## Abstract

Dealing with finite Markov chains in discrete time, the focus often lies on convergence behavior and one tries to make different copies of the chain meet as fast as possible and then stick together. There are, however, discrete finite (reducible) Markov chains, for which two copies started in different states can be coupled to meet almost surely in finite time, yet their distributions keep a total variation distance bounded away from 0, even in the limit as time tends to infinity. We show that the supremum of total variation distance kept in this context is \(\tfrac{1}{2}\).

## Keywords

Markov chain Non-Markovian coupling Total variation distance Coupling inequality## Mathematics Subject Classification (2010)

Primary 60J10 Secondary 60C05## 1 Introduction

When the long-time behavior of Markov chains is analyzed, one of the most common strategies is to couple several copies of the chain started in different states. In doing so, one standard approach is to define two copies of a Markov chain (started in different states) on a common probability space, correlated in such a way that they are likely to meet within some moderate time, and glue them together as soon as this happens.

This idea is so predominant that little attention was directed away from such couplings; in the standard reference [6], it was even claimed erroneously that *any* coupling of two Markov chains with the same transition probabilities can be modified so that the two chains stay together at all times after their first simultaneous visit to a single state. A counterexample to this statement was in fact given in [8]: If a coupling of two copies of the same Markov chain is changed in such a way that the second copy mimics the behavior of the first one once they meet, the altered individual process might no longer be a copy of the given chain.

For the sake of simplicity, we want to restrict our considerations to time-homogeneous Markov chains evolving in discrete time and on countable state spaces—except for Remark 1 and Theorem 4.3, where we discuss how the argument used to derive Theorem 4.1 applies to more general settings as well. So let \({{\mathbf {X}}}=(X_n)_{n\in {\mathbb {N}}_0}\) denote a Markov chain on a countable state space *S* with transition probabilities \(\{P(r,s)={\mathbb {P}}(X_{n+1}=s\,|\,X_n=r);\ r,s\in S,\ n\in {\mathbb {N}}_0\}\). While \({\mathcal {L}}(X_n)\) will be used as shorthand notation for the distribution of \(X_n\) in general, we will denote the distribution of \(X_n\) given \(X_0=x\), i.e., for a copy of the chain started in \(x\in S\), by \(P^n(x,.)\).

*segregation*of two states. Further, we consider the constant

*P*and states

*x*and

*y*, such that two copies of the chain corresponding to

*P*, one started in

*x*and the other in

*y*, can be coupled to meet a.s. in finite time. To put it briefly, the main result of this paper is that \(\kappa \) equals \(\tfrac{1}{2}\).

As a preparation, the second section deals with the concept of couplings in general and convergence of Markov chains. Much of this is standard, but there is also the lesser known but crucial distinction between Markovian and faithful couplings. Section 3 presents Häggström’s result (\(\kappa \ge 3-2\sqrt{2}\)) and puts the idea of segregating Markov chains into a broader context.

In Sect. 4, more precisely in Theorem 4.1, we prove that the value \(\tfrac{1}{2}\) is an upper bound on \(\kappa \).

In Sect. 5, a constructive and intuitively accessible example of a Markov chain is given, that segregates two states such that the total variation distance kept can be pushed arbitrarily close to \(\tfrac{1}{\mathrm {e}}\). This improves on the example in [3] and serves as a warm up for the more technical and implicit construction in the last section.

Finally, in Sect. 6 we introduce and employ the idea of *separation* to show that for any \(\varepsilon >0\), there exist Markov chains segregating two states *x* and *y* such that copies started in these states can be coupled to meet almost surely while their distributions \(P^n(x,.)\) and \(P^n(y,.)\) have a total variation distance of at least \(\tfrac{1}{2}-\varepsilon \) for all \(n\in {\mathbb {N}}\), see Theorem 6.2. Together with the upper bound from Sect. 4, this establishes our main result, Theorem 6.1, stating that \(\kappa =\tfrac{1}{2}\).

## 2 Preliminaries: Convergence and Couplings

In order to quantify the difference between two probability measures (such as the distributions of two copies of a Markov chain at a fixed time), there are quite a few distance measures. The so-called total variation distance is among the most common ones.

### Definition 1

*S*. The

*total variation distance*between the two measures is then defined as

This notion of distance is used in most of the standard convergence theorems on finite Markov chains as well (e.g., see Thm. 4.9 in [6]):

### Theorem 2.1

*S*. Then, there exists a unique limiting distribution \(\pi \) on

*S*, called the stationary distribution, as well as constants \(\alpha \in (0,1)\) and \(C>0\) such that

*n*converges to the same distribution \(\pi \) as

*n*tends to infinity, irrespective of its starting distribution, a standard way to measure the speed of convergence is the variation distance

*d*and \({\overline{d}}\), are non-increasing in

*n*and \({\overline{d}}\) is in addition submultiplicative, i.e., \({\overline{d}}(m+n)\le {\overline{d}}(m)\cdot {\overline{d}}(n)\). Submultiplicativity need not hold for

*d*, but can be verified for \(2\,d\) instead. Furthermore, it holds that \(d(n)\le {\overline{d}}(n)\le 2\,d(n)\). For proofs of the elementary facts just stated, we refer to Lemma 2.20 in [1]. Note that there

*S*is assumed to be finite, but the arguments immediately transfer to countable

*S*.

On the basis of the notion of distance *d*, the central concept of mixing time is defined, loosely speaking, as the time it takes until the effect of the starting distribution has begun to disappear substantially.

### Definition 2

*mixing time*by

As already mentioned, the tool that often makes proofs about convergence of Markov chains both short and elegant is the coupling approach. Let us therefore properly define this standard concept and then highlight which additional properties a coupling can have.

### Definition 3

We define a *coupling* of two copies of a Markov chain on *S* to be a process \(((X_n,Y_n))_{n\in {\mathbb {N}}_0}\) on \(S\times S\), with the property that both \((X_n)_{n\in {\mathbb {N}}_0}\) and \((Y_n)_{n\in {\mathbb {N}}_0}\) are Markov chains on *S* with the same transition probabilities (but possibly different starting distributions).

If the process \(((X_n,Y_n))_{n\in {\mathbb {N}}_0}\) is itself a Markov chain (not necessarily time-homogeneous), it is called a *Markovian coupling*.

In order to get good estimates on mixing times, it is often of importance to bring into line the long-term behavior of the chain started in different states. In order to do so, one wants to make sure that the two coupled chains stay together once they meet, more precisely: if \(X_m=Y_m,\) then \(X_n=Y_n,\) for all \(n\ge m\). Couplings with this property are sometimes called “*sticky*” couplings. As noted in the introduction, it is however not possible to modify every coupling in such a way that it becomes sticky by simply glueing together the two copies once they meet, see Prop. 3 in [8] for an example. The crucial property is the following:

### Definition 4

*faithful*if for all \(x_n,y_n,x_{n+1},y_{n+1}\in S\), \(n\in {\mathbb {N}}_0\):

It should be mentioned that the term “Markovian coupling” is used in [6] to describe what we just defined as faithful coupling. However, since we actually want to focus on couplings that are not faithful (but may still be Markov chains—as both the example in Sect. 5 and the one in [3] are), we want to make this distinction by adopting the definitions in [8] and deviate from the notions in [6].

*identical*copies of the same chain, without affecting the marginals. Exploiting this fact leads to the estimate

*x*and \({{\mathbf {Y}}}\) started in

*y*, where

*first meeting time*of the coupled chains (cf. Thm. 1 in [8]).

## 3 Chains that Meet and Separate

If two copies of a Markov chain are coupled, but the coupling is not sticky, clearly they can meet in one state and separate afterward. As mentioned above, if the coupling is not faithful (i.e., violates the conditions given in Definition 4), in some cases it cannot be transformed into a sticky coupling by simply letting the two copies coalesce once they meet. As a by-product, Häggström [3] observed an even stronger form of incompatibility of two coupled copies of a chain that meet. He gives an example of a finite reducible Markov chain with the following property: Two copies of the chain, started in different states *x* and *y*, can be coupled in such a way that they meet almost surely in finite time, while the total variation distance of their distributions never drops below a fixed positive value. More precisely, he shows (see Prop. 4.1 in [3]):

### Proposition 3.1

*x*and

*y*we have that

Note that for any Markov chain and any two states *x* and *y*, the sequence \((||P^n(x,.)-P^n(y,.)||_{\mathrm {TV}})_{n\in {\mathbb {N}}_0}\) is non-increasing. This, together with the fact that the total variation distance is always nonnegative, guarantees the existence of \(\lim _{n\rightarrow \infty }||P^n(x,.)-P^n(y,.)||_{\mathrm {TV}}\).

*x*and

*y*can be coupled such that their first meeting time is a.s. less than or equal to 2 (for the explicit calculations, see Prop. 4.1 in [3]). The copies will reach one of the two absorbing states (

*a*and

*b*) after two steps and the probability that the chain started in

*x*lands in

*a*is \(1-2p\,(1-p)\), in

*b*accordingly \(2p\,(1-p)\). By symmetry, for the chain started in

*y*it is precisely reversed.

So for \(n\ge 2\), \(P^n(x,.)\) and \(P^n(y,.)\) are unchanging and different if \(p\ne \frac{1}{2}\). Choosing \(p=\frac{1}{2}\sqrt{2}\) (or \(p=1-\frac{1}{2}\sqrt{2}\)) maximizes their total variation distance at \(3-2\sqrt{2}\approx 0.17153\).

As mentioned in the introduction, we will call Markov chains that have the property described in Proposition 3.1 to be *segregating the states x and y*. From the convergence theorem, we know that such things cannot happen for *irreducible* finite Markov chains (not even periodic ones). In addition to that, even if the chain is either reducible or infinite, a coupling that lets two copies started in different states meet almost surely, while the total variation distance of their distributions stays bounded away from 0 for all time, cannot be faithful due to the coupling inequality (2).

## 4 The Upper Bound

From the previous section, we know that there exist finite reducible Markov chains that *segregate two states x and y*. A natural question in this respect is how large a total variation distance between \(P^n(x,.)\) and \(P^n(y,.)\) can be kept, under the condition that two copies started in *x* and *y*, respectively, can be coupled to meet in finite time almost surely—in other words, the value of \(\kappa \) as defined in (1). The example in [3] shows \(\kappa \ge 3-2\sqrt{2}\); the following theorem establishes \(\tfrac{1}{2}\) as an upper bound.

### Theorem 4.1

*S*and two fixed states

*x*and

*y*. Further, we denote by \({\mathbf X}=(X_n)_{n\in {\mathbb {N}}_0}\) and \({{\mathbf {Y}}}=(Y_n)_{n\in {\mathbb {N}}_0}\) two coupled copies of the chain, started in

*x*and

*y*, respectively, and their first meeting time by \(\tau \). If \({{\mathbf {X}}}\) and \({{\mathbf {Y}}}\) can be coupled in such a way that \({\mathbb {P}}(\tau <\infty )=1\), it holds that

This result is an immediate implication of the following proposition:

### Proposition 4.2

### Proof

*n*if the corresponding set is empty. Note that \(\tau _x\) and \(\tau _y\) are stopping times for \(M_t\) and \(N_t\), respectively. Since \((M_t)_{t=0}^n\) and \((N_t)_{t=0}^n\) are bounded martingales, the Optional Stopping Theorem (see for example Cor. 17.7 in [6]) gives the estimates

### Remark 1

Reading carefully through the proof of Proposition 4.2, one may notice that the martingale argument used essentially does not require our general assumptions of time-homogeneity and countable state space. For time-inhomogeneous chains, \(M_t:={\mathbb {P}}\left( X_n \in A \,|\, X_t\right) \), can no longer be written as \(P^{n-t}(X_t, A)\) (likewise for \(N_t\)), but this does not impair the argument and we can again conclude \(\lim _{n\rightarrow \infty }||{\mathcal {L}}(X_n)-{\mathcal {L}}(Y_n)||_{\mathrm {TV}}\le \tfrac{1}{2}\), where \({\mathcal {L}}(X_n)\) denotes the distribution of \(X_n\). Given an uncountable state space, the first meeting time \(\tau \) is no longer measurable by default. If we add this as an extra condition, however, the above proof (with the minor modification that only measurable sets *A* are considered) extends to this setting as well.

### Theorem 4.3

*S*. Let \({\mathbf X}=(X_t)_{t\ge 0}\) and \({{\mathbf {Y}}}=(Y_t)_{t\ge 0}\) denote two coupled copies of the process, that are started in fixed states

*x*and

*y*respectively, and let \(\tau \) denote their first meeting time. Fix a time horizon \(T>0\) and assume that \(\{\tau \le T\}\) is measurable. If for all measurable sets \(A\subseteq S\), it is possible to choose versions of the martingales \((M_t)_{t\in [0,T]}\) and \((N_t)_{t\in [0,T]}\), as defined in (4), that are a.s. continuous from the right, while having the property that for all \(t\in [0,T]\), \(X_t=Y_t\) implies \(M_t=N_t\), it holds that

### Proof

Again, we fix some measurable subset \(A\subseteq S\) and define the martingales \((M_t)_{t\in [0,T]}\) and \((N_t)_{t\in [0,T]}\) as in (4), with the two additional properties stated in the theorem. Following the proof of Proposition 4.2 (literally, besides replacing *n* by *T*), we can still conclude that \(\{\tau \le T\}\cap (B_x\cap B_y)\) is a nullset. Continuity from the right of \((M_t)_{t\in [0,T]}\) and \((N_t)_{t\in [0,T]}\) implies \(M_{\tau _x}\le \frac{1}{2}\) on \(B_x^{\,\text {c}}\) and \(N_{\tau _y}\ge \frac{1}{2}\) on \(B_y^{\,\text {c}}\). Using the Optional Stopping Theorem for continuous-time martingales (see e.g., Thm. (3.2) in [7]), we can conclude just as above. \(\square \)

### Remark 2

A simple way to ensure the assumed properties of \((M_t)_{t\in [0,T]}\) and \((N_t)_{t\in [0,T]}\) is to consider a topology on *S* and to require two things: first, that the Markov process a.s. has right-continuous sample paths and second that the transition probabilities \({\mathbb {P}}(X_T\in A \,|\, X_t=x)\) are continuous in \(t\in [0, T)\) and \(x\in S\) for all measurable \(A\subseteq S\).

Besides these generalizations, the statement from Proposition 4.2 can also be used to get upper bounds on mixing times—similar to the usual approach, see for example Cor. 5.3 in [6]—replacing the coupling inequality (2) as starting point. In doing so, we pay by an additional factor \(\tfrac{1}{2}\) in front of \({\mathbb {P}}(\tau \le n)\), but can in return employ any kind of coupling, not only faithful ones. It remains to be seen whether this will ever turn out useful in practice as basically all standard coupling constructions are faithful. However, we want to mention at this point that non-Markovian couplings actually already proved to be useful in applications, cf. [4] for instance.

### Proposition 4.4

*x*and

*y*, respectively, such that the first meeting time \(\tau \) of the two coupled processes fulfills \({\mathbb {P}}(\tau \le n)\ge \alpha \). Then

### Proof

*d*, we get for any \(k\in {\mathbb {N}}:\)

## 5 A Simple Example that Narrows the Gap

In this section, we will present another finite state Markov chain that improves on the value of \(3-2\sqrt{2}\) established by Häggström [3]. To begin with, let us prepare a lemma, which will come in useful when the total variation distance in our example of a finite segregating Markov chain is to be assessed.

Consider a sequence of independent Bernoulli trials, each with success probability \(p<1\). The distribution of the number of successful attempts until *r* failures have occurred is called the negative binomial distribution with parameters *r* and *p* and commonly denoted by \(\mathrm {NB}(r,p)\).

### Lemma 5.1

### Proof

*S*, their total variation distance can be calculated as

Let us now use this lemma to establish the following result:

### Proposition 5.2

*x*and

*y*such that

### Proof

*S*comprises \(3m+5\) states, among which the two initial states

*x*and

*y*as well as the \(m+2\) absorbing states labeled \(0,\dots ,m\) and >.

*x*respectively

*y*, will hit an absorbing state after at most \(m+2\) steps and that on \(\{0,\dots ,m\}\) the distributions \(P^{m+2}(x,.)\) and \(P^{m+2}(y,.)\) coincide with \(\mathrm {NB}(1,p)\) and \(\mathrm {NB}(2,p)\), respectively. The probabilities to land in the state labeled > are

*m*large enough, more precisely such that \(\big (1-\frac{1}{m+1}\big )^{m+1}> \frac{1}{\mathrm {e}}-\varepsilon \), will establish the claim if we can present a coupling that ensures that the two copies started in

*x*and

*y*will meet with probability 1, either before or when they hit an absorbing state.

In order to establish such a coupling, let \({\mathbf Y}=(Y_n)_{n\in {\mathbb {N}}_0}\) be a copy of the MC started in *y*. The copy \({{\mathbf {X}}}=(X_n)_{n\in {\mathbb {N}}_0}\) started in *x* mimics all movements of \({{\mathbf {Y}}}\), with the delay of one step, until it finally hits an absorbing state: First, it will move downwards until the two processes meet—in particular this implies that its first step is downwards with probability 1, as \(x\ne y\). Then, once \(X_n=Y_n\) for some \(1\le n\le m+1\), the next step of the process \({{\mathbf {X}}}\) is to move to the right to an absorbing state, i.e., \(X_{n+1}=n-1\). If \({{\mathbf {Y}}}\) never moves to the right, neither does \({\mathbf X}\) and both finally end up in the state >.

First of all, we need to check whether the two coordinate processes are indeed copies of the MC given in Fig. 2: It is obvious from our construction that it suffices to verify this for the process \({{\mathbf {X}}}\). The way \({{\mathbf {X}}}\) is defined—to move downwards in the first step and then always imitate the previous move of \({{\mathbf {Y}}}\) until ending up in an absorbing state—gives the right marginals due to the structure of the MC: As all the non-absorbing states apart from *x* have the same transition probabilities (*p* downwards and \(1-p\) to the right), \({{\mathbf {X}}}\) performs indeed a random walk on the graph in Fig. 2 according to the transition probabilities of the MC. Note that there is just this one way \({{\mathbf {Y}}}\) can end up in an absorbing state before it meets \({{\mathbf {X}}}\), namely if it moves downwards only. Then, however, \({{\mathbf {X}}}\) copies this behavior and ends up in the state > as well, so the coupling guarantees \(\tau \le m+2\) with probability 1. This trivially implies the almost sure finiteness of the first meeting time \(\tau \) and in conclusion the claim that the MC segregates the two states *x* and *y*. \(\square \)

In order to get the idea of how faithfulness plays a crucial role in this context, it is worth noting that the coupling in our example—likewise the one in [3]—is in fact Markovian, but not faithful. Such couplings, however, have to be non-faithful as faithfulness would imply that the total variation distance necessarily tends to 0 as already mentioned at the end of Sect. 3.

## 6 Closing the Gap

In this last section, we want to further improve the lower bound \(\frac{1}{\mathrm {e}}\), established by the example from the previous section, in order to determine the true value of the constant \(\kappa \), defined in (1). These efforts amount to the following:

### Theorem 6.1

The value of \(\kappa \), denoting the supremum of \(\lim _{n\rightarrow \infty }||P^n(x,.)-P^n(y,.)||_{\mathrm {TV}}\) taken over all segregating Markov chains and segregated states *x* and *y*, is \(\tfrac{1}{2}\).

*S*with transition probabilities \(\{P(r,s);\ r,s\in S\}\), we are in fact thinking of a different chain, namely the reducible Markov chain \({{\mathbf {Y}}}=(Y_n)_{n\in {\mathbb {N}}_0}\) on the state space \(S\times \{0,\dots ,T\}\), with transition probabilities

*T*by making all states corresponding to time

*T*absorbing. Incidentally, the example given in [3] is also of this kind; it corresponds to a two state Markov chain with \(T=2\) (cf. Fig. 1).

So for the remainder of this article, we actually consider Markov chains in discrete, finite time and with finite state space only. With this simplification in mind, we want to prove the following:

### Theorem 6.2

*x*and

*y*and a positive integer

*T*such that

*x*and

*y*, respectively satisfying

Before proceeding with a proof of this theorem, we want to introduce an alternative way to view segregating couplings. Let \({\mathbf {X}}\) be any discrete-time Markov chain with a finite state space *S*, and fix two states \(x, y\in S\) and a positive integer *T*.

*meeting probability*

*x*,

*y*and

*T*by maximizing over all such couplings. Let \({\mathcal {X}}\) denote the subset of \(\{x\}\times S^{T}\) consisting of all possible trajectories of \({\mathbf {X}}=(X_t)_{t=0}^T\), and, respectively, \({\mathcal {Y}}\subseteq \{y\}\times S^{T}\) for \({\mathbf {Y}}=(Y_t)_{t=0}^T\). Observe that any coupling of \({\mathbf {X}}\) and \({\mathbf {Y}}\) is determined by the values \((p_{{{\mathbf {x}}}{{\mathbf {y}}}})_{{\mathbf {x}}\in {\mathcal {X}}, {\mathbf {y}}\in {\mathcal {Y}}}\), where \(p_{{{\mathbf {x}}}{{\mathbf {y}}}}\) denotes the probability of the event that both \({\mathbf {X}}={\mathbf {x}}=(x_t)_{t=0}^T\) and \({\mathbf {Y}}={\mathbf {y}}\). We will denote the maximal value of (6) by \({\mathcal {C}}_T(x,y)\) and call it the

*optimal meeting probability*.

While finding explicit couplings that maximize the meeting probability can quickly become cumbersome as the number of possible trajectories grows, it turns out that the problem of optimizing the meeting probability has a useful dual, which allows us to determine \({\mathcal {C}}_T(x,y)\) without having to deal with the couplings directly. This duality corresponds to the idea of max-flow min-cut and König’s theorem in combinatorial optimization.

### Definition 5

*S*, the (finite) state space of the considered Markov chain \({\mathbf {X}}\). We will refer to any such sequence as a

*separating sequence*. We define the

*separation*of any separating sequence as

*non-trivial*if both summands on the right-hand side in (7) are nonzero. We further define the

*optimal separation*\({\mathcal {S}}_T(x,y)\) as the maximum separation over all possible separating sequences.

*x*,

*y*respectively, and minimizing the upper bound by maximizing the separation \({\mathcal {S}}^{{\mathcal {A}}}_T(x,y)\) over all separating sequences yields

### Theorem 6.3

*T*, we have

### Proof

A simple way to prove the reverse inequality is to employ the max-flow min-cut theorem, in the same way it can be used to prove Strassen’s monotone coupling theorem. Starting from the sets \({\mathcal {X}}\) and \({\mathcal {Y}}\) as above, we build the following directed graph, which will be denoted by \(\vec {G}=(V,\vec {E})\):

First we let each \({\mathbf {x}}\in {\mathcal {X}}\) and \({\mathbf {y}}\in {\mathcal {Y}}\) be represented by a node. Then, we add two further nodes: a source *s* and a sink *t*. When it comes to the directed edges, there will be an arrow \((s,{\mathbf {x}})\) for all \({\mathbf {x}}\in {\mathcal {X}}\) and \(({\mathbf {y}},t)\) for all \({\mathbf {y}}\in {\mathcal {Y}}\). Additionally, we include the edge \(({\mathbf {x}},{\mathbf {y}})\), if the two trajectories \({\mathbf {x}}\in {\mathcal {X}}\) and \({\mathbf {y}}\in {\mathcal {Y}}\) share at least one state, i.e., \(x_t=y_t\) for some \(0\le t \le T\); in the sequel, we will write this as \({\mathbf {x}}\sim {\mathbf {y}}\).

Let us now consider the minimum cut problem on \(\vec {G}\). From the fact that the cut \((\{s\},V{\setminus }\{s\})\) has value 1, we know that we can focus on cuts that are not cutting any edges going in between \({\mathcal {X}}\) and \({\mathcal {Y}}\) when trying to find a minimal one. For a cut \((B,B^\text {c})\) of this kind, with \(s\in B\) and \(t\in B^\text {c}\) say, we note that \({\mathbf {x}}\sim {\mathbf {y}}\) can not occur for \({\mathbf {x}}\in {\mathcal {X}}\cap B\) and \({\mathbf {y}}\in {\mathcal {Y}}\cap B^\text {c}\), due to the assumption that no such edges are cut. Furthermore, the edges cut incident to *s* have at least the value \({\mathbb {P}}(X\in {\mathcal {X}}\cap B^\text {c})\), the ones incident to *t* at least \({\mathbb {P}}(Y\in {\mathcal {Y}}\cap B)\).

From the flow constraints, we know that all these probabilities are in [0, 1] and the resulting coupling satisfies \({\mathbb {P}}({\mathbf {X}}\sim {\mathbf {Y}})\ge 2-{\mathcal {S}}_T(x,y)\). The theorem then follows by combining this inequality with (8).\(\square \)

*x*,

*y*and

*T*as above, the following two statements are equivalent:

- (a)
The meeting probability under optimal coupling of two copies, started in

*x*,*y*respectively, is 1. - (b)
For all non-trivial separating sequences \({\mathcal {A}}= (A_t)_{t=0}^T\), we get \({\mathcal {S}}_T^{{\mathcal {A}}} (x,y) \le 1\).

### Example 1

Let \(0 < \alpha \le \frac{1}{2}\). Consider the Markov chain \({\mathbf {X}}\) with state space \(\{0, 1\}\) and transition probabilities \(P(0,1) = P(1,0) = \alpha \) as well as \(P(0,0)=P(1,1) = 1-\alpha \), and take \(x=0,\ y=1\). As mentioned above, the case where \(T=2\) is Häggström’s [3] example of a segregating Markov chain.

*t*is obviously best possible. It is immediate to check that its separation equals \(2(1-\alpha )^T\). Hence, the optimal meeting probability of two copies, started in states 0 and 1, respectively, is 1 if and only if \(2(1-\alpha )^T \le 1\). Using induction, one can easily check that for this chain we have

The next example is supposed to illustrate that reducible Markov chains obtained from irreducible and aperiodic finite chains in finite time, in the way described before Theorem 6.2, usually do segregate any two states:

### Example 2

*S*, and let

*x*and

*y*be any two states. Pick \(\varepsilon >0\) and \(n\in {\mathbb {N}}\) such that \(P^n(x', y') \ge \varepsilon \) for all \(x',y'\in S\). Then, for any non-trivial separating sequence \((A_t)_{t=0}^{nk}\), we have

*k*it follows that the optimal meeting probability during [0,

*T*] is 1. This shows that, unless \(||P^T(x,.)- P^T(y,.)||_{\mathrm {TV}}=0\), the Markov chain segregates the two states

*x*and

*y*, choosing

*T*sufficiently large.

*L*and transition probabilities given by \(P(0, 1) = P(L, L-1) = 1-P(0, 0) = 1-P(L, L) = \alpha \) as well as \(P(i, i+1) = P(i, i-1) = \frac{1}{2}\) for all \(0< i < L\), see Fig. 4. Such chains, with \(S=\{0, 1, \dots , L\}\) and the additional property that \(|X_{t+1}-X_t|\le 1\) a.s. for all

*t*, are commonly called finite

*birth-and-death chains*, cf. Section 2.5 in [6]. To begin with, our main interest lies in the optimal meeting probability of this chain given the starting states \(x=0\) and \(y=L\). We will then show that we can obtain segregation between 0 and

*L*with total variation arbitrarily close to \(\frac{1}{2}\) by choosing

*L*,

*T*and \(\alpha \) appropriately.

The qualitative behavior of this chain for small \(\alpha \) is easy to describe: Most of the time, the process is either at 0 or *L*. Occasionally, that is at rate \(\alpha \), the process takes one step inwards and typically moves around for order *L* steps before hitting one of the marginal states 0 or *L*, more precisely the expected time to reach \(\{0,L\}\), when starting in state 1 equals \(L-1\). With probability \(1-\frac{1}{L}\), it will return to the same side it started at, with probability \(\frac{1}{L}\) it will cross over to the opposite side (check the analysis of the so-called gambler’s ruin problem in Section 2.1 in [6] for the explicit calculations).

In preparation to the in-depth analysis of the separation of states 0 and *L*, let us collect a few general estimations for this chain in the following lemma, which will come in useful later on. For the sake of clarity, we will use the standard big *O* notation to represent error terms, i.e., for any nonnegative function *f* in *k* and \(\alpha \), the expression \(O(f(k,\alpha ))\) denotes a quantity that is bounded in absolute value by \(c\cdot f(k,\alpha )\), where the constant \(c>0\) does not depend on \(k,\ \alpha \) or *t*.

### Lemma 6.4

- (a)For any \(1 \le i \le L-1\) and \(t\ge 0\) we have$$\begin{aligned} \begin{aligned} P^t(0, i)=P^t(L, L-i)\le 2\alpha . \end{aligned} \end{aligned}$$
- (b)For any \(t\ge 0\) we have$$\begin{aligned} P^t(0, 0) = P^t(L, L) = \frac{1}{2} + \frac{1}{2} \left( 1-\frac{2\alpha }{L}\right) ^t + O(L\alpha ), \end{aligned}$$(10)$$\begin{aligned} ||P^t(0,.)- P^t(L,.)||_{\mathrm {TV}} = \left( 1-\frac{2\alpha }{L}\right) ^t + O(L\alpha ). \end{aligned}$$(11)
- (c)For any \(0\le k \le L-1\) and \(t\ge 0\) we have$$\begin{aligned} \begin{aligned}&{\mathbb {P}}\left( X_{t'} \le k \text { for all }0\le t' \le t\,|\, X_0=0\right) \\&\quad = {\mathbb {P}}\left( X_{t'} \ge L- k \text { for all }0\le t' \le t\,|\, X_0=L\right) \\&\quad = \left( 1 - \frac{\alpha }{k+1}\right) ^t + O(k\alpha ). \end{aligned} \end{aligned}$$(12)

### Proof

- (a)The first statement easily follows by induction on
*t*using the recursionwhich holds for any \(t\ge 0\) and any \(1\le i \le L-1\).$$\begin{aligned} P^{t+1}(0, i) = P^t(0, i-1)\, P(i-1, i) + P^t(0, i+1)\, P(i+1, i), \end{aligned}$$ - (b)To show the second claim, consider the sequence \(a_t = {\mathbb {E}}\left[ X_t \,|\, X_0=0\right] \). Using part (a), we know thatwhere the error terms are bounded by \(2\,L^2\alpha \) in absolute value. Furthermore, since \({\mathbb {E}}\left[ X_{t+1}-X_t\,|\, X_t\right] \) equals \(\alpha \) if \(X_t=0\), \(-\alpha \) if \(X_t=L\) and 0 otherwise, we can infer from (13)$$\begin{aligned} a_t = L\cdot P^t(0, L)+O(L^2\alpha )=L-L\cdot P^t(0, 0) + O(L^2\alpha ), \end{aligned}$$(13)which implies$$\begin{aligned} a_{t+1}&= a_t + \alpha \cdot P^t(0, 0) - \alpha \cdot P^t(0, L)\\&= a_t + \alpha \,\left( 1 - \frac{a_t}{L} \right) - \alpha \, \frac{a_t}{L} + O(L\alpha ^2), \end{aligned}$$where the error term is bounded by \(4\,L\alpha ^2\) in absolute value irrespectively of$$\begin{aligned} a_{t+1}-\frac{L}{2} = \left( 1-\frac{2\alpha }{L}\right) \cdot \left( a_t - \frac{L}{2}\right) + O(L\alpha ^2), \end{aligned}$$
*t*. Solving this recursion, using \(a_{0} = 0\) and \(\sum _{k=0}^{t-1}(1-\tfrac{2\alpha }{L})^k\le \tfrac{L}{2\alpha }\), yieldsThe estimate (10) immediately follows from (13), which together with part (a) implies (11).$$\begin{aligned} a_t = \frac{L}{2} - \frac{L}{2}\,\left( 1-\frac{2\alpha }{L}\right) ^t + O(L^2\alpha ). \end{aligned}$$ - (c)The case \(k=0\) is obvious, so we may assume \(k>0\). Let \(\tau \) denote the first time \(t\ge 0\) for which \(X_t = k+1\). Consider the sequence \(b_t = {\mathbb {E}}\left[ X_{t\wedge \tau }\,|\, X_0=0\right] \), where \(t\wedge \tau \) denotes the minimum of
*t*and \(\tau \). Note that part (a) implies \({\mathbb {P}}\left( X_{t\wedge \tau } = i \,|\, X_0=0\right) \le {\mathbb {P}}\left( X_t = i \,|\, X_0=0\right) \le 2\alpha \) for any \(i\in \{1,\dots ,k\}\) and furtherWith the same reasoning as in part (b), we end up in a similar situation with \(b_0=0\) and \(b_t\) satisfying the recursive formula$$\begin{aligned} \begin{aligned} b_t&= (k+1)\cdot {\mathbb {P}}\left( X_{t\wedge \tau } = k+1 \,|\, X_0=0\right) + O(k^2\alpha )\\&= k+1 - (k+1)\cdot {\mathbb {P}}\left( X_{t\wedge \tau } = 0 \,|\, X_0=0\right) + O(k^2\alpha ). \end{aligned} \end{aligned}$$(14)Solving it gives \(b_t = k+1 - (k+1)\,( 1 - \frac{\alpha }{k+1})^t + O(k^2 \alpha )\) and plugging this into (14) completes the proof of part (c), using$$\begin{aligned} b_{t+1}&= b_t + \alpha \cdot {\mathbb {P}}\left( X_{t\wedge \tau } = 0 \,|\, X_0=0\right) \\&=b_t + \alpha \left( 1 - \frac{b_t}{k+1}\right) + O(k \alpha ^2). \end{aligned}$$and noting that \(X_{t\wedge \tau }\le k\) if and only if \(X_{t'}\le k\) for all \(0\le t'\le t\).$$\begin{aligned} {\mathbb {P}}\left( 1\le X_{t\wedge \tau }\le k \,|\, X_0=0\right) =O(k\alpha ) \end{aligned}$$

Let us now take a closer look on the optimal separation of states 0 and *L* in this chain. To make our lives easier, we establish three auxiliary results showing that among the non-trivial separating sequences, there are very simple ones which are essentially best possible as *T* grows large.

### Proposition 6.5

Let *L* be fixed, and let \(\alpha =\alpha (T) = {\varTheta }\left( \frac{1}{T}\right) \). Then, for sufficiently large *T*, any non-trivial separating sequence \({\mathcal {A}}=(A_t)_{t=0}^T\) such that \({\mathcal {S}}^{{\mathcal {A}}}_T(0,L)>1\) (if such exist) must satisfy \(0\in A_t\) and \(L\notin A_t\) for all \(0\le t \le T\).

### Proof

*T*sufficiently large. Hence if \(0 \not \in A_{t_1}\) and \(L\in A_{t_2}\) for some \(0\le t_1, t_2 \le L\), then

*T*sufficiently large, either \(0\in A_t\) for all \(0\le t \le T\) or \(L\not \in A_t\) for all \(0\le t \le T\).

*T*sufficiently large any non-trivial separating sequence satisfies

Next, let \({\mathbf {X}}=(X_t)_{t=0}^{T+1}\) be a copy of the Markov chain started in \(X_0=0\). We define the process \({\mathbf {X}}'=(X'_t)_{t=0}^T\) as \(X'_t = X_{t+1}\) for all \(0\le t\le T\) if \(X_1=0\), and otherwise put \(X'_0=0\) and let this process evolve independently of \({\mathbf {X}}\). Clearly, this implies that \((X_t)_{t=0}^{T}\) and \({\mathbf {X}}'\) have the same distribution.

*t*. By the union bound, we find

*T*tends to infinity with our choice of \(\alpha ={\varTheta }(\tfrac{1}{T})\). So choosing \(\varepsilon >0\) small,

*T*large enough such that

### Proposition 6.6

### Proof

By symmetry, the same argument works for the chain started at *L* and the sequence of complementary sets \((S{\setminus } A_t)_{t=0}^T\). \(\square \)

### Proposition 6.7

### Proof

*f*is decreasing, \(f(0)=1\) and \(f(L)=0\). To simplify the notation, we additionally set \(f(L+1) := 0\) and write \(M:=\max _{0\le t\le T} X_t\). Considering only the first summands coming from each of the separation terms \({\mathcal {S}}^{{\mathcal {A}}^a}_T(0,L),\ 0\le a\le T\), we find

*k*, yields

*T*sufficiently large, the optimal separation \({\mathcal {S}}_T(0,L)\) is the maximum of 1 and

To finish the proof of Theorem 6.2, we need one more elementary estimation:

### Lemma 6.8

### Proof

Consequently, \(f_A\) can have at most one local maximum in (0, 1), which then lies at \(x=\tfrac{1}{2}\) for symmetry reasons. In conclusion, \(f_A\) either attains its maximum on (0, 1) at \(x=\frac{1}{2}\) or converges to its supremum on the boundary.\(\square \)

### Proof of Theorem 6.2

*L*in the chain \({\mathbf {X}}\) on [0,

*T*] is 1 if

*T*is chosen sufficiently large (after having fixed

*L*). Hence, by Theorem 6.3 and (11) we know that the chain \((X_t)_{t=0}^T\) segregates 0 and

*L*. From (11) we can further read off that for

*L*fixed

*L*large enough and then pick

*T*sufficiently large to make the right-hand side of (17) larger than \(\frac{1}{2}-\varepsilon \). This completes the proof. \(\square \)

### Remark 3

One downside of the implicit construction proving Theorem 6.2 is the fact that it does not give much information about the coupling involved. As the coupling will have to take into account the whole trajectories of the two individual copies, it is highly unlikely that the coupled process will have the Markov property. In this respect, it is still an open problem if the value of \(\frac{1}{\mathrm {e}}\), established in Sect. 5, can be pushed further (as supremum of achievable total variation distances that can be retained in segregating Markov chains), if we restrict ourselves to *Markovian* couplings.

*in discrete time*with two segregated states

*x*and

*y*such that

### Proposition 6.9

*S*and two states \(x,y\in S\). If two copies, \({\mathbf {X}}=(X_t)_{t\in {\mathbb {N}}_0}\) and \({\mathbf {Y}}=(Y_t)_{t\in {\mathbb {N}}_0}\), started in

*x*and

*y*, respectively, can be coupled to meet almost surely in finite time, it follows that

### Proof

- (i)
\(\big (f(t,X_t)\big )_{t\in {\mathbb {N}}_0}\) is a martingale with respect to the natural filtration of \({\mathbf {X}}\), and likewise for \({\mathbf {Y}}\).

- (ii)
\(f(0,x)-f(0,y) = \lim \limits _{n\rightarrow \infty } ||P^n(x,.)-P^n(y,.)||_{\mathrm {TV}}\).

*f*to be the limit of a pointwise converging subsequence of the uniformly bounded sequence of functions \((f_{n})_{n\in {\mathbb {N}}_0}\).

Then (ii) is immediate and since for all \(n\in {\mathbb {N}}_0\), \(f_n(t,X_t)\) is a martingale for \(0\le t\le n\), bounded by 0 and 1 from below and above, respectively, the conditional dominated convergence theorem ensures that \(f(t,X_t)\) inherits these properties.

## Notes

### Acknowledgements

First of all, we want to thank Jeff Steif for bringing this special kind of coupling to our attention. Then, we want to give thanks to him, Olle Häggström, Peter Hegarty and the anonymous referees for their valuable comments on earlier drafts.

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