# Some Comments on Quasi-Birth-and-Death Processes and Matrix Measures

Authors: Holger Dette; Bettina Reuther
Journal: Journal of Probability and Statistics
 (2010)

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## Abstract

We explore the relation between matrix measures and quasi-birth-and-death processes. We derive an integral representation of the transition function in terms of a matrix-valued spectral measure and corresponding orthogonal matrix polynomials. We characterize several stochastic properties of quasi-birth-and-death processes by means of this matrixmeasure and illustrate the theoretical results by several examples.


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## Body

## 1. Introduction

Let(Ω,ℱ,P,(Xt)t≥0) be a continuous-time two-dimensional homogeneous Markov process with state space(1.1)E={(i,j)∈ℕ0×{1,…,d}},d∈ℕ,d<∞
and infinitesimal generator(1.2)Q=(Qij)i,j=0,1,…=(B0A00C1TB1A1C2TB2A2C3TB3A30⋱⋱⋱),
where A0,A1,…,B0,B1,…,C1,C2,…∈ℝd×d. The transition rate from state (i,j) to state (k,ℓ) is given by the element in the position (j,ℓ) of the matrix Qik. Markov processes with an infinitesimal generator matrix of the form (1.2) are known as continuous-time quasi-birth-and-death processes. These models have many applications in the evaluation of communicating systems and queueing systems (see, e.g., [1–3]) and have been analyzed by many authors (see, e.g., [4–6]). The case d=1 corresponds to a “classical” birth-and-death process with a tridiagonal infinitesimal generator which has been investigated in great detail using the theory of orthogonal polynomials by Karlin and McGregor [7, 8]. Since this pioneering work several authors have used these techniques to derive interesting properties of birth-and-death processes in terms of orthogonal polynomials and the corresponding measure of orthogonality (see, e.g., [9, 10]).It is the purpose of the present paper to extend some of these results to quasi-birth-and-death processes with a generator of the form (1.2) using the theory of matrix measures and corresponding orthogonal matrix polynomials.We associate to a matrix of the form of (1.2) a sequence of matrix polynomials, recursively defined by(1.3)-xQn(x)=AnQn+1(x)+BnQn(x)+CnTQn-1(x)
with initial conditions Q-1(x)=0 and Q0(x)=Id. A matrix measure Σ={σij}i,j=1,…,d on the real line is a function for which Σ(A)={σij(A)}i,j=1,…,d is a symmetric and nonnegative definite matrix in ℝd×d for each Borel set A⊂ℝ, where the entries σij are finite signed measures. In Section 2 we formulate sufficient conditions on the infinitesimal generator (1.2) such that there exists a matrix measure Σ on the real line with (1.4)〈Qi,Qj〉=∫ℝQi(x)dΣ(x)QjT(x)=δijId,
that is, the matrix polynomials are orthonormal with respect to the matrix measure Σ (see [11]). In this case we derive an integral representation for the blocks of the transition function in terms of the orthogonal matrix polynomials Qi and the matrix measure Σ, which generalize the representation of Karlin and McGregor [7] to the case d>1. We also investigate relations between the Stieltjes transforms of random walk measures corresponding to two quasi-birth-and-death processes, where only a few blocks differ. In Section 3 we discuss several examples to illustrate the theory. Finally, in Section 4 the theoretical results are used to characterize α-recurrence of quasi-birth-and-death processes.


## 2. Quasi-Birth-and-Death Processes and Matrix Polynomials

The moments of the matrix measureΣ are defined by the d×d matrices (2.1)Sk=∫xkdΣ(x),k=0,1,…,
and throughout this paper we will only consider matrix measures with existing moments of all order. The “left’’ inner product with respect to Σ of two matrix polynomials Q and P is defined by (2.2)〈Q,P〉=∫Q(x)dΣ(x)PT(x).
If {Sn}n≥0 is a sequence of matrices such that the block Hankel matrices, (2.3)H̲2m=(S0⋯Sm⋮⋮Sm…S2m),m≥0,
are positive definite, then there exists a matrix measure Σ with moments Sn, n≥0, and a sequence of matrix polynomials {Qn(x)}n≥0 which is orthogonal with respect to Σ (see [12]). The following theorem characterizes the existence of a matrix measure Σ such that there is a sequence of matrix polynomials which is orthogonal with respect to Σ. The proof follows by similar arguments as presented in Theorem 2.1 of  [13] and is therefore omitted.Theorem 2.1.
Let the matricesAn, n≥0, and CnT,n≥1, in (1.2) be nonsingular and  Bn≥0, and assume that {Qn(x)}n≥0 is a sequence of matrix polynomials defined by recursion (1.3).
There exists a matrix measureΣ with positive definite block Hankel matrices H̲2m, m≥0, such that the sequence of matrix polynomials {Qn(x)}n≥0 is orthogonal with respect to Σ if and only if there is a sequence of nonsingular matrices {Rn}n≥0 with
(2.4)RnBnRn-1symmetric,∀n∈ℕ0,RnTRn=Cn-1⋯C1-1(R0TR0)A0⋯An-1,∀n∈ℕ.
Moreover,
(2.5)R0-1((R0T)-1)=(R0TR0)-1=S0,
and the matrices {R̃n}n≥0={UnRn}n≥0, where Un, n≥0, are orthogonal matrices and also satisfy condition (2.4).Note that condition (2.4) is crucial for our approach and is always satisfied in the case d=1. If d>1 it has to be checked in concrete examples, but—to our best knowledge—there do not exist any general conditions which imply (2.4). Some examples where (2.4) is satisfied are presented in Section 3. Several  other examples can be found in the papers of Grünbaum [14, 15], Grünbaum et al. [16], and Cantero et al. [17]. If condition (2.4) is satisfied, the corresponding measure Σ is called a spectral measure corresponding to {Qn(x)}n≥0 and the matrix Q in (2.4), respectively. The infinitesimal generator matrix (1.2) is called conservative if (2.6)(A0+B0)1=0,(An+Bn+CnT)1=0,∀n∈ℕ,
where 1=(1,1,…,1)T∈ℝd and 0=(0,0,…,0)T∈ℝd (see [18]). In this case there exists a transition function(2.7)P(t)=(Pii′(t))i,i′=0,1,…,
with d×d block matrices Pii′(t)∈ℝd×d, (2.8)P(0)=I,P′(0)=Q,
which satisfies the Kolmogorov forward differential equation(2.9)P′(t)=P(t)Q,∀t≥0
and the Kolmogorov backward differential equation (2.10)P′(t)=QP(t),∀t≥0.
The probability P(Xt=(i′,j′)∣X0=(i,j)) of going from state (i,j) to (i′,j′) in time t is given by the element in the position (j,j′) of the matrix Pii′(t).Note that there always exists a transition functionP(t) such that the Kolmogorov forward differential equation (2.9) is satisfied. The infinitesimal generator Q is called regular if there exists only one such transition function (see [18]). If additionally a spectral measure Σ corresponding to the generator matrix (1.2) exists, we can derive an integral representation for the block of the transition function P(t) in the position (i,j) in terms of the spectral measure and the corresponding matrix orthogonal polynomials, which generalizes the famous Karlin and McGregor representation.Theorem 2.2.
Assume that the conditions for the existence of the measureΣ in Theorem 2.1 are satisfied and that there exists a transition function P(t) which satisfies the Kolmogorov forward equation (2.9) for all t≥0. Then the following representation holds for the block Pij(t)∈ℝd×d in the position (i,j) of the transition function P(t):
(2.11)Pij(t)=(∫e-txQi(x)dΣ(x)QjT(x))(∫Qj(x)dΣ(x)QjT(x))-1.Proof.
LetQ(x)=(Q0T(x),Q1T(x),…)T denote the vector of orthogonal matrix polynomials Qi(x) with respect to the spectral measure Σ. Then the recursive relation (1.3) is equivalent to the matrix equation
(2.12)-xQ(x)=QQ(x).
Defining
(2.13)F(x,t):=P(t)Q(x),
we obtain the differential equation
(2.14)ddtF(x,t)=P'(t)Q(x)=P(t)QQ(x)=-xP(t)Q(x)=-xF(x,t),
and the condition P(0)=I yields
(2.15)F(x,0)=P(0)Q(x)=Q(x).
Hence, it follows that
(2.16)F(x,t)=e-txQ(x)=P(t)Q(x),
which implies (integrating with respect to dΣ(x)) that
(2.17)∫e-txQ(x)dΣ(x)QjT(x)=P(t)∫Q(x)dΣ(x)QjT(x).
Because of the orthogonality of the matrix polynomials Qn(x), n≥0, we obtain for the blocks Pij(t) of the transition function the representation
(2.18)Pij(t)=(∫e-txQi(x)dΣ(x)QjT(x))(∫Qj(x)dΣ(x)QjT(x))-1,∀i,j,
which completes the proof of Theorem 2.2.In what follows we present two results, which relate the Stieltjes transforms of the spectral measures of two quasi-birth-and-death processes, which have an infinitesimal generator of similar structure. The first result refers to the case where the entryB0 has been replaced by the matrix B¯0. The proof is similar to a corresponding result in [13] and is therefore omitted.Theorem 2.3.
Consider the infinitesimal generator defined by (1.2) and the matrix
(2.19)Q¯=(B¯0A00C1TB1A1C2TB2A2C3TB3A30⋱⋱⋱).
Let Σ be a spectral measure corresponding to the infinitesimal generator Q with positive definite block Hankel matrices such that the matrix R0B¯0R0-1 is symmetric and such that {Rn}n≥0 is a sequence of matrix polynomials which satisfies condition (2.4). Then there exists a spectral measure Σ¯ corresponding to Q¯. If the spectral measures Σ and Σ¯ are determined by their moments, then the Stieltjes transforms of the measures satisfy
(2.20)Φ(z)=∫dΣ(t)z-t={(∫dΣ¯(t)z-t)-1-S0-1(B¯0-B0)}-1.Given a sequence{Qn(x)}n≥0 of matrix polynomials defined by recursion (1.3), the corresponding associated sequence of matrix polynomials {Qn(k)(x)}n≥0 of order k,k≥1, is defined by a recursion of the form of (1.3), in which the matrices An,Bn, and Cn have been replaced by the matrices An+k,Bn+k, and Cn+k, respectively (see [19]). The following result gives a relation between the Stieltjes transform of the spectral measure corresponding to the sequence of matrix polynomials {Qn(x)}n≥0 and the Stieltjes transform of the spectral measure corresponding to {Qn(k)(x)}n≥0. The associated quasi-birth-and-death process will be denoted by (Xt(k))t≥0 with state space E defined by  (1.1) (throughout this paper we use the notation Xt(0):=Xt).Theorem 2.4.
Consider the infinitesimal generatorQ defined by (1.2) and the matrix
(2.21)Q(k)=(BkAk0Ck+1TBk+1Ak+1Ck+2TBk+2Ak+2Ck+3TBk+3Ak+30⋱⋱⋱).
The matrix Q(k) is called the associated matrix of order k,k≥1, corresponding to Q. Assume that Σ is a spectral measure corresponding to Q with positive definite block Hankel matrices, that is, there exists a sequence {Rn}n≥0 of nonsingular matrices, which satisfies condition (2.4) of Theorem 2.1. Then there exists a spectral measure Σ(k) corresponding to Q(k) with positive definite block Hankel matrices. If the measures are determined by their moments, then the Stieltjes transforms of the measures are related by
(2.22)∫dΣ(x)z-x=R0-1{{zId-Ek-1-DkRk∫dΣ(k)(x)z-xRkTDkT}-1zId-E0-D1{{zId-Ek-1-DkRk∫dΣ(k)(x)z-xRkTDkT}-1zId-E1-D2{{zId-Ek-1-DkRk∫dΣ(k)(x)z-xRkTDkT}-1zId-E2-⋯⋯-Dk-1{zId-Ek-1-DkRk∫dΣ(k)(x)z-xRkTDkT}-1Dk-1T}-1⋯⋯D2T{zId-Ek-1-DkRk∫dΣ(k)(x)z-xRkTDkT}-1}-1D1T{zId-Ek-1-DkRk∫dΣ(k)(x)z-xRkTDkT}-1}-1(R0T)-1,
where
(2.23)Dn+1=-RnAnRn+1-1,En=-RnBnRn-1,DnT=-RnCnTRn-1-1,
and the Stieltjes transforms of the matrix measures Σ(k) and Σ(k+1) are related by
(2.24)∫dΣ(k)(x)z-x=Rk-1{dΣ(k+1)(x)z-xzId+RkBkRk-1-RkAk∫dΣ(k+1)(x)z-xRk+1TRk+1Ck+1TRk-1}-1(RkT)-1.Proof.
Let the sequence of polynomials{Qn(x)}n≥0 be defined by recursion (1.3) with corresponding spectral measure Σ. Then the polynomials Wn(x):=RnQn(x) are orthonormal with respect to the matrix measure Σ and satisfy the three-term recurrence relation
(2.25)xWn(x)=Dn+1Wn+1(x)+EnWn(x)+DnTWn-1(x)
with initial conditions W-1(x)=0 and W0(x)=R0. From Theorem 1.2 and Lemma 1.3 in [20] it follows that
(2.26)∫dΣ(x)z-x=limn→∞R0-1{⋯-Dn{zId-En}-1DnT}-1⋯}-1D2TzId-E0-D1{⋯-Dn{zId-En}-1DnT}-1⋯zId-E1-D2{{zId-En}-1DnTzId-E2-⋯⋯-Dn{zId-En}-1DnT}-1⋯}-1D2T}-1D1T}-1(R0T)-1.
Assume that the sequence of polynomials {Qn(k)(x)}n≥0 is defined by recursion (1.3), where the matrices Bn,An, and Cn have been replaced by the matrices Bn+k,An+k, and Cn+k, respectively, that is
(2.27)-xQn(k)(x)=An+kQn+1(k)(x)+Bn+kQn(k)(x)+Cn+kTQn-1(k)(x),
with Q0(k)(x)=I and Q-1(k)(x)=0. Define An(k)=An+k,Bn(k)=Bn+k, Cn(k)=Cn+k, and Rn(k)=Rn+k, n≥0. From Theorem 2.1 we obtain the symmetry of the matrices
(2.28)-Rn(k)Bn(k)(Rn(k))-1=-Rn+kBn+kRn+k-1,∀n≥0
and the equation
(2.29)(Rn(k))TRn(k)=Rn+kTRn+k=Cn+k-1Cn+k-1-1⋯Ck+1-1Ck-1⋯C1-1R0TR0A0A1⋯Ak-1Ak⋯An+k-1=Cn+k-1Cn+k-1-1⋯Ck+1-1RkTRkAk⋯An+k-1=(Cn(k))-1(Cn-1(k))-1⋯(C1(k))-1(R0(k))TR0(k)A0(k)⋯An-1(k),∀n≥1.
Therefore, from Theorem 2.1 it follows that there exists a spectral measure Σ(k) with positive definite block Hankel matrices corresponding to the sequence of polynomials {Qn(k)(x)}n≥0.
The polynomialsWn(k)(x):=Rn(k)Qn(k)(x) are orthonormal with respect to the measure Σ(k) and satisfy the recursion
(2.30)xWn(k)(x)=Dn+1(k)Wn+1(k)(x)+En(k)Wn(k)(x)+(Dn(k))TWn-1(k)(x),W0(k)(x)=R0(k)=Rk,
where
(2.31)Dn+1(k)=Dn+k+1,En(k)=En+k,∀n≥0.
Therefore, it follows from Theorem 1.2 and Lemma 1.3 in [20] that
(2.32)∫dΣ(k)(x)z-x=limn→∞(R0(k))-1{(Dn(k))TzId-E0(k)-D1(k){⋯-Dn(k){zId-En(k)}-1(Dn(k))T}-1zId-E1(k)-D2(k){(Dn(k))TzId-E2(k)-⋯⋯-Dn(k){zId-En(k)}-1(Dn(k))T}-1…⋯-Dn(k){zId-En(k)}-1(Dn(k))T}-1}-1(D2(k))T}-1(D1(k))T}-1((R0(k))T)-1=limn→∞Rk-1{⋯-Dn+k{zId-En+k}-1Dn+kT}-1…}-1zId-Ek-Dk+1{⋯-Dn+k{zId-En+k}-1Dn+kT}-1…zId-Ek+1-Dk+2{{zId-En+k}-1Dn+kTzId-Ek+2-⋯⋯-Dn+k{zId-En+k}-1Dn+kT}-1…}-1Dk+2T}-1Dk+1T⋯-Dn+k{zId-En+k}-1Dn+kT}-1…}-1Dk+2T}-1}-1(RkT)-1.
A combination of (2.26) and (2.32) yields
(2.33)∫dΣ(x)z-x=R0-1{zId-E0-D1{⋯-Dk-1{zId-Ek-1-DkRk∫dΣ(k)(x)z-xRkTDkT}-1Dk-1T}-1zId-E1-D2{{zId-Ek-1-DkRk∫dΣ(k)(x)z-xRkTDkT}-1Dk-1TzId-E2-⋯⋯-Dk-1{zId-Ek-1-DkRk∫dΣ(k)(x)z-xRkTDkT}-1Dk-1T}-1⋯⋯D2T⋯-Dk-1{zId-Ek-1-DkRk∫dΣ(k)(x)z-xRkTDkT}-1Dk-1T}-1}-1D1T⋯D2T⋯-Dk-1{zId-Ek-1-DkRk∫dΣ(k)(x)z-xRkTDkT}-1Dk-1T}-1}-1}-1(R0T)-1,
and from  (2.32) and (2.23) we obtain
(2.34)∫dΣ(k)(x)z-x=Rk-1{zId-Ek-Dk+1Rk+1∫dΣ(k+1)(x)z-xRk+1TDk+1T}-1(RkT)-1=Rk-1{zId+RkBkRk-1-RkAk∫dΣ(k+1)(x)z-xRk+1TRk+1Ck+1TRk-1}-1(RkT)-1,
which completes the proof of the theorem.Remark 2.5.
Note that in the literature, many queueing models are considered, where the matricesCn do not have full rank (see [21]). Following the arguments used in Remark 2.7 in [13] the conditions
(2.35)RnBn=EnRn,n≥0,Cn+1Rn+1TRn+1=RnTRnAn,n≥1
are sufficient for the existence of a spectral measure Σ corresponding to Q, where {En}n≥0 is a sequence of symmetric matrices and
(2.36)∫Qi(x)dΣ(x)QjT(x)=δijRjTRj.
In other words, the assumption of nonsingularity of the matrices Cn can be relaxed. The same arguments as those used in Theorem 2.2 then imply that
(2.37)Pij(t)RjTRj=∫e-txQi(x)dΣ(x)QjT(x).


## 3. Examples

Example 3.1.
Dayar and Quessette [3] considered a queuing system consisting of an M/M/1-system and an M/M/1/d-1-system. Both queues have Poisson arrival processes with rate λi, i=1,2, and exponential service distributions with rate μi, i=1,2, and it was assumed that γ=λ1+λ2+μ1+μ2. The level represents the length of queue 1, which is unbounded, and the phase represents the length of queue 1, which can range from 0 to d-1. The process is of interest because of its level geometric stationary distribution. This system can be described by a homogeneous Markov process X(t)=(L1(t),L2(t))t∈ℝ+ with state space E=ℕ×{0,…,d-1}, where L1(t) and L2(t) denote the length of the first queue at time t and the length of the second queue at time t, respectively. The entries of the corresponding infinitesimal generator (1.2) have the form
(3.1)B0=(-(λ1+λ2)λ2μ2-(γ-μ1)λ2⋱⋱⋱μ2-(γ-μ1)λ2μ2-(λ1+μ2)),Bi=(-(γ-μ2)λ2μ2-γλ2⋱⋱⋱μ2-γλ2μ2-(γ-λ2)),i≥1,Ai=λ1Id, i≥0, and CiT=μ1Id, i≥1. It is easy to see that Q is conservative. A straightforward calculation shows that the conditions of Theorem 2.1 are satisfied with the matrices
(3.2)R0=diag(1,λ2μ2,(λ2μ2)2,…,(λ2μ2)d-1),Ri=(λ1μ1)iR0,i∈ℕ.
This implies the existence of a spectral measure.Example 3.2.
In general, the spectral distribution can only be identified in special cases. Even if the Stieltjes transform can be determined, its inversion is usually difficult (see, e.g., [22, Chapter 3]). We now present an example where the spectral measure can be found explicitly. To be precise consider a homogeneous Markov process (Xt)t≥0 with infinitesimal generator (1.2), where
(3.3)B0=(-γβ1β2-γβ1⋱⋱⋱β2-γβ1β2-γ),γ≠0,Bi=(-δβ1β2-δβ1⋱⋱⋱β2-δβ1β2-δ),i≥1,δ≠0,Ai=α1Id, i≥0, and CiT=α2Id, i≥1. A generator matrix of this form can be associated to a queueing model which consists of d different M/M/1-systems. Each M/M/1-system has a Poisson arrival process with rate α1 and an exponential service time distribution with rate α2. If the customer is situated in system i, then it changes to the system i-1 and i+1 with the rate β2 and β1, respectively. This model can be described by the two-dimensional homogeneous Markov process (Nt,St)t≥0 with state space E=ℕ0×{0,…,d-1}, where ℕ0={0,1,2,…}, Nt denotes the number of customers in the whole model at time t, and St denotes the number of the system at time t.Ifβ1≠0 and β2≠0 the conditions of Theorem 2.1 are satisfied with
(3.4)R0=diag((β2β1)(d-1)/2,(β2β1)(d-2)/2,…,(β2β1)1/2,1),Rn=(α1α2)nR0,n≥1.
This implies the existence of a spectral measure Σ corresponding to Q. In order to determine the measure explicitly, note that the matrices in (2.23) have the form
(3.5)D:=Dn=-α1α2Id,n≥1,E0=(γ-β1β2-β1β2γ-β1β2⋱⋱⋱-β1β2γ-β1β2-β1β2γ),E:=En=(δ-β1β2-β1β2δ-β1β2⋱⋱⋱-β1β2δ-β1β2-β1β2δ),n≥1.
The eigenvalues of the matrix E are given by (3.6)λk=δ+2β1β2cos(kπd+1),k=1,…,d,
with corresponding eigenvectors given by u(k)=(u1(k),…,ud(k))T, where (3.7)uj(k)=2d+1sin(kjπd+1),j,k=1,…,d.
With the notations H:=diag(λ1-z,…,λd-z) and U:=(u(1),…,u(d)), it follows that (3.8)E-zId=UHUT,UTU=Id.
Let Q¯ be the infinitesimal generator obtained from Q by replacing the first diagonal block B0 by block B1 (which coincides with all other blocks Bi, i≥2), and denote by Σ¯ the spectral measure corresponding to Q¯. From [23] we obtain for the Stieltjes transform Φ¯(z) of the matrix measure Σ¯(3.9)Φ¯(z)=-12D-2(E-zId)1/2{Id+{Id-4D2(E-zId)-2}1/2}(E-zId)1/2=-12α1α2UH1/2{Id+{Id-4α1α2H-2}1/2}H1/2UT,
and Theorem 2.3 gives the Stieltjes transform Φ(z) of the measure Σ. Moreover, the results in [23, page 318] also show that the support of the spectral measure is given by (3.10)supp(Σ)={x∈ℝ:D-1/2(xId-E)D-1/2hasaneigenvaluein[-2,2]}=[-2α1α2+δ+2β1β2cos(πdd+1),2α1α2+δ+2β1β2cos(πd+1)].
Note that supp(Σ)⊂[0,∞) if δ≥α1+α2+β1+β2.


## 4.α-Recurrence

The decay parameter of continuous-time quasi-birth-and-death processes was introduced by van Doorn [19]. To be precise assume that (Xt)t≥0 is an irreducible quasi-birth-and-death process with state space (1.1) and infinitesimal generator Q defined by (1.2), where (4.1)B01+A01<0.
Then the decay parameter α of the process (Xt)t≥0 is defined by (4.2)α=sup{s≥0:ejT∫0∞estPii′(t)dtej′<∞},(i,j),(i′,j′)∈E.
A state (i,ℓ)∈E is called α-recurrent(4.3)eℓT∫0∞eαtPii(t)dteℓ=∞,
where eℓ=(0,…,0,1,0,…,0)T∈ℝd denotes the ℓth unit vector. The process (Xt)t≥0 is called α-recurrent if and only if some state (and then all states in E) is α-recurrent. The process (Xt)t≥0 is called α-positive if and only if for some state (i,ℓ)∈E (and then for all states in E)(4.4)eℓTlimt→∞eαtPii(t)eℓ>0.
The following results characterize α-recurrence of the process (Xt)t≥0 in terms of the spectral measure Σ, the corresponding orthogonal polynomials Qj(x), and the blocks of the infinitesimal generator. Throughout this section it will be assumed that condition (2.4) of Theorem 2.1 is satisfied.Theorem 4.1.
Assume that the conditions of Theorem2.1 are satisfied with a spectral measure supported in the interval [α,∞) and that there exists a transition function, which satisfies the Kolmogorov forward differential equation (2.9). The process (Xt)t≥0 is α-recurrent if and only if for some state (i,ℓ)∈E (and then for all states in E)
(4.5)eℓT(∫Qi(x)dΣ(x)QiT(x)x-α)(∫Qi(x)dΣ(x)QiT(x))-1eℓ=∞.Proof.
With representation (2.11) and Fubini's Theorem, condition (4.3) is equivalent to
(4.6)eℓT(∫∫0∞e(α-x)tdtQi(x)dΣ(x)QiT(x))(∫Qi(x)dΣ(x)QiT(x))-1eℓ=∞,
which implies (4.5).In the following we define for a matrix measureΣ with existing moments the d×d matrices ζ0=0 and ζk=(Sk-1-Sk-1-)-1(Sk-Sk-)∈ℝd×d, where S2n-S2n- and S2n-1-S2n-1- denote the Schur complement of S2n and S2n-1 in the matrix H̲2n and (4.7)H̲2n-1=(S1⋯Sn⋮⋮Sn…S2n-1),
respectively (see [24]). The next result gives a representation of the Stieltjes transform of the spectral measure Σ in terms of the quantities ζj and the blocks of the generator matrix (1.2).Note thatsupp(Σ)⊂[α,∞) is crucial for our approach and in general difficult to check. Consider for example the case of recurrence (i.e., α=0), then it follows from the results of Duran and Lopez-Rodriguez [25] that the spectral measure Σ can be found as weak accumulation points of a sequence of discrete measures with support precisely on (4.8)Δn={x∣detQn(x)=0}.
A straightforward calculation shows that the set Δn coincides with the eigenvalues of the matrix (4.9)-(B0A0C1TB1A1⋱⋱⋱Cn-1TB1)
and consequently all bounds on eigenvalues of these matrices will yield bounds on the support of spectral measure.Corollary 4.2.
Assume that conditions (2.4) of Theorem 2.1 are satisfied. Let {Qn(x)}n≥0 denote the corresponding orthogonal matrix polynomials defined by recursion (1.3). Assume that the corresponding spectral measure Σ is supported in the interval [0,∞) and that it is determined by its moments. Then the Stieltjes transform of the measure Σ can be represented as
(4.10)∫dΣ(x)z-x=limn→∞{zId-{Id-{zId-⋯-{zId-ζ2n+1T}-1ζ2nT}-1…}-1ζ2T}-1ζ1T}-1S0.
In particular, the following representations hold:
(4.11)∫dΣ(x)x=limn→∞∑j=0n+1(ζ2j+1Tζ2j-1T⋯ζ1T)-1(ζ2jTζ2j-2T⋯ζ2T)S0(4.12)=limn→∞∑j=0n+1Tj+1-1Aj-1CjTTj-1Tj-1Aj-1-1Cj-1TTj-2Tj-1-1⋯T0T1-1A0-1T0S0,
where Tj=Qj(0), j≥0.Proof.
From Lemma3.3 in [24] it follows that the monic orthogonal matrix polynomials {P̲n(x)}n≥0 with respect to a matrix measure Σ supported in [0,∞) satisfy the recursive relation
(4.13)xP̲n(x)=P̲n+1(x)+(ζ2n+1T+ζ2nT)P̲n(x)+ζ2nTζ2n-1TP̲n-1(x),
with P̲-1(x)=0,P̲0(x)=Id,ζ0=0, and ζk=(Sk-1-Sk-1-)-1(Sk-Sk-), where the matrices
(4.14)Δ2n≔〈P̲n,P̲n〉=(S0ζ1⋯ζ2n)T
are positive definite. Then the polynomials
(4.15)Pn(x):=Δ2n-1/2P̲n(x),n≥0,
are orthonormal with respect to the matrix measure Σ and satisfy the recursion
(4.16)xPn(x)=An+1Pn+1(x)+BnPn(x)+AnTPn-1(x)
with P-1(x)=0,P0(x)=S0-1/2, and
(4.17)An+1=Δ2n-1/2Δ2n+21/2,Bn=Δ2n-1/2(ζ2nT+ζ2n+1T)Δ2n1/2,AnT=Δ2n-1/2ζ2nTζ2n-1TΔ2n-21/2.
From Theorem 1.2 in [20] it follows that
(4.18)Fn(z)=(Pn+1(z))-1P̃n+1(1)(z)=S01/2{⋯-An{zId-Bn}-1AnT}-1}-1zId-B0-A1{⋯-An{zId-Bn}-1AnT}-1zId-B1-A2{{zId-Bn}-1AnTzId-B2-⋯⋯-An{zId-Bn}-1AnT}-1}-1⋯A1T}-1S01/2,
where P̃n(1)(z) denote the first associated polynomials for Pn(z) defined by recursion (4.13) with initial conditions P̃0(1)(z)=0,  P̃1(1)(z)=ζ1T. An application of Markov's Theorem (see [26]), (4.17), and (4.18) now yields
(4.19)∫dΣ(x)z-x=limn→∞Fn(z)=limn→∞{⋯-{zId-ζ2nT-ζ2n+1T}-1ζ2nTζ2n-1T}-1⋯ζ4Tζ3T}-1zId-ζ1T-{⋯-{zId-ζ2nT-ζ2n+1T}-1ζ2nTζ2n-1T}-1zId-ζ2T-ζ3T-{{zId-ζ2nT-ζ2n+1T}-1ζ2nTζ2n-1TzId-ζ4T-ζ5T⋯⋯-{zId-ζ2nT-ζ2n+1T}-1ζ2nTζ2n-1T}-1⋯ζ4Tζ3T}-1ζ2Tζ1T}-1S0=limn→∞{zId-{Id-{zId-⋯-{zId-ζ2n+1T}-1ζ2nT}-1⋯}-1ζ2T}-1ζ1T}-1S0.
If z=0, then we obtain from (4.19) and   (1.3) in [27]
(4.20)∫dΣ(x)-x=-limn→∞∑j=0n+1Xj+1-1ζ2jTζ2j-1TXj-1Xj-1ζ2j-2Tζ2j-3TXj-2Xj-1-1⋯X1X2-1ζ2TS0,
where X0=Id, X1=-ζ1T, and
(4.21)Xn+1=-(ζ2n+1T+ζ2nT)Xn-ζ2nTζ2n-1TXn-1,n≥1.
An induction argument yields Xn=(-1)nζ2n-1Tζ2n-3T⋯ζ1T,n≥1, and the first representation in (4.11) follows. For the second part we note that the polynomials Q̲n(x):=(-1)nA0⋯An-1Qn(x),n≥0, have leading coefficient Id and because of (1.3) they satisfy the recursion
(4.22)Q̲n+1(x)=xQ̲n(x)+A0⋯An-1BnAn-1-1⋯A0-1Q̲n(x)-A0⋯An-1CnTAn-2-1⋯A0-1Q̲n-1(x).
A comparison with the polynomials P̲n(x) in (4.13) now yields
(4.23)A0⋯An-1BnAn-1-1⋯A0-1=-(ζ2nT+ζ2n+1T),A0⋯An-1CnTAn-2⋯A0=ζ2nTζ2n-1T.
Define Tn:=Qn(0),n≥0. Then (4.23)  imply
(4.24)Tn=An-1-1⋯A0-1ζ2n-1Tζ2n-3T⋯ζ1T,∀n≥0.
Therefore, we can define the polynomials Q̂n(x):=Tn-1Qn(x). From (1.3) it follows that these polynomials satisfy the recurrence relation
(4.25)xQ̂n(x)=ÂnQ̂n+1(x)+B̂nQ̂n(x)+ĈnTQ̂n-1(x)
with
(4.26)Ân=Tn-1AnTn+1,B̂n=Tn-1BnTn,ĈnT=Tn-1CnTTn-1,
and Ân+B̂n+ĈnT=0. Consequently we obtain from (4.23)  that
(4.27)Â0⋯Ân-1B̂nÂn-1-1⋯Â0-1=-(ζ2nT+ζ2n+1T),Â0⋯Ân-1ĈnTÂn-2-1⋯Â0-1=ζ2nTζ2n-1T,
and hence
(4.28)ζ2n+1T=Â0⋯ÂnÂn-1-1⋯Â0-1,ζ2nT=Â0⋯Ân-1ĈnTÂn-1-1⋯Â0-1.
Equation (4.11) finally yields
(4.29)∫dΣ(x)x=limn→∞∑j=0n+1Âj-1ĈjTÂj-1-1⋯Â1-1Ĉ1TÂ0-1S0=limn→∞∑j=0n+1Tj+1-1Aj-1CjTTj-1Tj-1Aj-1-1Cj-1TTj-2Tj-1-1⋯T0T1-1A0-1T0S0,
which completes the proof of the theorem.In the following, theα-recurrence condition will be represented in terms of properties of the spectral measure, the corresponding orthogonal matrix polynomials, and the blocks of the infinitesimal generator (1.2). For this purpose, consider the process (Xt,α)t≥0 with state space E defined in (1.1) and infinitesimal generator matrix(4.30)Qα=(B0,αA0,α0C1,αTB1,αA1,αC2,αTB2,αA2,αC3,αTB3,αA3,α0⋱⋱⋱),
where (4.31)An,α:=Qn-1(α)AnQn+1(α),n≥0,Bn,α:=Qn-1(α)BnQn(α),n≥0,Cn,αT:=Qn-1(α)CnTQn-1(α),n≥1.
The corresponding sequence {Qn,α(x)}n≥0 of matrix polynomials satisfies the recurrence relation (4.32)-xQn,α(x)=An+1,αQn+1,α(x)+Bn,αQn,α(x)+Cn,αTQn-1,α(x)
with initial conditions Q-1,α(x)=0, Q0,α(x)=Id. If conditions (2.4) of Theorem 2.1 are satisfied, then the matrix Qα can be symmetrized with the matrices (4.33)Rn,α=RnQn(α),n≥0.
An induction argument shows the representation(4.34)Qn,α(x)=Qn-1(α)Qn(x+α),n≥0,
and therefore (4.35)∫Qn,α(x)dΣα(x)Qm,αT(x)=0,n≠m,
where the matrix measure Σα is defined by (4.36)Σα(0,x]=Σ(α,α+x].
If representation (2.11) holds, it is easy to see that(4.37)eαtP00(t)=∫e-txdΣα(x)S0-1,
and the following remark is a consequence of Theorem 4.1.Remark 4.3.
Assume that the conditions of Theorem4.1 are satisfied and that Σ is a corresponding spectral measure supported in the interval [α,∞). The process (Xt)t≥0 is α-recurrent if and only if
(4.38)ejT∫0∞dΣα(x)xS0-1ej=ejT∫α∞dΣ(x)x-αS0-1ej=∞
for some j∈{1,…,d}. The process is α-positive if
(4.39)eℓTlimt→∞eαtP00(t)eℓ>0
for some ℓ∈{1,…,d}. This is the case if and only if the measure eℓTdΣ(x)S0-1eℓ has a jump in the point x=α.Theorem 4.4.
Assume that the conditions of Theorem2.1 are satisfied and that the corresponding matrix measure Σ is supported in the interval [α,∞) and determined by its moments. The process (Xt)t≥0 is α-recurrent if and only if for some state (0,ℓ)∈E (and then for all states in (0,k)∈E)
(4.40)eℓT∑j=0∞Hj+1-1Aj-1CjTHj-1Hj-1Aj-1-1Cj-1THj-2⋯C1TH1-1A0-1H0S0eℓ=∞,
where Hj:=Qj(α), j≥0.Proof.
Because condition (2.4) holds for the polynomials {Qn(x)}n≥0, this condition is also fulfilled for the polynomials {Qn,α}n≥0 with Rn,α:=RnQn(α),n≥0. From   (4.34) it follows that Qj,α(0)=Id for all j≥0. Therefore we obtain with   (4.12)
(4.41)∫dΣα(x)x=∑j=0∞Aj,α-1Cj,αTAj-1,α-1Cj-1,αT⋯C1,αTA0,α-1S0.
From the representation Aj,α-1Cj,αT=Qj+1(α)Aj-1CjTQj-1(α), j≥0, it follows from Remark 4.3 that the state (0,ℓ) is α-recurrent if and only if
(4.42)eℓT∑j=0∞Hj+1-1Aj-1CjTHj-1Hj-1Aj-1-1Cj-1THj-2⋯C1TH1-1A0-1H0S0eℓ=∞,
where Hj=Qj(α),j≥0.Remark 4.5.
In the cased=1, the results of Theorems 4.1 and 4.4 reduce to known results in the scalar case (see Theorem 5.2(ii), (iii), (vii) in [10]).Remark 4.6.
Assume that the conditions of Theorem4.4 are satisfied, and let Σ(1) be a spectral measure corresponding to the sequence of associated matrix polynomials {Qn(1)(x)}n≥0.(1)
The state(0,ℓ)∈E is α-recurrent if and only if
(4.43)eℓT∫dΣ(x)x-αS0-1eℓ=eℓT{-αId-B0-A0∫dΣ(1)(x)x-αR1TR1C1T}-1eℓ=∞.(2)
The state(0,ℓ)∈E is α-positive if and only if(4.44)eℓTlimt→∞eαtP00(t)eℓ=limz→0zeℓT∫dΣ(x)(z+α)-xS0-1eℓ=eℓTlimz→0{z+αzId+1z(B0-A0∫dΣ(1)(x)(z+α)-xR1TR1C1T)}-1>0.
Note that  conditions (4.3) and (4.4) reduce to recurrence and positive recurrence if α=0. Therefore, with Theorem 4.2 we obtain the following conditions for recurrence and positive recurrence of a quasi-birth-and-death process.Corollary 4.7.
Assume  that the conditions of Theorem2.1 are satisfied and that the corresponding matrix measure Σ is supported in the interval [0,∞) and determined by its moments. The following statements hold.  (1)
The state(i,ℓ)∈E is recurrent if and only if
(4.45)eℓT(∫Qi(x)dΣ(x)QiT(x)x)(∫Qi(x)dΣ(x)QiT(x))-1eℓ=∞,
where eℓ=(0,…,0,1,0,…,0)T. In particular, the state (0,ℓ)∈E is recurrent if and only if
(4.46)eℓT∫0∞dΣ(x)xS0-1eℓ=∞.(2)
The state(0,ℓ) is recurrent if and only if
(4.47)eℓT∑j=0∞Tj+1-1Aj-1CjTTj-1Tj-1Aj-1-1Cj-1TTj-2Tj-1-1⋯T0T1-1A0-1T0S0eℓ=∞
with Tj=Qj(0),j≥0.(3)
The state(0,ℓ) is positive recurrent if and only if the matrix measure eℓTdΣ(x)S0-1eℓ has a jump in the point x=0.Remark 4.8.
(1) LetΣ(1) be a spectral measure supported in [0,∞) corresponding to the associated polynomials {Qn(1)(x)}n≥0 introduced in Theorem 2.4. Then, a combination of Theorem 2.4 and Corollary 4.7 shows that the state (0,ℓ)∈E is recurrent if and only if
(4.48)eℓT∫dΣ(x)xS0-1eℓ=-limz→0eℓT∫dΣ(x)z-xR0TR0eℓ=eℓT{-B0-A0∫dΣ(1)(x)xR1TR1C1T}-1eℓ=∞.
An induction argument shows that
(4.49)Qn(1)(x)=-Q̃n+1(1)(x)S0-1A0,n≥0,
where Q̃n(1)(x) are the first associated polynomials corresponding to Qn(1)(x), and Qn(1)(x) are the associated polynomials of order k=1 corresponding to Qn(x). Therefore it follows for the Stieltjes transform of the spectral measure corresponding to the associated orthogonal polynomials that
(4.50)∫dΣ(1)(x)x=limn→∞∑j=0n+1A0-1S0Zj+1-1Aj+1-1Cj+1TZj-1Zj-1Aj-1⋯⋯A2-1C2TZ1-1A1-1Z0(R1TR1)-1,
where Zj:=Q̃j+1(1)(0).
(2)  A straightforward calculation yields(4.51)eiTΣ({0})ej=limz→0zeiTΦ(z)ej.
From Theorem 2.4 it follows that the state (0,ℓ)∈E is positive recurrent if the condition
(4.52)eℓTlimt→∞P00(t)eℓ=eℓTlimz→0z∫dΣ(x)z-xS0-1eℓ=eℓTlimz→0zR0-1{zId+R0B0R0-1-R0A0∫dΣ(1)(x)z-xR1TR1C1TR0-1}-1R0eℓ=eℓTlimz→0{Id+1z(B0-A0∫dΣ(1)(x)z-xR1TR1C1T}-1eℓ>0
holds.



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*Source: 730543-2010-08-24.xml*