1. INTRODUCTION
Since Daley's [2] article,
it is well known that the stochastically monotone Markov chain
{Jn}n≥1 has the
following property. Let
be monotone and γ(n) = Cov[f
(J1),f (Jn)].
Then, a map n → γ(n) is decreasing. Recall that a
real-valued homogeneous discrete-time Markov chain
{Jn}n≥1 is
stochastically monotone if its one-step transition probability function
P(Jn+1 >
x|Jn = y) is
nondecreasing in x for every fixed y. Daley's
result was extended in Bergmann and Stoyan [1]. Hu and Joe [3] (see also Joe [5,
Thms. 8.3, 8.4, and 8.7]) stated conditions for the concordance ordering
of bivariates of Markov chains under some monotonicity assumptions. The most
general result was given in Hu and Pan [4]. If both the Markov chain and its time-reversed
counterpart are stochastically monotone, then
γ(n1,…,nm)
= E [f
(Jn1,…,Jnm)]
is decreasing in
(n1,…,nm)
coordinatewise for each m ≥ 2 and supermodular f.
We refer to Müller and Stoyan [10] for a review of a recent work in this
area. The monotonicity of a Markov chain is also sufficient for
association of it (Lindqvist [7]).
However, monotonicity is not necessary; see Lindqvist's article
for a counterexample.
In our article, we define a family of stationary and homogeneous
Markov chains which are not necessary monotone. For that family, we
derive the monotonicity of bivariates w.r.t. supermodular order and,
hence, monotonicity of covariances. Therefore, we show that an answer
to a question which appears in [9]
is negative.
The article is organized as follows. In Section 2, we collect some
needed definitions and preliminary results. In Section 3, we define the
family of Markov chains and prove the main result. Section 4 is devoted
to examples and counterexamples. Some additional comparison results for
our class are given in Section 5.
2. PRELIMINARIES
Define for 1 ≤ i ≤ m and ε > 0 a
difference operator Δiε by
for given
u1,…,um. A
function
is called supermodular if for all 1 ≤ i <
j ≤ m and
εi,εj > 0,
for all u =
(u1,…,um). For
smooth functions, the above condition is equivalent to
for all 1 ≤ i < j ≤ n. The
standard examples of supermodular functions are u1
× ··· ×
um,
−max(u1,…,um),
min(u1,…,um),
and h(u1 + ··· +
um), 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,…,πN), 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,…,πN).
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
{Jn}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:
1. We have
TABT−1 =
TAT−1TBT−1
≥ 0 from the assumptions on A and
B.
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) =
(pij(k))i,j=1N,
k ≥ 1, the k-step probability matrix for
{Jn}n≥1. From Lemma
3.3, we have that
for all r,s = 1,…,N. Therefore,
(J1,Jn+1)
<sm
(J1,Jn). █
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.
5. ADDITIONAL COMPARISON RESULTS
Observe that for a ≤ a′ and stochastic
matrices A and B with the same invariant
distributions, we have a′A + (1 −
a′)B <sm aA +
(1 − a)B if A <sm
B. This can be interpreted as follows. Let
be the stationary homogeneous Markov chains with transition matrices
A and B, respectively. Define
,
where
are Bernoulli random variables with
,
independent of everything else. Then,
,
provided
.
If we have some additional monotonicity properties, we can have a
comparison of the whole sequences
{Zn}n≥1 and
.
However, the above consideration cannot be rewritten for the random variables
assuming their values in {1,…,K}. This can be done if the
transition matrices belong to
.
We refer to Marshall and Olkin [8] for
the concept of majorization.
Proposition 5.1: Assume the following:
Then,
Proof: Define for l = 1,…,N and
r = 1,…,K the vectors
where for j = 1,…,N,
vj(l,r) are
defined as
According to assumption (a), these vectors have decreasing
coordinates. Moreover, because of assumption (b), we have
for each fixed l. Now, take
From Marshall and Olkin [8,
p.125], we have
for every l = 1,…,N. Because
have decreasing coordinates, majorization order implies
.
Observe now that the jth coordinates of
can be written as
respectively. Since
,
we have for each k, l
which implies comparison of matrices by (2.3). █
Acknowledgments
The author is grateful to Ryszard Szekli for many helpful comments.
This work was partially done during the author's stay at
University of Ottawa.
