# Large Deviation Rates for the Continuous-Time Supercritical Branching Processes with Immigration

Authors: Juan Wang; Xiaojuan Wang
Journal: Journal of Mathematics
 (2022)

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## Abstract

LetYt;t≥0 be a supercritical continuous-time branching process with immigration; our focus is on the large deviation rates of Yt and thus extending the results of the discrete-time Galton–Watson process to the continuous-time case. Firstly, we prove that Zt=e−mtYt−emt+1−1/em−1ea+m is a submartingale and converges to a random variable Z. Then, we study the decay rates of PZt−Z>εast⟶∞ and PYt+v/Yt−emv>ε|Z≥αast⟶∞ for α>0 and ε>0 under various moment conditions on bk;k≥0 and aj;j≥0. We conclude that the rates are supergeometric under the assumption of finite moment generation functions.


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## Body

## 1. Introduction

Suppose thatYt;t≥0 is a continuous-time branching process with immigration. Its generating matrix Q=qjk;j,k∈Z+ is defined as follows:(1)qjk≔jbk−j+1+ak−j,ifj≥0,k≥j,0,otherwise,where(2)ak≥0k≠0,0<−a0=∑k≠0ak<∞,bk≥0k≠1,0<−b1=∑k≠1bk<∞.Throughout this paper, we always suppose thatb0=0, m≔∑k=1∞kbk<∞, and a≔∑k=0∞kak<∞.Set(3)Y0=1,Yt=Mt+Kt,t>0,where Mt and Kt separately represent the total number of the original individual’s offspring and the offspring of the individual from immigrants at the time t. If Kt≡0 for all t≥0 and Y0=1, then the process degenerates to the continuous-time branching process Y^t;t≥0. It is known from Athreya and Ney [1] or Harris [2] that e−mtY^t;t≥0 is a martingale and converges to a r.v. W w.p.1 as t⟶∞. Therefore, Y^t+v/Y^t converges to emv w.p.1 as t⟶∞ for arbitrary fixed v≥0, and hence(4)limt⟶∞PY^t+vY^t−emv>ε=0,∀ε>0.For Yt;t≥0, define(5)Zt=e−mtYt−emt+1−1em−1ea+m≔Ut+It,t≥0,where Ut=Mt/emt and It=Kt/emt−emt+1−1/em−1ea+m−mt.In this paper, leta=m/em−1<∞, we will demonstrate that Zt is a submartingale and converges to a r.v. Z almost surely since Ut converges to a r.v. U a.s. and It converges to a r.v. I a.s.The above discussion of the decay rates is not only of its own meaning but also of some significance to the algorithm tree structure in computer science (Karp and Zhang [3] and Miller [4]). Heyde [5] discussed that there exists a constant sequence Dn which makes Wn=Zn/Dn converge to a random variable W w.p.1. Athreya [6] considered similar decay rates for the Galton–Watson process. Seneta [7] researched the supercritical Galton–Watson process with immigration, and Riotershtein [8] considered multitype branching processes with immigration in a random environment. Liu and Zhang [9] studied the decay rates of PYn+1/Yn−m>ε for the supercritical branching processes with immigration. Recently, Sun and Zhang [10] considered the convergence rates of harmonic moments for supercritical branching processes with immigration. Chen and He [11] investigated the lower deviation and moderate deviation probabilities for maximum of a branching random walk. Abraham et al. [12] analyzed the stationary continuous state branching processes. The model considered in this paper involves continuous-time and immigration, while the models considered in the above references do not involve continuous-time. Such change in conditions may affect the large deviation rates and thus is mathematically interesting. We need to find a new method to investigate the effect of continuous-time and immigration.Based on the previous results, it is natural to develop the large deviation rates for the continuous-time branching processes with immigration. In this paper, we are aiming to discuss the decay rates of(6)PZt−Z>ε,PYt+vYt−emv>ε|Z≥α,ast⟶∞,under various moment conditions on bk;k≥0 and ak;k≥0 for any α>0 and ε>0. We conclude that the rates of (6) are always supergeometric under the assumption of finite moment generation functions. The specific conclusions are presented in Section 3.


## 2. Preliminaries

Before the investigation, we present the generating functionsBw and Aw of the known sequences bk;k≥0 and ak;k≥0 as(7)Bw=∑k=0∞bkwk,Aw=∑k=0∞akwk.Clearly,Bw is well defined and finite at least on −1,1 with B1=0, so 1 is the solution of Bw=0. Then, m=B′1=∑k=0∞kbk is the mean birth rate of Mt. Moreover, Aw is similar to Bw.We present some preliminaries in this section.Lemma 1.
Letη>0 and j,k≥0. If(8)limt⟶∞eηtpjkt=ϕjk≥0,exists, then for arbitrary b>0,(9)limt⟶∞e−bt∫0teη+bvpjkvdv=b−1ϕjk.Proof.
Omitted.Lemma 2.
limt⟶∞e−b1+a0tp1kt=ρk exists for any i≥1, and ρk≤ρ1=1k≥0. Furthermore, Qw=∑k=1∞ρkwk satisfies the functional equation:(10)b1+a0Qw=BwQ′w+AwQw,0≤w≤1,with condition Q0=0.Proof.
By the Kolmogorov forward equation,(11)p1k′t=∑i=1kp1itibk−i+1+ak−i,k≥1.
Fork=1,(12)p11′t=p11tb1+a0.
That is,(13)p11t=eb1+a0t.
Hence,(14)ρ1=limt⟶∞e−b1+a0tp11t=1.
Fork=2,(15)p12′t=p11tb2+a1+p12t2b1+a0.
So,(16)e−2b1+a0tp12t=b2+a1∫0tp11ve−2b1+a0vdv.
By Lemma1,(17)ρ2=limt⟶∞e−b1+a0tp12t=b2+a1limt⟶∞eb1t∫0tp11ve−2b1+a0vdv=−ρ1b2+a1b1≤ρ1.
It follows from (11) that(18)e−kb1+a0tp1kt=∑i=1k−1ibk−i+1+ak−i∫0tp1ive−kb1+a0vdv.
Hence,(19)ρk=limt⟶∞e−b1+a0tp1kt=1−k−1b1∑i=1k−1ibk−i+1+ak−iρi≤ρ1.
Finally, we can obtain the following equality from (11):(20)∑k=0∞e−b1+a0tpjk′tωk=Bω∑k=1∞e−b1+a0tpjktωk−1+Aω∑k=0∞e−b1+a0tpjktωk,and moreover,(21)∑k=0∞e−b1+a0tpjktωkt′+b1+a0∑k=0∞e−b1+a0tpjktωk=Bω∑k=1∞e−b1+a0tpjktkωk−1+Aω∑k=0∞e−b1+a0tpjktωk.
Taking the limit ast⟶∞, we have(22)b1+a0Qω=BωQ′ω+AωQω.
The proof is completed.Proposition 1.
Letςt be the σ-algebra generated by Yt;t≥0 and EM1logM1<∞. Then, Zt;t≥0 is a submartingale and converges to a r.v. Z almost surely.Proof.
According to the definition ofZt, we conclude the following inequalities:(23)EZt+v|Zt=Ee−mt+vYt+v−emt+v+1−1em−1ea+m|Yt=e−mt+vE∑j=1Ytξt,jv+Kv−emt+v+1−1em−1ea+m|Yt=e−mt+vemvYt+ea+mv−emt+v+1−1em−1ea+m=e−mtYt+eav+m−eav−emt+a+2m+ea+m−mvem−1≥e−mtYt−emt+1−1em−1ea+m=Zt,which implies Zt;t≥0 is a submartingale. We know that Ee−mtYt<∞ if and only if EM1logM1<∞. Hence, EZt<∞. Thus, Zt is an integrable submartingale and converges to a r.v. Z almost surely.
DefineGw,t=EwYt|Y0=1, w≥0 with initial condition Gw,0=w, where Yt is a continuous-time branching processes with immigration with Y0=1. Denote kFw=w and(24)Gw≔Gw,1=EwY1|Y0=1=Hw,1Fw,1≔HwFw.
We will study the properties ofkw and Gw in the subsequent contents. Let ϖ=supw≥0;Gw<∞. Apparently, Gw is strictly increasing with w∈0,ϖ and ϖ≥1. Furthermore, we have kw≥w for 0≤w≤1 since Fw≤w and kw≤w for 1≤w≤ϖ since Fw≥w. Therefore, the iterates kw,n of k are nonincreasing with w∈1,ϖ and nondecreasing with w∈0,1 (with respect to n).Proposition 2.
Letv0>1. If Gw0,v0<∞ for some w0>1, then for 1≤w≤Fw0,v0, kw,t↓1 as t↑∞ and(25)Rw,t≡emtkw,t−1↓Rw,ast↑∞,where Rw is the unique solution of(26)RFw,v0=emv0Rw,for1≤w≤Fw0,v0,subject to(27)R1=0,R′1=1,(28)0<Rw<∞,for1<w≤Fw0,v0.Proof.
Note thatFw,t≥w for w≥1, we know that Fw,v0+t=FFw,t,v0≥Fw,v0 for w≥1, t≥0. Hence, k⋅,v0+t is well defined on 0,Fw0,v0 for t≥0 and kw,v0+t≥1 for w≥1.
Sincew=Fkw,v0+t,v0+t=FFkw,v0+t,t,v0≥Fkw,v0+t,v0, we have kw,v0+t≤kw,v0 for any 1≤w≤Fw0,v0, which implying kw,t↓ as v0≤t↑. On the other hand,(29)Fkω,v0+t,v0=kω,t,that is,(30)kω,v0+t=kkω,t,v0.
Denotec≔limt⟶∞kw,t. Putting t↑∞ yields c=kc,v0. Thus, we obtain c=1, i.e., kw,t↓1 as t↑∞.
Now, there existsa∈kw,t,1 for v,t>0 such that(31)Rw,v+tRw,t=emvkkw,t,v−1kw,t−1=emvkw′a,v=emvFw′a,v<1.
Hence,Rw=limt⟶∞Rw,t exists.
Furthermore,Rw satisfies (26) and (27). For w∈Fw0,v0,1,(32)RFw,v0,t+v0=emt+v0kFw,v0,t+v0−1=emt+v0kkFw,v0,v0,t−1=emt+v0kw,t−1.
Lettingt↑∞ yields(33)RFw,v0=emt+v0Rw.
For another,(34)R1=limt⟶∞R1,t=0,R′1=1,since Rw′1,t=1 for t>v0. Thus, 0<Rw<∞.
Finally, it is easily seen that the solution of (26) and (27) is unique.Proposition 3.
Letl≡Gw0<∞ for some w0=eθ0>1. Then,(35)Gkw0,t≤Gw0∏l=1t−1Hkw0,l,t≥2,where t is the round of t.Proof.
Fort=2, noting kw0>1 yields(36)Gkw0,2=GFkw0Hkw0=Gw0Hkw0.
Assume that inequality (35) holds for t−1, i.e.,(37)Gkw0,t−2,t−1≤Gw0∏l=1t−2Hkw0,l.
Then,(38)Gkw0,t−1,t=Gfkw0,t−1,t−1Hkw0,t−1=Gkw0,t−2,t−1Hkw0,t−1≤Gw0∏l=1t−1Hkw0,l,holds for all t≥2.
Therefore, (35) is proved.


## 3. Main Results and Proofs

Theorem 1.
Assume thatBθ0<∞ and Aθ0<∞ for some θ0>0. Then, for arbitrary small ε>0,(39)limt⟶∞e−b1+a0t⋅PYt+vYt−emv>ε|Y0=1=∑l=1∞ϕv,l,ερl<∞,where ϕv,l,ε=PM¯lv+Kv/l−emv>ε,M¯lv=∑j=1lMjv/l.Proof.
Note thatYt+v could be defined as(40)Yt+v=∑j=1Ytξt,jv+Kv,where Kt;t≥0 and ξt,jv;t≥0;j≥1 are independent and identical distributed, ξt,jv;t≥0,j≥1 are i.i.d. random variables with the same law as Y^v;v≥0. Hence,(41)PYt+vYt−emv>ε|Y0=1=P∑j=1Ytξt,jv+KvYt−emv>ε|Y0=1=∑l=1∞PYt=l|Y0=1⋅P∑j=1lξt,jv+Kvl−emv>ε=∑l=1∞PYt=l|Y0=1ϕv,l,ε.
Indeed, for any fixedv>0,(42)ϕv,l,ε=P∑j=1lξt,jv+Kvl−emv>ε+P∑j=1lξt,jv+Kvl−emv<−ε≤P∑j=1lξt,jv>lemv+ε2+PKvl>ε2+P∑j=1lξt,jv<lemv−ε≤Pα∑j=1lξt,jv>αlemv+ε/2+PKvl>ε2+Pβ∑j=1lξt,jv>βlemv−ε≔I1+I2+I3,where α∈1,θ0 and β∈0,1 are arbitrary constants. Thus,(43)ϕv,l,ε≤fα,vα−emv+ε/2l+I2+fβ,vβ−emv−εl,where fs,v=∑k=0∞p1kvsk.
Since it is assumed thatBθ0<∞ and Aθ0<∞, we have fα,v<∞ and it follows that there exist α0∈1,θ0 and β0∈0,1 such that(44)0<fα0,vα0−emv+ε/2<1,0<fβ0,vβ0−emv−ε<1,for any ε∈0,1 and v>0. Hence, there exist cj,λj∈0,1j=1,3 such that Ij≤cjλjv,εl for all l≥1. By Markov’s inequality, we can prove that there exist c2>0 and λ2∈0,1 such that I2≤c2λ2v,εl. Therefore, by Lemma 2,(45)limt⟶∞e−b1+a0t⋅PYt+vYt−emv>ε|Y0=1=limt⟶∞∑l=1∞e−b1+a0t⋅PYt=1|Y0=1ϕv,l,ε=∑l=1∞ϕv,l,ερl<∞.
The proof is completed.Theorem 2.
Assume that for fixedε>0 and v>0, there exist constants Cεv and r>0 such that mr>−b1+a0 and ϕv,l,ε≤l−rCεv for all l≥1. Then, (39) holds.Proof.
Under the above assumptions, there exists anotherC˜ε>0 such that ϕv,l,ε≤C˜εl+1−r for all l>0 since l−r is equivalent to l+1−r as l⟶∞. In general, we denote the positive constants C˜ε by Cε. Therefore,(46)kl,t≔ϕv,l,εeb1+a0t⋅PYt=l≤Cεl+1r⋅PYt=leb1+a0t≕k˜l,t.
With a simple modification of the convergence theorem, it can be proved that(47)limt⟶∞∑l=0∞k˜l,t=limt⟶∞Cεeb1+a0tEYt+1−r<∞,since(48)e−b1+a0t⋅PYt+vYt−emv>ε=∑l=1∞ϕv,l,εPYt=leb1+a0t.
For allY≥0,(49)EY−r=1Γr∫0∞Ee−tYtr−1dt,and hence(50)EYt+1−reb1+a0t=1Γr∫0∞Ee−tYt+1tr−1eb1+a0tdt=1Γr∫01Gv,t−logvr−1eb1+a0tdv.
SinceQv,t=Gv,t/eb1+a0t↑Qv, the proof shall be completed if we prove(51)∫01Qvkvdv<∞,where kv=logvr−1. Indeed, for any fixed v0∈0,1, let vt be the inverse function of Fv0,t at v0, then we have vt↑1 as t↑∞. Note that Hv,1QFv,1=eb1+a0Qv, thus we can write(52)In=∫vnvn+1Qvkvdv=∫vnvn+1QFv,1Hv,1eb1+a0kvdv=∫vn−1vnQwkwHF−1wkF−1wF−1w′kweb1+a0dw≔∫vn−1vnQwkwDwdw,where(53)Dw=hF−1wkF−1wF−1w′kweb1+a0.
Sincelimw⟶1F−1w′=m−1, it follows promptly that Dw satisfying(54)limw⟶1Dw=e−b1+a0+mr.
By hypothesiseb1+a0+mr>1 and hence for any λ∈e−b1+a0+mr,1, we can pick an n0 such that Dw<λ for all w≥tn0. Hence, for n≥n0,(55)In≤λIn−1,which implies(56)∑n=n0∞In≤In0∑j=0∞λj<∞,and hence(57)∫tn01Qvkvdv<∞.
Applying∫0tQvkvdv<∞ for 0<t<1 together with (57) implies (51).Corollary 1.
Assume thatEY2r+δ1<∞ for some r≥1 and δ>0 such that mr>−b1+a0. Then, (39) holds.Proof.
By Markov’s inequality, we have(58)v,l,ε=PY¯lv+Kvl−emv>ε≤ElY¯lv+Kv/l−emv2rε2rlr.
According to the assumption,(59)C˜ε=suplElY¯lv+Kvl−emv2r,is finite, and then there exists Cε s.t. ϕv,l,ε≤Cεl−r for all l≥1. Applying Theorem 2, the proof is completed.Theorem 3.
Assume thatEeθ0Yv|Y0=1<∞ for some v>0 and θ0>0. Then, there exists θ1>0 satisfying(60)C1=supt≥0Eeθ1Zt<∞.Proof.
In general, we suppose thatK=Gw0<∞ for w0=eθ0>1. Firstly, we prove that(61)L=∏l=1∞hkw0,l<∞.
Since∑l=1∞hkw0,l−1<∞ is equivalent to ∏l=1∞hkw0,l<∞, we have(62)∑l=1∞hkw0,l−1≤∑l=1∞h′kw0kw0,l−1=h′kw0⋅∑l=1∞kw0,l−1≔C5<∞.
Hence, (61) is proved since kw0,l−1∼Rw0/eml. By Proposition 3, note that for any t≥1,(63)Gv,t≤KL,if0≤v≤kw0,t−1.
Indeed, if0≤v≤kw0,t−1, by Proposition 3,(64)Gv,t≤Gkw0,t−1,t≤Gw0⋅∏l=1t−1hkw0,l≤KL.
But, by definition ofZt,(65)EeθZt|Y0=1=Eeθe−mtYt−emt+11/em−1ea+m|Y0=1≤Geθe−mt,t,and hence the above inequality becomes(66)EeθZt|Y0=1≤KL,since θ≤emtlogkw0,t−1. Hence, kw0,t↓1 as t↑∞ since w0>1. Finally, by Proposition 2,(67)limt⟶∞emtlogkw0,t−1=limt⟶∞emtkw0,t−1−1=emRw0,which is positive and finite. Therefore, we can find θ1>0 such that(68)C1=supt≥0Eeθ1Zt<∞.Theorem 4.
Assume thatEeθ0Yv|Y0=1<∞ for some v>0 and θ0>0. Then, there exist constants C2 and λ>0 satisfying(69)PZt−Z>ε≤C2e−λε2/3emt/3.Proof.
According to Theorem3 and the construction of Zt, we know that there exists θ1>0 such that ϕθ=EeθZ=EeθU+I<∞ for all θ≤θ1 which implies ϕ1θ=EeθU<∞,ϕ2θ=EeθI<∞ for all θ≤θ1. If Ui;i≥1 are i.i.d. copies of U and Sn=∑i=1nUi−1, I and I1 have the identical distribution, then for any θ≤θ1,(70)EeθSn+I1+ea+m/n=Eeθ∑i=1nUi−1+I1+ea+m/n=ϕ1θne−θ/nn⋅EeθI1+ea+m/n=1+1nϕ1θ/ne−θ/n−1θ2/nθ2n⋅EeθI1+ea+m/n.
Note that(71)limw⟶0ϕ1we−w−1w2=12VarU<∞,we write(72)limw≤1ϕ1we−w−1w2≕c,EeθI1+ea+m/n≤EeθI1+ea+m≕C.
Now, letθ2=minθ1,1, then(73)supθ≤θ2ϕ1θne−θ/nn≤ec≕L<∞,since 1+x/nn≤ex for x>0. Therefore,(74)EeθSn+I1+ea+m/n≤CL≕Q<∞.
Observe that(75)Z−Zt=limv⟶∞Zt+v−Zt=limv⟶∞e−mtYt+vemv−Yt−ea+m1em+1e2m+⋯+1evm+1ev+1m+⋯+ea+m1emv+1+⋯=limv⟶∞e−mtYt+vemv+ea+m1emv+1+⋯+limv⟶∞e−mt−Yt−ea+m1em+1e2m+⋯+1evm+1emv+1+⋯=limv⟶∞e−mt∑j=1Ytξt,jv+Kvemv−ea+memv+1−1evmem−1+e−mt−Yt−ea+mem−1+ea+memem−1=e−mt∑j=1YtUj+I1−Yt+ea+m,where ξt,jv is the population size at time s + t of the jth particle among the Yt particles, Uj is the limit random variable in the descent line of the jth original parent at time t, and I1 and I are identically distributed. Putting the conditional independence into consideration,(76)PZ−Zt≥ε|ξt=ψYt,emtε,where(77)ψl,r=P∑j=1lUj−1+I1+ea+m≥r.
However,(78)PSl+I1+ea+m≥r=PSl+I1+ea+ml≥rl≤Eeθ2Sk+I1+ea+m/le−θ2r/l≤C5e−θ−θ2r/lrl.
Thus,(79)PZ−Zt≥ε=EψYt,emtε≤C5Ee−θ2emtε/Yt=C5Ee−θ2emt/2ε1/Zt+ea+m1+1/em+1/e2m+⋯+1/em.
For arbitraryλ>0,(80)Ee−λ1/Zt+ea+m1+1/em+1/e2m+⋯+1/em=λ∫0∞e−λyP1Zt+ea+m1+1/em+1/e2m+⋯+1/em≤ydy=λ∫0∞e−λyPZt+ea+m1+1em+1e2m+⋯+1etm≥y−2dy=λ∫0∞e−λyPeθ1Zt+ea+m1+1/em+1/e2m+⋯+1/em≥eθ1y−2dy≤λC6∫0∞e−λye−θ1y−2dy=C6∫0∞e−ye−θ1λ2y−2dy.
Hence,(81)PZ−Zt≥ε≤C5C6∫0∞e−ye−θ1λt2y−2dy,where λt=θ2εemt/2. But, for any λ>0,(82)Iλ=∫0∞e−ye−λ2y−2dy=∫0lλe−ye−λ2y−2dy+∫lλ∞e−ye−λ2y−2dy≤e−λ2/lλ+e−lλ.
Now, chooselλ=λ2/3, we have Iλ≤2e−λ2/3. Therefore,(83)PZ−Zt≥ε≤2C5C6e−θ1θ2εemt/22/3=C7e−λε2/3emt/2,where λ=θ1θ22/3. Similar arguments still hold for PZ−Zt≤ε.
Under the conditionZ≥αα>0, the following theorem proves that the decay rate of PYt+v/Yt−emv>ε is also supergeometric.Theorem 5.
Assume thatEeθ0Yv|Y0=1<∞ for some v>0 and θ0>0. Then, there exist constants λ>0 and C3 such that for any α>0, ε>0, we have 0<Iε<∞ satisfying(84)PYt+vYt−emv>ε|Z≥α≤C3e−αηIεemt+C2e−λα1−η2/3emt/3,for 0<η<1.
In particular, whenη=1/2,(85)PYt+vYt−emv>ε|Z≥α≤C4e−λα/22/3emt/3.Proof.
Note that(86)PYt+vYt−emv>ε|Z≥α=PYt+vYt−emv>ε,Z≥α⋅1PZ≥α≔αt1+αt2pα,where 0<η<1, pα=1/PZ≥α, αt1=PYt+v/Yt−emv>ε,Zt≤αη,Z≥α, and αt2=PYt+v/Yt−emv>ε,Zt≥αη,Z≥α. Clearly, as shown in Theorem 4,(87)αt1≤PZ−Zt≥α1−η≤C7e−λα1−η2/3emt/3.
On the other hand,(88)αt2≤PYt+vYt−emv>ε,Zt≥αη=∑lPYt+vYt−emv>ε,Zt≥αη|Yt=l⋅PYt=l=∑lPYt+vl−emv>ε,l≥αηemt+emt+1−1em−1ea+m⋅PYt=l≤∑l≥αηemtP∑j=1lξt,jv+Kvl−emv>ε⋅PYt=l≤∑l≥αηemtP∑j=1lξt,jv−emvl+Kvl>ε⋅PYt=l≤∑l≥αηemtP∑j=1lξt,jv−emvl>ε2⋅PYt=l+PKvl>ε2⋅PYt=l≤C8e−αηI1εemt+∑l≥αηemtPKvl>ε2⋅PYt=l.
The existence ofC8 and I1ε follows from Chernoff type bounds since Eeθ1Yv<∞. Moreover,(89)PKvl>ε2≤Eeθ0Kv⋅e−θ0lε/2≤C9e−θ0εαηemt/2.
Therefore, we can chooseIε and constant C10 satisfying(90)αt2≤C10e−αηIεemt.
Hence,(91)PYt+vYt−emv>ε|Z≥α≤pαC7e−λα1−η2/3emt/3+C10e−αηIεemt.
In particular, forη=1/2, there exists C4 such that(92)PYt+vYt−emv>ε|Z≥α≤C4e−λα/22/3emt/3,since the first term αt1 tends to 0 faster than αt2.


## 4. Concluding Remarks

In this paper, we studied the evolutionary structure of a supercritical branching process with immigration through the behavior of conditional limits under various notions of “information” about the current population size. A natural next question concerns the large deviation rates of high-dimensional branching processes under other “information” notions. These and other related issues will be investigated in subsequent papers.



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