A family of quasi-birth-and-death processes coming from the theory of orthogonal matrix polynomials
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A family of quasi-birth-and-death processes coming from the theory of orthogonal matrix polynomials1 Manuel Domínguez de la Iglesia Department of Mathematics, K. U. Leuven International Workshop on Orthogonal Polynomials and Approximation Theory September 8-12, 2008, Universidad Carlos III de Madrid 1joint work with F. A. Grünbaum
Introduction The family of processes Probabilistic aspects Outline 1Introduction Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes 2The family of processes 3Probabilistic aspects Karlin-McGregor formula Recurrence The invariant measure Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Outline 1Introduction Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes 2The family of processes 3Probabilistic aspects Karlin-McGregor formula Recurrence The invariant measure Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Random walks Transition probability matrix P= b0a0 c1b1a1 c2b2a2 ......... ,bn⩾0,an,cn>0,an+bn+cn=1 Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Random walks Transition probability matrix P= b0a0 c1b1a1 c2b2a2 ......... ,bn⩾0,an,cn>0,an+bn+cn=1 ··· 0 1 2 Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Random walks Transition probability matrix P= b0a0 c1b1a1 c2b2a2 ......... ,bn⩾0,an,cn>0,an+bn+cn=1 ··· b0 b1b2 0 1 2 Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Random walks Transition probability matrix P= b0a0 c1b1a1 c2b2a2 ......... ,bn⩾0,an,cn>0,an+bn+cn=1 ··· a0a1 b0 b1b2 0 1 2 Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Random walks Transition probability matrix P= b0a0 c1b1a1 c2b2a2 ......... ,bn⩾0,an,cn>0,an+bn+cn=1 ··· a0a1 c1c2 b0 b1b2 0 1 2 Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Properties n-step transition probability matrix: Prob {Ei→Ejin nsteps}=Pn ij =X k1,k2,...,kn−1 Pik1Pk1k2···Pkn−1j Recurrence Transient Recurrent ◮Positive or ergodic ◮Null Invariant distribution or measure A non-null vector π= (π0,π1,π2,...)with non-negative components πP=π Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Karlin and McGregor (1959): integral representation of Pn Karlin-McGregor formula Pn ij =Z1 −1 tnqi(t)qj(t)dω(t)Z1 −1 qj(t)2dω(t) Invariant measure or distribution π= (π0,π1,π2,...)with πP=π ⇒πi=a0a1···ai−1 c1c2···ci =1 R1 −1q2 i(t)dω(t)=1 kqik2 Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Orthogonal matrix polynomials Krein (1949): orthogonal matrix polynomials (OMP) Orthogonality: weight matrix W. Matrix valued inner product: hP,QiW=ZR P(t)dW(t)Q∗(t)∈CN×N,P,Q∈CN×N[t] Using Gram-Schmidt we get a family of OMP (Qn)n. One gets a three term recurrence relation tQn(t) = AnQn+1(t) + BnQn(t) + CnQn−1(t),n⩾0,det(An)6=0. Jacobi operator (block tridiagonal) t Q0(t) Q1(t) Q2(t) . . . = B0A0 C1B1A1 C2B2A2 ......... Q0(t) Q1(t) Q2(t) . . . Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Orthogonal matrix polynomials Krein (1949): orthogonal matrix polynomials (OMP) Orthogonality: weight matrix W. Matrix valued inner product: hP,QiW=ZR P(t)dW(t)Q∗(t)∈CN×N,P,Q∈CN×N[t] Using Gram-Schmidt we get a family of OMP (Qn)n. One gets a three term recurrence relation tQn(t) = AnQn+1(t) + BnQn(t) + CnQn−1(t),n⩾0,det(An)6=0. Jacobi operator (block tridiagonal) t Q0(t) Q1(t) Q2(t) . . . = B0A0 C1B1A1 C2B2A2 ......... Q0(t) Q1(t) Q2(t) . . . Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Orthogonal matrix polynomials Krein (1949): orthogonal matrix polynomials (OMP) Orthogonality: weight matrix W. Matrix valued inner product: hP,QiW=ZR P(t)dW(t)Q∗(t)∈CN×N,P,Q∈CN×N[t] Using Gram-Schmidt we get a family of OMP (Qn)n. One gets a three term recurrence relation tQn(t) = AnQn+1(t) + BnQn(t) + CnQn−1(t),n⩾0,det(An)6=0. Jacobi operator (block tridiagonal) t Q0(t) Q1(t) Q2(t) . . . = B0A0 C1B1A1 C2B2A2 ......... Q0(t) Q1(t) Q2(t) . . . Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Orthogonal matrix polynomials Krein (1949): orthogonal matrix polynomials (OMP) Orthogonality: weight matrix W. Matrix valued inner product: hP,QiW=ZR P(t)dW(t)Q∗(t)∈CN×N,P,Q∈CN×N[t] Using Gram-Schmidt we get a family of OMP (Qn)n. One gets a three term recurrence relation tQn(t) = AnQn+1(t) + BnQn(t) + CnQn−1(t),n⩾0,det(An)6=0. Jacobi operator (block tridiagonal) t Q0(t) Q1(t) Q2(t) . . . = B0A0 C1B1A1 C2B2A2 ......... Q0(t) Q1(t) Q2(t) . . . Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Orthogonal matrix polynomials Krein (1949): orthogonal matrix polynomials (OMP) Orthogonality: weight matrix W. Matrix valued inner product: hP,QiW=ZR P(t)dW(t)Q∗(t)∈CN×N,P,Q∈CN×N[t] Using Gram-Schmidt we get a family of OMP (Qn)n. One gets a three term recurrence relation tQn(t) = AnQn+1(t) + BnQn(t) + CnQn−1(t),n⩾0,det(An)6=0. Jacobi operator (block tridiagonal) t Q0(t) Q1(t) Q2(t) . . . = B0A0 C1B1A1 C2B2A2 ......... Q0(t) Q1(t) Q2(t) . . . Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Quasi-birth-and-death processes Transition probability matrix P= B0A0 C1B1A1 C2B2A2 ......... , (An)ij ,(Bn)ij ,(Cn)ij ⩾0,det(An),det(Cn)6=0 X j (An)ij + (Bn)ij + (Cn)ij =1,i=1,...,N Particular case: pentadiagonal matrix P= b0a0 c1b1 d00 a1d1 0 e2c2 0e3 b2a2 c3b3 d20 a3d3 0 0e4c4 0e5 b4a4 c5b5 d40 a5d5 ... ............ Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Quasi-birth-and-death processes Transition probability matrix P= B0A0 C1B1A1 C2B2A2 ......... , (An)ij ,(Bn)ij ,(Cn)ij ⩾0,det(An),det(Cn)6=0 X j (An)ij + (Bn)ij + (Cn)ij =1,i=1,...,N Particular case: pentadiagonal matrix P= b0a0 c1b1 d00 a1d1 0 e2c2 0e3 b2a2 c3b3 d20 a3d3 0 0e4c4 0e5 b4a4 c5b5 d40 a5d5 ... ............ Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Network ··· ··· 1 3 5 2 4 6 Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Network ··· ··· b0 b1 b2b4 b3b5 1 3 5 2 4 6 Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes Karlin-McGregor formula Pn ij =Z1 −1 tnQi(t)dW(t)Q∗ j(t)Z1 −1 Qj(t)dW(t)Q∗ j(t)−1 Invariant measure or distribution Non-null vector with non-negative components π= (π0;π1;···)≡(π0 1,π0 2,...,π0 N;π1 1,π1 2,...,π1 N;···) such that πP=π ⇒πj i=? Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Outline 1Introduction Random walks Orthogonal matrix polynomials Quasi-birth-and-death processes 2The family of processes 3Probabilistic aspects Karlin-McGregor formula Recurrence The invariant measure Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects The family of processes (N=2) Conjugation W(t) = T∗f W(t)T where T= 1 1 0−α+β−k+2 β−k+1 Grünbaum-MdI (2008) f W(t) = tα(1−t)βkt +β−k+1(1−t)(β−k+1) (1−t)(β−k+1) (1−t)2(β−k+1) t∈(0,1),α,β > −1, 0 <k< β +1 Pacharoni-Tirao (2006) Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects We consider the family of OMP (Qn(t))nsuch that Three term recurrence relation tQn(t) = AnQn+1(t) + BnQn(t) + CnQn−1(t),n=0,1,... where the Jacobi matrix is stochastic Choosing Q0(t) = Ithe leading coefficient of Qnis Γ(β+2)Γ(α+β+2n+2) Γ(α+β+n+2)Γ(β+n+2) k+n k−n(α+β+2n+2) (α+β+n+2)(α+β−k+2) 0(n+α+β−k+2)(α+β+2n+2) (α+β+n+2)(α+β−k+2)! Moreover, the corresponding norms are diagonal matrices: kQnk2 W=Γ(n+α+1)Γ(n+1)Γ(β+2)2(n+α+β−k+2) Γ(n+α+β+2)Γ(n+β+2)× n+k k(2n+α+β+2)0 0(n+α+1)(n+k+1) (β−k+1)(2n+α+β+3)(n+α+β+2)! Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects We consider the family of OMP (Qn(t))nsuch that Three term recurrence relation tQn(t) = AnQn+1(t) + BnQn(t) + CnQn−1(t),n=0,1,... where the Jacobi matrix is stochastic Choosing Q0(t) = Ithe leading coefficient of Qnis Γ(β+2)Γ(α+β+2n+2) Γ(α+β+n+2)Γ(β+n+2) k+n k−n(α+β+2n+2) (α+β+n+2)(α+β−k+2) 0(n+α+β−k+2)(α+β+2n+2) (α+β+n+2)(α+β−k+2)! Moreover, the corresponding norms are diagonal matrices: kQnk2 W=Γ(n+α+1)Γ(n+1)Γ(β+2)2(n+α+β−k+2) Γ(n+α+β+2)Γ(n+β+2)× n+k k(2n+α+β+2)0 0(n+α+1)(n+k+1) (β−k+1)(2n+α+β+3)(n+α+β+2)! Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects We consider the family of OMP (Qn(t))nsuch that Three term recurrence relation tQn(t) = AnQn+1(t) + BnQn(t) + CnQn−1(t),n=0,1,... where the Jacobi matrix is stochastic Choosing Q0(t) = Ithe leading coefficient of Qnis Γ(β+2)Γ(α+β+2n+2) Γ(α+β+n+2)Γ(β+n+2) k+n k−n(α+β+2n+2) (α+β+n+2)(α+β−k+2) 0(n+α+β−k+2)(α+β+2n+2) (α+β+n+2)(α+β−k+2)! Moreover, the corresponding norms are diagonal matrices: kQnk2 W=Γ(n+α+1)Γ(n+1)Γ(β+2)2(n+α+β−k+2) Γ(n+α+β+2)Γ(n+β+2)× n+k k(2n+α+β+2)0 0(n+α+1)(n+k+1) (β−k+1)(2n+α+β+3)(n+α+β+2)! Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects The choice of the leading coefficient is motivated by the fact that Qn(1)eN=eN where eN= (1,1,··· ,1)T. Consequently, the Jacobi matrix is stochastic: 1·Qn(1)eN=AnQn+1(1)eN+BnQn(1)eN+CnQn−1(1)eN q q eN=(An+Bn+Cn)eN Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects The choice of the leading coefficient is motivated by the fact that Qn(1)eN=eN where eN= (1,1,··· ,1)T. Consequently, the Jacobi matrix is stochastic: 1·Qn(1)eN=AnQn+1(1)eN+BnQn(1)eN+CnQn−1(1)eN q q eN=(An+Bn+Cn)eN Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Particular case α=β=0, k=1/2 An= (2n+1)(n+2)2 2(2n+3)2(n+1)0 2(n+2) (2n+5)(2n+3)2 n+3 2(2n+5) Bn= 1 2−4n2+8n−1 2(2n+1)2(2n+3)2 n+2 (2n+3)2(n+1) 2(n+1) (2n+1)(2n+3)2 1 2−1 (2n+3)2 Cn= n2(2n+3) 2(2n+1)2(n+1) n (n+1)(2n+1)2 0n 2(2n+1) Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Particular case α=β=0, k=1/2 Pentadiagonal Jacobi matrix: P= 5 9 2 9 2 9 2 9 7 18 4 45 3 10 5 36 1 18 107 225 3 50 27 100 1 6 4 75 23 50 6 175 2 7 14 75 2 75 597 1225 4 147 40 147 1 5 6 245 47 98 8 441 5 18 81 392 3 196 1955 3969 5 324 175 648 ............... Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Karlin-McGregor formula Recurrence The invariant measure The invariant measure Invariant measure The row vector π= (π0;π1;···) πn=1 kQnk2 W1,1 ,1 kQnk2 W2,2 ,··· ,1 kQnk2 WN,N,n⩾0 is an invariant measure of P Particular case N=2, α=β=0, k=1/2: πn=2(n+1)3 (2n+3)(2n+1),(n+1)(n+2) 2n+3,n⩾0 π=2 3,2 3;16 15,6 5;54 35,12 7;128 63 ,20 9;250 99 ,30 11;432 143,42 13;686 195,56 15;··· Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Karlin-McGregor formula Recurrence The invariant measure The invariant measure Invariant measure The row vector π= (π0;π1;···) πn=1 kQnk2 W1,1 ,1 kQnk2 W2,2 ,··· ,1 kQnk2 WN,N,n⩾0 is an invariant measure of P Particular case N=2, α=β=0, k=1/2: πn=2(n+1)3 (2n+3)(2n+1),(n+1)(n+2) 2n+3,n⩾0 π=2 3,2 3;16 15,6 5;54 35,12 7;128 63 ,20 9;250 99 ,30 11;432 143,42 13;686 195,56 15;··· Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Karlin-McGregor formula Recurrence The invariant measure The shape of the invariant measure (N=2) The invariant measure πsuch that πP=πis given by π= (π0;π1;···) where πn,n⩾0, is a 2-dimensional vector. We have lim n→∞ πn= (∞,∞),if β > −1 2, 4 π(2k,1−2k),if β= −1 2, (0,0),if −1< β < −1 2. Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Karlin-McGregor formula Recurrence The invariant measure The shape of the invariant measure (N=2) The invariant measure πsuch that πP=πis given by π= (π0;π1;···) where πn,n⩾0, is a 2-dimensional vector. We have lim n→∞ πn= (∞,∞),if β > −1 2, 4 π(2k,1−2k),if β= −1 2, (0,0),if −1< β < −1 2. Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Karlin-McGregor formula Recurrence The invariant measure For β > −1/2 0 1 2 3 4 5 6 7 0.5 0.6 0.7 0.8 0.9 1 1.1 Π1n Π2n Figure: α= −0.8,β= −0.4,k=0.3 Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Karlin-McGregor formula Recurrence The invariant measure For β= −1/2 0 1 2 3 4 5 6 7 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Π1n Π2n Figure: α= −0.92,β= −0.5,k=0.3 0 1 2 3 4 5 6 7 0.5 0.55 0.6 0.65 0.7 0.75 0.8 Π1n Π2n Figure: α= −0.9,β= −0.5,k=0.25 Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Karlin-McGregor formula Recurrence The invariant measure For −1< β < −1/2 0 1 2 3 4 5 6 7 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Π1n Π2n Figure: α= −0.9,β= −0.6,k=0.2 Manuel Domínguez de la Iglesia A family of QBD processes
Summary A stochastic block tridiagonal matrix Pgives rise to a quasi-birth-and-death process. The probabilistic aspects of these processes can be greatly simplified if we have the explicit expression of the weight matrix W(t). We start from a rich group theoretical situation that yields W(t)as well as an stochastic Jacobi matrix P. Therefore, we have a nonhomogeneous quasi-birth-and-death process depending on 4 parameters, α,β,k,N, where we can study recurrence. Also we have an explicit expression of the invariant measure π. F. A. Grünbaum and M. D. de la Iglesia, Matrix valued orthogonal polynomials arising from group representation theory and a family of quasi-birth-and-death processes, SIMAX 30 (2008), 741–761.
Introduction The family of processes Probabilistic aspects Karlin-McGregor formula Recurrence The invariant measure Summary A stochastic block tridiagonal matrix Pgives rise to a quasi-birth-and-death process. The probabilistic aspects of these processes can be greatly simplified if we have the explicit expression of the weight matrix W(t). We start from a rich group theoretical situation that yields W(t)as well as an stochastic Jacobi matrix P. Therefore, we have a nonhomogeneous quasi-birth-and-death process depending on 4 parameters, α,β,k,N, where we can study recurrence. Also we have an explicit expression of the invariant measure π. F. A. Grünbaum and M. D. de la Iglesia, Matrix valued orthogonal polynomials arising from group representation theory and a family of quasi-birth-and-death processes, SIMAX 30 (2008), 741–761. Manuel Domínguez de la Iglesia A family of QBD processes
Introduction The family of processes Probabilistic aspects Karlin-McGregor formula Recurrence The invariant measure Summary A stochastic block tridiagonal matrix Pgives rise to a quasi-birth-and-death process. The probabilistic aspects of these processes can be greatly simplified if we have the explicit expression of the weight matrix W(t). We start from a rich group theoretical situation that yields W(t)as well as an stochastic Jacobi matrix P. Therefore, we have a nonhomogeneous quasi-birth-and-death process depending on 4 parameters, α,β,k,N, where we can study recurrence. Also we have an explicit expression of the invariant measure π. F. A. Grünbaum and M. D. de la Iglesia, Matrix valued orthogonal polynomials arising from group representation theory and a family of quasi-birth-and-death processes, SIMAX 30 (2008), 741–761. Manuel Domínguez de la Iglesia A family of QBD processes