EXPONENTIAL ERGODICITY FOR SINGLE-BIRTH PROCESSES

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1 J. Appl. Prob. 4, (2004) Prnted n Israel Appled Probablty Trust 2004 EXPONENTIAL ERGODICITY FOR SINGLE-BIRTH PROCESSES YONG-HUA MAO and YU-HUI ZHANG, Beng Normal Unversty Abstract An explct, computable, and suffcent condton for exponental ergodcty of snglebrth processes s presented. The correspondng crteron for brth death processes s proved usng a new method. As an applcaton, some suffcent condtons are obtaned for exponental ergodcty of an extended class of contnuous-tme branchng processes and of multdmensonal Q-processes, by comparson methods. Keywords: Exponentally ergodc; sngle-brth process; brth death process 2000 Mathematcs Subect Classfcaton: Prmary 60J25; 60J75; 60J80. Introducton Consder a contnuous-tme, rreducble Marov chan wth transton probablty matrx P(t) = (p (t)) on a countable state space Z + ={0,, 2,...} wth statonary dstrbuton (π > 0 : Z + ). In the study of the theory of Marov chans, there are tradtonally three types of ergodcty: ordnary ergodcty (or postve recurrence), exponental ergodcty, and strong ergodcty (or unform ergodcty). The man purpose of ths paper s to deal wth the second of these for sngle-brth processes, whch are also called upwardly sp-free processes (see [], [2]): lm t e βt p (t) π =0for some β>0. The Q-matrx of a sngle-brth process (q :, Z + ) s as follows: q,+ > 0 and q,+ = 0 for all Z + and 2. Throughout the paper, we consder only totally stable and conservatve Q-matrces: q = q = = q < for all Z +. Defne and q () n = q n for = 0,...,n (n Z + ) m 0 =, m n = ( + q 0 q n,n+ n =0 =0 ) q n () m, n, F n (n) =, F n () = n q n () q F (), 0 <n, n,n+ = d 0 = 0, d n = ( n ) + q n () q d, n. n,n+ Receved 6 May 2003; revson receved 20 February Postal address: Department of Mathematcs, Beng Normal Unversty, Beng 00875, P. R. Chna. Emal address: maoyh@bnu.edu.cn Emal address: zhangyh@bnu.edu.cn 022

2 Exponental ergodcty for sngle-brth processes 023 Then m n = q n tae a smple form: 0 + d n for all n Z +. For brth death processes (a,b ), these quanttes m n = µ n b n µ[0,n], n = b 0 µ n b n, d n = µ n b n µ[,n], n, where µ 0 =,µ = b 0 b b /a a 2 a ( ), and µ[, ] = = µ. The man advantage of sngle-brth processes s that the ext boundary conssts of at most a sngle extremal pont and so explct crtera are expected. We gve here the crtera for several classcal problems (see [5], [6], [8], [9], [20]). Frst, the process s unque (regular) f and only f R := n=0 m n =. Next, assume that the Q-matrx s rreducble; then the process s recurrent f and only f n=0 F n (0) =. In the regular case, t s ergodc f and only f d := sup Z+ ( n=0 d n )/( n=0 F n (0) )<, and t s strongly ergodc f and only f sup n=0 Z+ (F n (0) d d n )<. The four crtera are all explct (dependng on the Q-matrx (q ) only, wthout usng test functons) and computable. Ths advantage maes sngle-brth processes a useful tool when studyng more complcated processes (see [5, Chapters 3 and 4] and [6]). Now, t s natural to loo for an explct crteron of exponental ergodcty for ths class of processes. Meanwhle, there are a number of prncpal nvestgatons nto the exponental ergodcty of brth death processes (see [8], [], [4], [7], [8]) and such a crteron has been obtaned recently by Mu-Fa Chen (see [7, Theorem 3.5]). But the dffculty s that sngle-brth processes are n general rreversble. In ths paper, we gve a partal answer. In fact, Theorem. s a generalzaton of the crteron for brth death processes. For research on the ergodcty of nonhomogeneous Marov chans, see [2]. One of the most mportant problems for possble applcatons s the boundng of the rate of convergence (see [7], [], [2], [4], [7], [8]). So the lower bound of the rate of exponental convergence for sngle-brth processes s studed n Theorem.. Denote the rate of exponental convergence by α = sup{α : p (t) π =O(e αt ) as t for all, Z + }. Theorem.. Let the sngle-brth Q-matrx be regular and rreducble. If q := nf 0 q > 0 and M := sup >0 = q,+ <, (.) then the process s exponentally ergodc and α (4M) q 0. In addton, f q 0 nf >0 q then α (4M). The condton (.) s necessary for the exponental ergodcty of brth death processes (a,b ). Equvalently, δ := sup µ b >0 µ <. (.2) = Now we dscuss exponental ergodcty for a class of multdmensonal Q-processes. In [5, Theorem 4.58] and [6], a method that reduces the multdmensonal problems to onedmensonal ones s proposed. By eepng the dea n mnd, some suffcent condtons for strong ergodcty of multdmensonal Q-processes are obtaned.

3 024 Y.-H. MAO AND Y.-H. ZHANG Theorem.2. Let E be a countable set and let (q(x, y) : x,y E) be a conservatve Q-matrx. Suppose that there exsts a partton {E } of E such that =0 E = E wth E 0 ={θ}, where θ E s a reference pont. Next, suppose that () f q(x,y) > 0 and x E, then y + E for all 0; () for all x E and all 0, q(x,y) > 0; y E + () for all 0, C := sup{q(x) : x E } <. Defne a conservatve Q-matrx (q :, Z + ) as follows: { } sup q(x,y) : x E f = +, y E { q = } nf q(x,y) : x E f <, y E 0, otherwse f =. Moreover, suppose that both (q(x, y)) and (q ) are rreducble and that (q ) s regular. If M<, where M s as defned n Theorem., then the (q )-process and the (q(x, y))-process are both exponentally ergodc. The remander of the paper s organzed as follows. In the next secton, the proofs of Theorems. and.2 are gven and some examples are llustrated along wth some remars. As applcatons, n Secton 3, some suffcent condtons for exponental ergodcty for an extended class of tme-contnuous branchng processes are presented. 2. Proofs of Theorems. and.2 In ths secton, we present the proofs of Theorems. and.2 n detal. 2.. Proof of Theorem. In vew of Theorem 4.45(2) of [5], the condton q>0s ndeed necessary. We dvde the rest of the proof nto three parts. (a) From [5, Theorem 4.45(2)], the sngle-brth process s exponentally ergodc f and only f, for some λ wth 0 <λ<q for all Z +, the system of nequaltes q y λy,, (2.) has a nonnegatve fnte soluton (y ). We need to construct a soluton (g ) to the equaton (2.) forafxedλ wth 0 <λ<q. Frst, defne an operator II (f ) = f =+ f q,+,.

4 Exponental ergodcty for sngle-brth processes 025 Ths s an analogue of the operator I(f)used many tmes n [7]. It ndcates a ey pont n ths proof, whch comes from the study of the frst egenvalue. Next, defne ϕ = q 0,. Then ϕ s ncreasng n and ϕ = q 0. Let f = cq 0 q0 ϕ for some c>. Then f s ncreasng and f = cq 0. Fnally, defne g = f II (f ). Then g s ncreasng and g = = f q,+ By Lemma 3.6 of [7], t follows that g = cq 0 q0 2Mcq 0 q0 f q 2 = c>. =+ ϕ /2 + ϕ q,+ 2Mcq 0 <,. Let g 0 =. Then g < for all 0. We now determne λ usng (2.). When =, we get λ (c )c II (f ). When 2, we should have that λg q () =0 For ths, t suffces that λg q () =0 = q,+ = f. =+ = = f q,+ f q,+ f q,+ q,+ q,+ q,+ =+ =+ =+ f q,+ f q,+ f q,+. (2.2) In other words, for (2.), we need only λ f /g = II (f ) for all 2 and λ (c )c II (f ). Then we can tae any λ such that ( ) ( ) c 0 < λ < λ(c) := II (f ) nf c II (f ) q, (2.3) 2 provded the rght-hand sde of (2.3) s postve or, equvalently, sup 2 II (f ) <. To prove the last property, defne another operator I (f ) = f + f =+ f q,+,,

5 026 Y.-H. MAO AND Y.-H. ZHANG whch s exactly the one used many tmes n [7]. By the proporton property, we get sup II (f ) sup I (f ). By Lemma 3.6 of [7] and the condton M<, t follows that I (f ) = ϕ+ ϕ =+ ϕ q,+ 2M q 0 ( ϕ + ϕ ) ϕ + 4M for all. Therefore, sup II (f ) 4M < as requred. We have thus constructed a soluton (g ) to (2.) wth g < for all. Ths mples the exponental ergodcty of the sngle-brth process. By the defnton of,wehaveq,+ q ( ) for all. Hence, nf II (f ) M nf >0 q, so we can rewrte (2.3) as ( ) ( ) c 0 < λ < λ(c) := II (f ) nf c II (f ) q 0. (2.4) 2 So, by (2.4) and the above dscusson, we obtan that the exponental convergence rate α lm λ(c) = nf II (f ) q 0 (4M) q 0. c If q 0 nf >0 q then q 0 M. Hence, we have α (4M). For the remander of ths proof, we consder brth death processes only. (b) Let σ 0 = nf{t at or followng the frst umpng tme such that X t = 0}. Suppose that the process s exponentally ergodc. By [5, Theorem 4.44(2)], there exsts a λ wth 0 <λ<q for all such that E 0 e λσ 0 <. Defne e 0 (λ) = 0 e λt P [σ 0 >t] dt, Z +. Then E e λσ 0 = λe 0 (λ)+. By [5, p. 48], e 0 (λ) < for all. Furthermore, E e λσ 0 < for all. Note that f the startng pont s not 0, then σ 0 s equal to the frst httng tme: Then E e λτ 0 < for all. Defne m (n) τ 0 = nf{t >0 : X(t) = 0}. >E e λτ 0 = = E τ0 n. The Taylor expanson n=0 λ n n! m(n) (2.5) leads us to estmate the moments m (n). By a result due to Wang (see [5, p. 525]), we have m () = µ b =+ µ, m (n) = n µ b =+ µ m (n ), n 2. (2.6)

6 Exponental ergodcty for sngle-brth processes 027 Obvously, m (n) m (n) m (n) n µ b f. By (2.6), t follows that = µ m (n ) ( n µ b = µ ) m (n ), n 2, and Hence, by nducton, m () µ b µ. = ( m (n) n! µ b ) n µ, n. = Combnng ths wth (2.5), we have whch mples that ( λ µ b n= λ µ b ) n µ <, = µ <. = Tang the supremum over, we obtan δ λ <. Hence, the necessty s proved. (c) To complete the proof of the theorem, t suffces to show that nf q = 0 δ =. (2.7) From Proposton 5.3 of [9], we now that, f nf q = 0, then the frst egenvalue (spectral gap) λ s 0. For brth death processes, by Theorem 3.5 of [7], we have (4δ) λ ( µ )δ, so the proof of (2.7) s trval. Remar 2.. For brth death processes, Theorem 9.2 of [5] tells us that the frst egenvalue λ concdes wth the exponental convergence rate α. So Theorem 3.5 of [7] gves us the estmates of the exponental convergence rate, and at the same tme t does ndeed gve us an explct crteron for exponental ergodcty of brth death processes for the frst tme. In the proof of Theorem., we have presented another proof of the crteron. Remar 2.2. In the above proof, not only do we prove the exponental ergodcty, but also we obtan an ncreasng soluton (whch s very mportant) to the equaton (2.) for sngle-brth processes. In partcular, for a brth death process, once the process s exponentally ergodc, an ncreasng soluton to (2.) s obtaned as follows: g 0 =, g = f II (f ) = µ b =+ µ f,, where f = ca b0 ϕ and ϕ = (µ b ) for all and some c>.

7 028 Y.-H. MAO AND Y.-H. ZHANG Remar 2.3. Snce brth death processes are specal cases of sngle-brth processes, by (.2) and the crteron for strong ergodcty of sngle-brth processes, t seems that a more reasonable suffcent condton should be M := sup 0 ( d d ) <. For brth death processes, snce d d / = =+ (q,+ ) = =+ µ for all 0, t follows that M = M = δ. Example 2.. Let q n,n+ = for all n 0, q 0 =, q n,n 2 = for all n 2, and q = 0 for other =. Then the sngle-brth process s exponentally ergodc and not strongly ergodc. Moreover, we have α (4C), where C s gven n (2.8), below. We prove ths asserton n detal as follows. Obvously, the process s regular and recurrent. By computaton, we now that {F n (0) } are Fbonacc numbers: F n (0) = [A n+ ( B) n+ ]=:F n, n 0, 5 where A = ( 5 + )/2 and B = ( 5 )/2. Note that Fbonacc numbers have the property that n =0 F n = F n+2 F for n 0. So d n = F n+ F for n 0. By the facts that AB =, A B =, and A + B = 5, t s not dffcult to show that n=0 d A> n=0 = F n+3 (n + 3)F A, n. F n+2 F So d = A. In addton, we can prove that d n / n A, mplyng that ˆd := sup n 0 (d n / n ) = A = d. Therefore, sup (F n (0) 0 n=0 d d n) = sup 0 n=0 = sup 0 ( (F n A F n+ + F ) ( + ) + ( + B2 )( ( B) + ) 5( + B) ) =, whch mples that the process s not strongly ergodc. As ponted out by a referee, the absence of strong ergodcty for the process seems to be well nown. Ths follows from the sp-free property and from boundedness of the Q-matrx. Note that F n (A n+ )/ 5 A n / 5 for n. Thus, A(A 2 ( B/A) B 2 5/A ) M sup A 2 (A 2 + B 4 + 5B) =: C<. (2.8) >0 A So, by Theorem., t follows that the process s exponentally ergodc and α (4M) (4C). In addton, (A M 2 ( B/A) + B 2 5/A + )( ( B) + ) = sup 0 ( B/A) + C<.

8 Exponental ergodcty for sngle-brth processes 029 Example 2.2. () Tae a n = b n for n. Then the process s exponentally ergodc f and only f sup >0 = /a <. () Tae a n = νb n for n, where ν>. Then the process s exponentally ergodc f and only f sup >0 = /(ν a ) = sup >0 /(ν a + )<. If, n addton, nf >0 a > 0, then the process must be exponentally ergodc Proof of Theorem.2 By Theorems 3.9 and 4.58 of [5], the (q(x, y))-process s regular. At present, (q ) s a regular, rreducble sngle-brth Q-matrx. By Theorem., the (q )-process s exponentally ergodc. To prove exponental ergodcty of the (q(x, y))-process, we need only to show that the equatons q(x, y)(g(y) g(x)) λg(x), x = θ, y =x q(θ,y)g(y) < y =θ have a fnte soluton (g(y)) wth g for some λ>0. For ths, by the assumptons and Remar 2.2, let (g ) be an ncreasng soluton to (2.) wth g for all 0, and tae g(x) = g for x E, 0. Now, for x = θ, there exsts exactly one such that x E. Hence, on the one hand, by the defnton of (q ), y =x q(x, y)(g(y) g(x)) = y E q(x,y)(g g ) + q (g g ) + q,+ (g + g ) = = q (g g ) λg = λg(x) and, on the other hand, q(θ,y)g(y) = y =θ We have thus constructed a soluton, as desred. y E q(θ,y)g = q 0 g <. 3. Applcatons y E + q(x,y)(g + g ) In ths secton, we dscuss exponental ergodcty for an extended class of tme-contnuous branchng processes. The orgnal branchng process can be descrbed as follows. Let α>0 and let (p : Z + ) be a probablty dstrbuton. Then the process has death rate αp 0 : (for ) and growth rate αp + : + (for, Z + ). Note that the process absorbs at state 0. In [0], an extended class of branchng processes wth the followng Q-matrx

9 030 Y.-H. MAO AND Y.-H. ZHANG s ntroduced (see also [3], [4]): q 0, > = 0, q 0, = = 0, r p 0, =,, q = r p +, = +,,, r ( p ), =, 0 otherwse for, Z +, (3.) where r > 0 for all and 0 <q 0 := q 0 <. The typcal case we are nterested n s where q 0 = p or p + ( ) and r s a polynomal wth degree θ ( ). Defne M = = p and Ɣ = = p +. It s easy to chec that Ɣ = M + p 0. Hence, M < f and only f Ɣ<p 0, and M = f and only f Ɣ = p 0. Let ν = p 0 /Ɣ. Based on the same comparson dea as n [0], some suffcent condtons for exponental ergodcty are obtaned as follows. Theorem 3.. Let (q ) be the rreducble Q-matrx gven by (3.). Assume that M. If sup = >0 /r < and q 0 ( ) <, (3.2) r = = then the (q )-process s exponentally ergodc. In addton, under the same assumpton, f sup >0 ν r sup > = =+ ν r <, ν r ν, (3.3) and ν q 0 ν <, (3.4) r = = then the (q )-process s exponentally ergodc. Proof. Frst, under the assumpton, bytheorems.2 and.3 of [0], we now that the process s unque and recurrent. Next, consder the brth death process ( p (t)) wth a = b = r p 0 for and b 0 some postve constant. By Example 2.2() and the assumpton, the process ( p (t)) s exponentally ergodc; furthermore, by Remar 2.2, we have the followng ncreasng soluton (g ) to (2.) for some λ wth 0 <λ<q: g 0 =, g = cr,. r Snce g + g = cr =+ =+ r,,

10 Exponental ergodcty for sngle-brth processes 03 as, g + g s decreasng. Thus, on the one hand, for, q (g g ) = q, (g g ) + q,+ (g + g ) = r p 0 (g g ) + = r p + (g + g ) = = r p 0 (g g ) + r Ɣ(g + g ) a (g g ) + b (g + g ) λg, and, on the other hand, by (3.2), we have = q 0 g <. By these facts, t follows that (p (t)) s exponentally ergodc. Assume now that M <. Consder the brth death process ( ˆp (t)) wth a = r p 0, b = r Ɣ for, and b 0 some postve constant. Snce Ɣ<p 0, by Example 2.2() and the assumpton, the process ( ˆp (t)) s exponentally ergodc. As above, we have the followng ncreasng soluton (g ) to (2.): g 0 =, g = cr ν =+ ν r,. Note that g + g = cr ν + ν,. r =+ By (3.3), we now that g + g s decreasng as. Thus, t follows smlarly that q (g g ) a (g g ) + b (g + g ) λg,. = By (3.3) and (3.4), we have = q 0 g <. Puttng these facts together, t follows that (p (t)) s exponentally ergodc. From Theorem 3., we obtan the followng corollary. Corollary 3.. Let (q ) be the rreducble Q-matrx gven by (3.). Assume that M. If δ := sup = >0 /r < and = q 0 <, then the (q )-process s exponentally ergodc. In addton, under the same assumpton, f sup = >0 /ν r <, /r s decreasng n 2, and = q 0 <, then the (q )-process s exponentally ergodc. Proof. As n the frst part of the proof of Theorem 3., by Lemma 3.6 of [7], we obtan that g 2(δp0 )ca b0 = 2δcr µ b ϕ+ + 2δcr. By the assumptons (rather than by (3.2)), we obtan = q 0 g <. The frst asserton follows easly. By the assumptons, both (3.3) and (3.4) hold. Thus, the second asserton follows from Theorem 3..

11 032 Y.-H. MAO AND Y.-H. ZHANG Example 3.. Let (q ) be the rreducble Q-matrx gven by (3.), where r = θ and M. () By [3] and [20, Theorem.2], we now that f M = and θ > 2, then the process s strongly ergodc, and when M <, the process s strongly ergodc f and only f θ>. Note that, by [3] and [3, Corollary 2.2], f M = and 0 <θ, then the process cannot be strongly ergodc. () Assume that = q 0 <. By Corollary 3., the process s exponentally ergodc provded that ether M = and θ 2, or M < butθ 2. Remar 3.. By [5, Corollary 4.49], Theorem.4() of [0] tells us that, f M <, lm sup /r < and = q 0 <, then the process s exponentally ergodc. Compared wth Corollary 3., ths result does not need the decreasng property. However, applyng t n Example 3., we obtan only that the process s exponentally ergodc, provded that M < and θ. Remar 3.2. In fact, we can prove that f M < and the r are ncreasng, then the process s strongly ergodc f and only f = /r <. Acnowledgements The authors would le to acnowledge Professor Mu-Fa Chen for valuable dscussons durng our wor on the paper, and a referee s helpful comments on the frst verson of the paper. Ths research was supported n part by NSFC (grants 020, , 00003), 973 Proect and RFDP (grant ). References [] Anderson, W. J. (99). Contnuous-Tme Marov Chans. Sprnger, New Yor. [2] Andreev, D. B. et al. (2002). On ergodcty and stablty estmates for some nonhomogeneous Marov chans. J. Math. Sc. 2, [3] Chen, A.-Y. (2002). Ergodcty and stablty of generalsed Marov branchng processes wth resurrecton. J. Appl. Prob. 39, [4] Chen, A.-Y. (2002). Unqueness and extncton propertes of generalsed Marov branchng processes. J. Math. Anal. Appl. 274, [5] Chen, M.-F. (992). From Marov Chans to Non-Equlbrum Partcle Systems. World Scentfc, Sngapore. [6] Chen, M.-F. (999). Sngle brth processes. Chnese Ann. Math. B 20, [7] Chen, M.-F. (2000). Explct bounds of the frst egenvalue. Sc. Chna A 43, [8] Chen, M.-F. (200). Explct crtera for several types of ergodcty. Chnese J. Appl. Statst. 7, [9] Chen, M.-F. (2004). Egenvalues, Inequaltes and Ergodc Theory. Sprnger, New Yor. [0] Chen, R.-R. (997). An extended class of tme-contnuous branchng processes. J. Appl. Prob. 34, [] Kma, M. (992). Evaluaton of the decay parameter for some specalzed brth death processes. J. Appl. Prob. 29, [2] Kma, M. (993). Quas-lmtng dstrbutons of Marov chans that are sp-free to the left n contnuous tme. J. Appl. Prob. 30, [3] Ln, X. and Zhang, H.-J. (2002). The convergence property of branchng process. Preprnt (n Chnese). [4] Van Doorn, E. (985). Condtons for exponental ergodcty and bounds for the decay parameter of a brth death process. Adv. Appl. Prob. 7, [5] Wang, Z.-K. (996). Introducton to Stochastc Processes, Vol. 2. Beng Normal Unversty Press (n Chnese). [6] Yan, S.-J. and Chen, M.-F. (986). Multdmensonal Q-processes. Chnese Ann. Math. B 7, [7] Zefman, A. I. (99). Some estmates of the rate of convergence for brth and death processes. J. Appl. Prob. 28, [8] Zefman, A. I. (995). Upper and lower bounds on the rate of convergence for nonhomogeneous brth and death processes. Stoch. Process. Appl. 59, [9] Zhang, J. K. (984). On the generalzed brth and death processes. I. Acta Math. Sc. 4, [20] Zhang, Y.-H. (200). Strong ergodcty for sngle-brth processes. J. Appl. Prob. 38,