (2), we have

(24) \begin{equation} \rho(\mu, \nu) = W_s(\mu, \nu) + W_s(\nu, \mu).\end{equation}

We can understand s as a ‘directed semimetric’ that fails symmetry and positive definiteness but obeys $s(x,x)=0$ and the triangle inequality. The ‘directed Wasserstein semimetric’ $W_s$ inherits these properties. The sum of this directed Wasserstein semimetric and its reversed deviation creates a metric, as in (24). The inequalities (19) and (20), which we combined to prove Theorem 1, are just bounds on $W_s(\mu, \nu)$ and $W_s(\nu, \mu)$ . For example, (19) tells us that

\begin{equation*} W_s(\mu Q^t, \mu' Q^t) \leqslant (1 - \varepsilon)^j + \frac{B^{j-1}}{\delta^{-t}(1)} (\mu \times \mu')(W).\end{equation*}

The discussion above helps us understand the relationship between the order-theoretic mixing condition used in this article and the Wasserstein distance mixing condition in [Reference Butkovsky4]. In the latter, the notion of d-small sets is introduced in order to study Wasserstein distance convergence rates for distributions: for transition kernel Q, a Borel set C is called d-small if there exists an $\varepsilon > 0$ such that $W_d(Q_x, Q_{x^{\prime}}) \leqslant (1-\varepsilon) d(x, x^{\prime})$ for all $(x, x^{\prime}) \in C \times C$ . Here d is an arbitrary ground metric on $\mathbb X$ and $W_d$ is defined as in (22). By analogy, we replace d with s from (23) and call C s-small if there exists an $\varepsilon > 0$ such that

\begin{equation*} W_s(Q_x, Q_{x^{\prime}}) \leqslant (1-\varepsilon) s(x, x^{\prime}) \quad \text{for all } (x, x^{\prime}) \in C \times C.\end{equation*}

Fixing $x, x^{\prime} \in C$ and using the definition of s, we can equivalently write this as

(25) \begin{equation} \inf_{\pi} \pi(\mathbb G^c) \leqslant (1-\varepsilon) \unicode{x1D7D9}\{x \npreceq x^{\prime}\},\end{equation}

where $\mathbb G$ is as defined in (1) and the infimum is over all $\pi \in \mathscr {C} \, (Q_x, Q_{x^{\prime}})$ . Rearranging and using the definition of $\alpha$ in (3), we can also write (25) as

(26) \begin{equation} \alpha(Q_x, Q_{x^{\prime}}) \geqslant \unicode{x1D7D9}\{x \preceq x^{\prime}\} + \varepsilon \unicode{x1D7D9}\big\{x \npreceq x^{\prime}\big\} \quad \text{for all } (x, x^{\prime}) \in C \times C.\end{equation}

When Q is increasing, as required in Theorem 1, we can use (4) to obtain $\alpha(Q_x, Q_{x^{\prime}}) = 1$ whenever $x \preceq x^{\prime}$ . In this case, (26) is equivalent to $\alpha(Q_x, Q_{x^{\prime}})\geqslant \varepsilon$ whenever $(x, x^{\prime}) \in C \times C$ . Thus, the requirement that C is s-small is equivalent to the condition that we can extract a positive $\varepsilon$ in (9).

6.1. Stochastic recursive sequences

The preceding section showed that Theorem 1 reduces to existing results for bounds on total variation distance when the partial order $\preceq$ is the identity order. Next we illustrate how Theorem 1 can lead to new results in other settings. To this end, consider the process

(27) \begin{equation} X_{t+1} = F(X_t, \xi_{t+1})\end{equation}

where $(\xi_t)_{t \geqslant 1}$ is an independent and identically distributed (i.i.d.) shock process taking values in some space $\mathbb Y$ , and F is a measurable function from $\mathbb X \times \mathbb Y$ to $\mathbb X$ . The common distribution of each $\xi_t$ is denoted by $\varphi$ . We suppose that F is increasing, in the sense that $x \preceq x^{\prime}$ implies $F(x,y) \preceq F(x^{\prime},y)$ for any fixed $y \in \mathbb Y$ . We let Q represent the transition kernel corresponding to (27), so that $Q(x,B) = \varphi \{ y \in \mathbb Y \, : \, F(x,y) \in B \} $ for all $x \in \mathbb X$ and $B \in \mathcal B$ . Since F is increasing, the kernel Q is increasing. Hence, Theorem 1 applies. We can obtain a lower bound on $\varepsilon$ in (9) by calculating

(28) \begin{equation} e \,:\!=\, \inf \left\{ \int \int \unicode{x1D7D9}\{ F(x^{\prime}, y') \preceq F(x, y) \} \varphi(\mathop{}\!\mathrm{d} y) \varphi(\mathop{}\!\mathrm{d} y') \, : \, (x, x^{\prime}) \in C \times C \right\} \! . \end{equation}

To see this, fix $(x,x^{\prime}) \in C \times C$ and let $\xi$ and $\xi'$ be drawn independently from $\varphi$ . Since $X = F(x, \xi)$ is a draw from $Q(x, \cdot)$ and $X' = F(x^{\prime}, \xi)$ is a draw from $Q(x^{\prime}, \cdot)$ , we have $e \leqslant \mathbb P\{X' \preceq X\} \leqslant \alpha(Q(x, \cdot), Q(x^{\prime}, \cdot))$ . As this inequality holds for all $(x,x^{\prime}) \in C \times C$ , we obtain $e \leqslant \varepsilon$ .

To illustrate how these calculations can be used, consider the TCP window size process (see, e.g., [Reference Bardet, Christen, Guillin, Malrieu and Zitt2]) with embedded jump chain $X_{t+1} = a (X_t^2 + 2E_{t+1})^{1/2}$ . Here $a \in (0,1)$ and $(E_t)$ is i.i.d. exponential with unit rate. If $C = [0, c]$ , then drawing E and E’ as independent standard exponentials and using (28), we obtain

\begin{equation*} e = \inf_{0 \leqslant x, y \leqslant c} \mathbb P \big\{ a (y^2 + 2E')^{1/2} \leqslant a (x^2 + 2E)^{1/2} \big\} = \mathbb P\{ c^2 + 2E' \leqslant 2E \} .\end{equation*}

Since $E' - E$ has the Laplace-(0,1) distribution, we can use $e \leqslant\varepsilon$ to get

\begin{equation*} 1 - \varepsilon \leqslant 1 - e = \mathbb P\{ c^2 + 2E' > 2E \} = \mathbb P\{ E' - E > c^2/2 \} = \tfrac{1}{2} \exp(\! - c^2/2 ).\end{equation*}

6.2. Example: when minorization fails

We provide an elementary scenario where Theorem 1 provides a usable bound, while the minorization based methods described in Section 5.1 do not. Let $\mathbb Q$ be the rational numbers, let $\mathbb X = \mathbb R$ , and assume that

\begin{equation*} X_{t+1} = \frac{X_t}{2} + \xi_{t+1}, \quad \text{where $\xi_t$ is i.i.d. on $\{0,1\}$ and $\mathbb P\{\xi_t = 0\} = 1/2$}.\end{equation*}

Let C contain at least one rational number and one irrational number. Let $\mu$ be a measure on the Borel sets of $\mathbb R$ obeying $\mu(B) \leqslant Q(x, B) = \mathbb P\{x/2 + \xi \in B\}$ for all $x \in C$ and Borel sets B. If x is rational then $x/2 + \xi \in \mathbb Q$ with probability 1, so $\mu(\mathbb Q^c)\leqslant Q(x, \mathbb Q^c) = 0$ . Similarly, if x is irrational then $x/2 + \xi \in \mathbb Q^c$ with probability 1, so $\mu(\mathbb Q) \leqslant Q(x, \mathbb Q) = 0$ . Hence, $\mu$ is the zero measure on $\mathbb R$ . Thus, we cannot take a $\hat \varepsilon > 0$ and probability measure $\nu$ obeying the minorization condition (21). On the other hand, if we let $C = [0, 1]$ , the value e from (28) obeys $e = \mathbb P\{1/2 + \xi \leqslant \xi'\} = \mathbb P\{\xi' - \xi \geqslant1/2 \} = \frac{1}{4}$ . Since $e \leqslant \varepsilon$ (see the discussion after (28)), the constant $\varepsilon$ in (9) is positive.

6.3. Example: wealth dynamics

Many economic models examine wealth dynamics in the presence of credit market imperfections (see, e.g., [Reference Antunes and Cavalcanti1]). These often result in dynamics of the form

(29) \begin{equation} X_{t+1} = \eta_{t+1} \, G(X_t) + \xi_{t+1}, \quad (\eta_t) \stackrel {\textrm{i.i.d.}} {\sim} \varphi, \quad (\xi_t) \stackrel {\textrm{i.i.d.}} {\sim} \psi.\end{equation}

Here $(X_t)$ is some measure of household wealth, G is a function from $\mathbb R_+$ to itself, and $(\eta_t)$ and $(\xi_t)$ are independent $\mathbb R_+$ -valued sequences. The function G is increasing, since greater current wealth relaxes borrowing constraints and increases financial income. We assume that there exists a $\kappa < 1$ such that $\mathbb E \,\eta_t G(x) \leqslant \kappa x$ for all $x \in \mathbb R_+$ , and, in addition, that $\bar \xi \,:\!=\, \mathbb E \xi_t < \infty$ .

Let Q be the transition kernel corresponding to (29). With $V(x) = x$ , we have

\begin{equation*} QV(x) = \mathbb E [ \eta_{t+1} \, G(x) + \xi_{t+1} ] \leqslant \kappa x + \bar \xi = \kappa V(x) + \bar \xi .\end{equation*}

Fixing $d \in \mathbb R_+$ and setting $C = \{V \leqslant d\} = [0, d]$ , we can obtain e in (28) via

\begin{equation*} e = \mathbb P \{\eta' G(d) + \xi' \leqslant \eta G(0) + \xi\} \quad \text{when } \quad (\eta', \xi', \eta, \xi) \sim \varphi \times \psi \times \varphi \times \psi.\end{equation*}

This term, which provides a lower bound for $\varepsilon$ , will be strictly positive under suitable conditions, such as when $\psi$ has a sufficiently large support. By the discussion in Section 3.3, the drift condition (10) holds with $W(x,x^{\prime}) = 1 + V(x) + V(x^{\prime})$ and $\gamma$ set to $(1 + \kappa d + 2\bar \xi)/(1 + d)$ . The function W is bounded above by $2d+1$ on $C \times C$ , so we can set $B = 2d+1$ . With $\gamma$ and B so defined, the bound in Corollary 1 is valid.

Notice that for this model, we cannot compute useful total variation or Wasserstein bounds without adding more assumptions.