2. PRELIMINARIES

Define for 1 ≤ i ≤ m and ε > 0 a difference operator Δ_i^ε by

for given u₁,…,u_m. A function

is called supermodular if for all 1 ≤ i < j ≤ m and ε_i,ε_j > 0,

for all u = (u₁,…,u_m). For smooth functions, the above condition is equivalent to

for all 1 ≤ i < j ≤ n. The standard examples of supermodular functions are u₁ × ··· × u_m, −max(u₁,…,u_m), min(u₁,…,u_m), and h(u₁ + ··· + u_m), where

is convex.

This class of functions induces the so-called supermodular order. For arbitrary random vectors

, we write

if

for all supermodular φ such that the respective expectations are finite. Similarly, for stationary random sequences

, we write

if for all m ≥ 1,

Supermodular ordering is a dependence ordering in the sense of Joe [5]. In particular, if

, then

Consider now two stationary homogeneous Markov chains

with the state space {1,…,N}, the same stationary distribution π = (π₁,…,π_N), and transition matrices

, respectively. We will write

if

. Here, in the sequel, we will assume that all matrices have dimension N × N.

For the supermodular ordering of

, we have the following characterization (cf. Hu and Pan [4]).

Lemma 2.1: Let Π be a diagonal matrix diag(π₁,…,π_N). Then,

if and only if

for all r,s ∈ {1,…,N}.

Condition (2.2) is equivalent to a concordance ordering of bivariates; that is,

for every r and s in the state space. Note that for the double stochastic matrices

, condition (2.2) is equivalent to

Moreover, supermodular ordering for the double stochastic matrices can be characterized in the following way. Let

Note that

The matrix T was used in Keilson and Kester [6] for obtaining stochastic monotonicity. A matrix A is stochastic monotone if and only if T⁻¹AT ≥ 0, where 0 is a matrix which consists of zeros.

Lemma 2.2: Assume that

are double stochastic. Then,

if and only if

.

Proof: Inequality

is equivalent to

for all r and s, which means

by Eq. (2.3). █

3. MAIN RESULT

We introduce a class of Markov chains which are not necessary monotone.

Definition 3.1: We say that a matrix A belongs to a class

if it is stochastic and TAT⁻¹ ≥ 0. Equivalently, we say that a stationary homogeneous Markov chain belongs to

if its transition probability matrix does.

Let a and b be N-dimensional vectors. We say that a precedes b in the partial sum ordering

. Now, denote by a_·i, i = 1,…,N, the columns of a matrix A. Then,

if and only if

Denote now by

a class of double stochastic matrices. It turns out that for the Markov chains driven by

, the following monotonicity property holds.

Theorem 3.1: Let {J_n}_n≥1 be a stationary homogeneous Markov chain with a transition probability matrix

. Then, for any n ≥ 1,

Corollary 3.1: Under conditions of Theorem 3.1, we have for the functions f and g both either nonincreasing or nondecreasing,

The proof of Theorem 3.1 consists of the sequence of lemmas.

Lemma 3.1:

.

Proof:

Lemma 3.2: Assume that

. If

, then

.

Proof: We have

The first term in brackets is greater than 0 due to assumptions on A; the second one has the same property because of Lemma 2.2. Therefore,

and using Lemma 2.2 once more, we obtain the comparison result. █

Lemma 3.3: Assume that

. Then, A² <_sm A. Moreover, for every n ≥ 1, Aⁿ⁺¹ <_sm Aⁿ.

Proof: Note that for any double stochastic matrix A, we have A <_sm I, where I is an identity matrix. Moreover,

. From Lemma 3.2, we have A² <_sm A. Assume now that Aⁿ <_sm Aⁿ⁻¹. Because of Lemma 3.1, Aⁿ and Aⁿ⁻¹ belong to

. Moreover, they are double stochastic. Therefore, Lemma 3.2 applies and we have AAⁿ <_sm AAⁿ⁻¹. █

Proof of Theorem 3.1: Denote by P^(k) = (p_ij^(k))_i,j=1^N, k ≥ 1, the k-step probability matrix for {J_n}_n≥1. From Lemma 3.3, we have that

for all r,s = 1,…,N. Therefore, (J₁,J_n+1) <_sm (J₁,J_n). █

Corollary 3.1 follows from stationarity of a Markov chain and the fact that for the functions f and g both either nonincreasing or nondecreasing, the function φ(x,y) = f (x)g(y) is supermodular.

4. EXAMPLES AND COUNTEREXAMPLES

Example 4.1: Let p and ε > 0 be such that p ≥ 2ε and p + ε ≤ 1. Then, the following matrix belongs to

and is not monotone:

Example 4.2: Lemma 3.3 fails if one of the assumptions is removed. Let

Then,

, and it is not true that A² <_sm A. The above matrix was used in Lindqvist [7] as a counterexample of a Markov chain which is associated but not stochastically monotone.

Now, let

This matrix belongs to

but is neither double stochastic nor monotone. It is not true that A² <_sm A.