Tails in a fixed-point problem for a branching process with state-independent immigration
aa r X i v : . [ m a t h . P R ] D ec Tails in a fixed-point problem for a branching processwith state-independent immigration
Sergey FossHeriot-Watt University andNovosibirsk State University ∗ Masakiyo MiyazawaTokyo University of Science † December 4, 2018
The authors congratulate Guy Fayolle on the occasion of hisbirthday and wish him many more happy and productive years!
Abstract
We consider a fixed-point equation for a non-negative integer-valued randomvariable, that appears in branching processes with state-independent immigration.A similar equation appears in the analysis of a single-server queue with a homoge-neous Poisson input, feedback and permanent customer(s).It is known that the solution to this equation uniquely exists under mild firstand logarithmic moments conditions. We find further the tail asymptotics of thedistribution of the solution when the immigration size and branch size distributionsare heavy-tailed. We assume that the distributions of interest are dominantly vary-ing and have a long tail. This class includes, in particular, (intermediate, extended)regularly varying distributions.We consider also a number of generalisations of the model.
Keywords: heavy tail asymptotics, branching process, state-independent im-migration, fixed-point equation, single-server feedback queue, long tail, domi-nantly varying tail, (intermediate) regularly varying tail
We are interested in the following fixed-point equation for a non-negative integer-valuedrandom variable X , X = st A + X X i =1 B i , (1.1) ∗ Research of S. Foss is supported by RSF research grant No. 17-11-01173 † Research of M. Miyazawa is supported by JSPS KAKENHI Grant No. JP16H02786 st ” represents the equality in distribution, A and B i are nonnegative integer-valued random variables independent of X , and B i for i = 1 , , . . . are i.i.d randomvariables that do not depend on A . In what follows, we drop the subscript i from B i ifits distribution is only concerned.Equation (1.1) arises in a branching process with state independent immigration, andits solution is subject to the stationary distribution of this branching process if it ex-ists. Thus, the distribution of X is the stationary distribution of a relatively simplediscrete-time Markov chain with state space Z + ≡ { , , . . . } . Although its existence anduniqueness are well studied (e.g., see [15]) and its moments can be inductively computedif they are finite, little is known about the distribution itself. Its tail asymptotics havebeen studied recently in [4] in the case of regularly varying distributions with a limitedrange of their parameters, and [3] have extended their results onto the case of two-levelprocesses.This is contrasted with a reflecting random walk on Z + , whose stationary distributionis the distributional solution of X satisfying the following fixed point equation: X = st max(0 , X + ξ ) , (1.2)where X and ξ are mutually independent. Various aspects of the solution of this fixedpoint equation and of related one- and multidimensional problems have been studied inthe queueing literature and, in particular, using the modern theory of random walks whereGuy Fayolle and his colleagues have made a great contribution, see their books [9, 10].The fixed-point equation (1.1) is highly nonlinear compared with (1.2), which causesa difficulty to see how does the distribution of X look like. In this paper, we are focusingon the study of its tail asymptotics, with assuming A and B to have heavy-tailed distri-butions, that are both dominantly varying and long-tailed. For this, we introduce a newapproach which allows to go beyond the regular variation. In particular, in the regularlyvarying case, we extend the results of [4], with providing simpler proofs (see our TheoremsTheorem 2.1 and Theorem 2.2, case (I)), and consider another situation (Theorem 2.2,case (II)). This paper is close to [12] where the tail asymptotics for related objects havebeen studied.There is another type of fixed point equations, which arise in internet page rankingproblems (see [1, 16]) and in a queueing problem (see [2]). The heavy tail asymptoticsare also studied for their solutions. However, it should be noted that those fixed pointequations are essentially different from (1.1).In what follows, we say that two (strictly) positive functions f ( x ) and g ( x ) are asymp-totically equivalent (at infinity) and write f ( x ) ∼ g ( x ) if lim x →∞ f ( x ) /g ( x ) = 1. We write f ( x ) & g ( x ) if lim inf x →∞ f ( x ) /g ( x ) ≥ . For two random variables X and Y , equality X = st Y means that X and Y are identically distributed.We list below some known classes of heavy-tailed distributions which are used for areference distribution G either on R + or on R . We use the same notation G for thedistribution and for its distribution function. Let G ( x ) = 1 − G ( x ) be the tail distributionfunction.1. G belongs to the class L of long-tailed distributions if, for some (or, equivalently, for2ny) y and as x → ∞ , G ( x + y ) G ( x ) → G ( x + y ) ∼ G ( x )).2. G belongs to the class S of subexponential distributions if G ∈ L and if G ∗ G ( x ) ∼ G ( x ). For distributions on the positive half-line, the 2nd condition implies the 1st.3. G belongs to the class S ∗ of strong subexponential distributions if G belongs to L ,if m + G ≡ R ∞ G ( x ) dx is finite and if R x G ( y ) G ( x − y ) dy ∼ m + G G ( x ).4. G belongs to the class D of dominantly varying distributions if there exists y > x →∞ G ( yx ) G ( x ) >
0. Then the same holds for any y > G belongs to the class IRV of intermediate regularly varying distributions iflim y ↓ lim inf x →∞ G ( yx ) G ( x ) = 1 . (1.3)6. G belongs to the class E RV of extended regularly varying distributions if there aresome α + , α − > y − α − ≤ lim inf x →∞ G ( yx ) G ( x ) ≤ lim sup x →∞ G ( yx ) G ( x ) ≤ y − α + , ∀ y ≥ . (1.4)7. G belongs to the class RV of regularly varying (at infinity) distributions if, for some α > G ( x ) = x − α L ( x ) , (1.5)where L ( x ) is a slowly varying (at infinity) function, i.e. L ( cx ) ∼ L ( x ), for any c > E RV \ RV is givenin Appendix C. Note that the definitions of
E RV and RV also can be used for positivevalued functions on [0 , ∞ ] instead of G . We recall some basic properties of heavy-taileddistributions we refer to in the paper. More properties and details may be found, e.g., inthe books [5, 11]. In particular, E RV is well studied in [6, 8].First, note the following relations between the introduced classes of heavy-tailed dis-tributions:
RV ⊂ E RV ⊂ IRV ⊂ L ∩ D ⊂ S ∗ ⊂ S ⊂ L , (1.6)where each of the inclusions is strict. Further, each of these classes in closed with respectto the tail equivalence: if G belongs to a class, and if G ( x ) ∼ G ( x ), then G belongsto the same class. We also need the following property: if G belongs to one of theclasses RV , IRV , L ∩ D , S ∗ or S and if G ( x ) = o ( G ( x )) as x → ∞ , then, for any fixed j = 1 , , . . . , we have G ∗ G ( ∗ j )2 ( x ) ∼ G ( x ) (we may informally say that the tail of G ( j )2 is “ignorable”) and, therefore, G ∗ G ( ∗ j )2 belongs to the same class.The following result will be repeatedly used in our proofs.3 roposition 1.1 (Theorem 1 in [14]) Let S n = P n ξ i , n = 1 , , . . . be the sums of i.i.d.r.v.’s with negative mean, S = 0 and M n = max ≤ k ≤ n S n . Let σ ≤ ∞ be any stopping time with respect to { ξ n } . If the commondistribution G of the ξ ’s is strong subexponential, then lim x →∞ P ( M σ > x ) G ( x ) = E σ ≤ ∞ . As we have already said, it is hard to derive an analytically tractable solution of thedistribution of X , so we consider its tail asymptotics. To exclude a trivial exception, weassume throughout the paper that0 < P ( A = 0) < , P ( B = 0) < . (2.1)Let a = E ( A ) and b = E ( B ). The following fact is well-known. Lemma 2.1 (Theorem and Corollary 2 of [15] and Theorem 3.1 of [17]) ( I ) If b > , then (1 . has no solution. ( II ) If < b ≤ , then the solution X of (1 . has a proper distribution if and only if Z − E ( x A ) E ( x B ) − x dx < ∞ . (2.2) In particular, if < b < , then the condition (2 . can be replaced by E (log max( A, < ∞ , (2.3) which obviously holds if a < ∞ , and the solution X is unique in distribution. Remark 2.1
Seneta ([17], Theorem 3.1) proved uniqueness of the distribution of X interms of generating functions in Theorem 3.1. In particular, he showed that the probabilitygenerating function E ( s X ) is regularly varying as s ↑ . In this paper, we assume that the stability conditions b < G such thatlim x →∞ P ( A > x ) G ( x ) = c , lim x →∞ P ( B > x ) G ( x ) = c , (2.4)for some constants c , c ≥ c + c >
0. We consider the following three cases:(i) c , c >
0. Here P ( A > x ) ∼ c c P ( B > x ).(ii) c = 0 , c >
0. Here P ( A > x ) = o ( P ( B > x )).4iii) c > , c = 0. Here P ( B > x ) = o ( P ( A > x )).Note that if b is finite, then a must be finite in the cases (i) and (ii). However, in the case(iii), a is allowed to be either finite or infinite. We consider these situations separately. Remark 2.2
In what follows, we consider the “sub-critical” case only: < b < . Inthis case, the two tail asymptotics, of P ( A > x ) and of P ( B > x ) , may contribute to thatof P ( X > x ) . The case b = 1 is more involved, and it is a subject of a future research. Inthis case, the tail of a null-recurrent random walk may also contribute to the asymptoticsof P ( X > x ) . We need another condition on the distribution G . For c >
1, let T c ( x ) = ∞ X n =0 G ( c n x ) , x > . We note the following fact.
Lemma 2.2 If b < and (2.3) holds, then T c ( x ) is finite for all x > and c > . Proof. T c ( x ) is obviously finite if G has a finite mean. Otherwise, we must have (iii)because b <
1, then there is an x > ǫ > c G ( x ) ≤ (1 + ε ) P ( A > x )for each x ≥ x , and therefore, by (2.3), T c ( x ) ≤ εc ∞ X n =0 P ( A > c n x ) ≤ εc ∞ X n =0 P (log max( A, > n log c + log x ) < ∞ . For each x >
0, we choose n such that c n x ≥ x , then T c ( x ) = n − X n =0 G ( c n x ) + ∞ X n = n G ( c n − n c n x ) ≤ n − X n =0 G ( c n x ) + ∞ X n =0 G ( c n x ) < ∞ . Hence, the lemma is proved.Then the extra condition is: with c = 1 /b ,lim c ↑ c lim sup x →∞ T c ( x ) /T c ( x ) = lim c ↓ c lim inf x →∞ T c ( x ) /T c ( x ) = 1 . (2.5)The condition (2.5) is not easy to check. So, we provide below a sufficient condition forit to hold. Lemma 2.3
The condition (2.5) is satisfied if G ∈ IRV and if G has the Karamataupper index c + ( G ) < , that is, c + ( G ) ≡ inf (cid:26) c ∈ R ; lim sup x →∞ G ( λx ) G ( x ) ≤ λ c uniformly in λ ∈ [1 , Λ] , for all Λ > (cid:27) < . In particular, if G ∈ E RV , then (2.5) holds, and the function T /b ( x ) also belongs to theclass E RV of non-increasing functions.
5e prove this lemma in Appendix A. An extended regularly varying distribution is aspecial case of the distributions with c + ( G ) < L ∩ D thatsatisfy (2.5), and this is a subject of further studies.
Theorem 2.1
Assume that (2 . holds and that a < ∞ and < b ≡ E ( B ) < . Assumethat either one of conditions (i), (ii) or (iii) holds.If G is strong subexponential, then, for any d > /b , P ( X > x ) & (1 − b ) c + ac − b T d ( x ) . (2.6) Further, if G ∈ D ∩ L , then, for any < d < /b , P ( X > x ) . (1 − b ) c + ac − b T d ( x ) . (2.7) If, in addition, (2.5) is satisfied, then P ( X > x ) ∼ (1 − b ) c + ac − b T /b ( x ) . (2.8) In particular, (2.8) holds if G has an extended regularly varying distribution (1 . . Inthis case, X also has an extended regularly varying distribution. Furthermore, if G has aregularly varying distribution (1 . , then X also has a regularly varying distribution withthe same parameter α , and P ( X > x ) ∼ (1 − b ) c + ac (1 − b )(1 − b α ) G ( x ) . (2.9) Theorem 2.2
Assume again conditions (2 . and < b ≡ E ( B ) < to hold. Assumethat G ∈ D ∩ L , and that condition (iii) holds. Assume now that a = ∞ , but that (2.3)holds. Assume further that either ( I ) random variable B has finite variance σ = Var B and C := lim inf x →∞ xG ( x ) ∈ (0 , ∞ ] or ( II ) the distribution H I ( x ) ≡ − min(1 , R ∞ x P ( B > u ) du ) is subexponential and lim sup x →∞ H I ( x ) /G ( x ) < ∞ . (2.10) If (2.5) is satisfied, then the asymptotic (2.8) holds again (here we use the conventionthat ac = 0 for c = 0 and a = ∞ ). In particular, this is the case for G ∈ E RV , andtherefore (2.9) holds for G ∈ RV . We prove these theorems in Section 3. 6 emark 2.3
The asymptotics (2 . hold if and only if P ( x < X ≤ x/b ) ∼ (1 − b ) c + ac − b G ( x ) . (2.11) This follows from the fact that T /b ( x ) − T /b ( x/b ) = G ( x ) . Let D = (1 − b ) c + ac − b , then (2 . implies that P ( X > x ) − P ( X > x/b ) = D ( T /b ( x ) − T /b ( x/b ))(1 + o (1)) = DG ( x )(1 + o (1)) , which proves (2 . . On the other hand, if (2 . holds, then summing up (2 . byreplacing x by x/b n and applying (2 . to each term yield (2 . . It is notable that thecondition (2 . is not needed for the equivalence of (2 . and (2 . . Remark 2.4
The asymptotics (2 . have a natural interpretation in the terms of thePrinciple of a Single Big Jump (PSBJ): for the sum to be large, either one of the sum-mands or the counting random variable must be large. Namely, we may rewrite equation (1 . as an a.s. equation: X (1) = A + X (2) X i =1 B i , (2.12) where r.v.’s X (1) and X (2) have the same distribution, and all r.v.’s on the right aremutually independent. Then X (1) > x if either A > x or one of B i > x , for i = 1 , . . . , X (2) ,or if X (2) is “sufficiently large”. The latter means that, due to the Strong Law of LargeNumbers, we have to have P X (2) B i ≈ bX (2) > x , which leads to X (2) > x/b . Therefore, P ( X (1) > x ) ≈ P ( A > x ) + E X (2) · P ( B > x ) + P ( X (2) > x/b ) . Then the induction argument completes the derivation. However, this is not a proof, andjust an explanation of the phenomenon.
Remark 2.5
Note that the two conditions a = ∞ and lim inf x →∞ xG ( x ) > do hold if G is a regularly varying distribution with parameter α ∈ [0 , . So, the case (I) of ourTheorem 2.2 extends the corresponding result of [4]. In addition, these two conditionsmay hold for other distributions from the class D ∩ L which are not regularly varying –for example, for distributions from the class
E RV . Remark 2.6
We do not know how restrictive are the conditions of Theorem 2.2, seeSection 5 for related comments.
In the rest of this section, we discuss stochastic models where the fixed-point equation(1.1) arises. In what follows, we assume (2.1), (2.3) and that 0 < b < a < ∞ , andtherefore (1.1) has a unique distributional solution, by Lemma 2.1.Define X n inductively for n = 0 , , . . . by X n = (cid:26) , n = 0 ,A n + P X n − i =1 B i,n , n ≥ , (2.13)7here A n and B i,n have the same distributions as A and B , respectively, and { A n } and { B i,n } are sequences of i.i.d. random variables. Clearly, { X n ; n = 0 , , . . . } is a Markovchain with state space Z + , and a branching process with immigration { A n } . The followinglemma is a direct consequence of (2.13) and Lemma 2.1. Lemma 2.4
Let the Markov chain { X n ; n = 0 , , . . . } be defined by (2 . . ( I ) The distribution of X n is stochastically non-decreasing in n , i.e. P ( X n +1 > x ) ≥ P ( X n > x ) , for all n and x , and X n converges to the solution X of (1 . in distributionas n → ∞ . ( II ) If < b < and a < ∞ , then E ( X n ) is non-decreasing in n and lim n →∞ E ( X n ) = a − b < ∞ , (2.14) and therefore E ( X ) = a/ (1 − b ) < ∞ for the solution X of (1 . , by the uniqueness ofLemma 2.1. For completeness, we provide a proof of this lemma in Appendix B.The branching process { X n ; n = 0 , , . . . } also arises in a feedback single server queuewith Poisson arrivals and with one (or more) permanent customer(s). In this model, allthe arriving customers receive service in the First-Come-First-Served order. There aretwo types of customers, the first (“target”) customer and all other customers. The targetcustomer arrives at time 0 at the empty system and, after each service completion, withprobability one joins the end of the queue again, for another service. Any other customer,after its service completion, either joins again the end of the queue (for another service),with probability p , or leaves the system, with probability 1 − p . If we denote by X n thenumber of customers observed by the tagged customer at its n -th return to the queue,then a sequence { X n } forms a time-homogeneous Markov chain which converges to thestationary distribution (under natural assumptions), and a random variable X with thatdistribution satisfies the fixed-point equation. See Section 2.2 of [12] for related details.Similarly, we may consider a sub-critical branching process with k permanent par-ticles. We may rephrase a particle as a customer. Each ordinary customer produces(independently of everything else) a random number of offspring with distribution G andeither stays in the system, with probability p , or leaves the system (dies), with probability q = 1 − p , while each permanent customer produces (again independently of everythingelse) a random number of offspring with distribution G and stays in the system. (Again,one can view this system as a single-server queueing system with Poisson input stream,where customers are served one-by-one and where customers arriving during service ofany customer are viewed as its “offspring”).Let ξ be a random variable with distribution G , and let Y be the population in thesteady state. Here we assume that E ( ξ )+ p < Y = st k + Y − k X i =1 α i + Y X i =1 ξ i , (2.15)8here α i are Bernoulli random variables with parameter p , random variables ξ i havedistribution G , and all random variables on the right of equation (2.15) are mutuallyindependent. Let X = Y − k , and let A = k X i =1 ξ i, , B i = α i + ξ i, , where ξ i, and ξ i, are i.i.d. and distributed as ξ , then (2.15) is the exactly same formof a fixed point equation as (1.1). Thus, we can get the tail asymptotic (2.8) for X andtherefore for Y = X + k under the assumptions of Theorem 2.1. Proof of Theorem 2.1 . We first consider the case (i). Here since b is finite, a is finitetoo, because of the tail equivalence. Hence, E ( X ) < ∞ by Lemma 2.4. We also note that G is strongly subexponential by (1.6). For 0 < y < x , let I − ( x, y ) = P A + X X i =1 B i > x, X ≤ y ! ,I + ( x, y ) = P A + X X i =1 B i > x, X > y ! , then the fixed-point equation (1.1) may be written as P ( X > x ) = I − ( x, y ) + I + ( x, y ) . First, we find the tail asymptotics for I − ( x, y ) as x → ∞ , for any fixed y . Here weneed only the strong subexponentiality assumption. We may apply Theorem 3.37 of [11]and obtain that, as x → ∞ , I − ( x, y ) = P A { X ≤ y } + X { X ≤ y } X i =1 B i > x = y X j =0 P ( X = j ) P A + j X i =1 B i > x ! ∼ y X j =0 P ( X = j )( c + c j ) G ( x ) ∼ E (( c + c X ) { X ≤ y } ) G ( x ) . (3.1)Here ( E ) is the indicator of event E , it takes value 1 if the event occurs and 0, otherwise.We next establish the lower and upper bounds for I + ( x, y ). We start with the upperbound. Here we assume in addition that G ∈ D ∩ L . We choose a sufficiently small ε > < b + ε <
1. Consider a random walk S n = P ni =1 ( B i − ( b + ε/ S = 0. It has a negative drift: E B i − ( b + ε/
2) = − ε/ < . Let d = ( b + ε ) − , then I + ( x, y ) ≤ P ( X > d x ) + P A + X X i =1 B i > x, y < X ≤ d x ! . (3.2)In turn, the second term in the RHS may be bounded above by P A + X X i =1 B i > x, y < X ≤ d x ! ≤ P ( A + S X + ( b + ε/ X > x, y < X ≤ d x ) ≤ P (cid:18) A + max ≤ n ≤ X S n + ( b + ε/ d x > x, y < X ≤ d x (cid:19) ≤ P (cid:18) A + max ≤ n ≤ X S n > ε/ b + ε x, X > y (cid:19) ≤ P (cid:18) A { X > y } + max ≤ n ≤ X { X>y } S n > ε/ b + ε x (cid:19) = P ( X > y ) P (cid:18) A + max ≤ n ≤ e X S n > ε/ b + ε x (cid:19) ≤ P ( X > y ) (cid:18) P (cid:18) A > ε/ b + ε x (cid:19) + P (cid:18) max ≤ n ≤ e X S n > ε/ b + ε x (cid:19)(cid:19) ∼ P ( X > y ) E (cid:16) c + c e X (cid:17) G (cid:18) ε/ b + ε x (cid:19) = E (( c + c X ) { X > y } ) G (cid:18) ε/ b + ε x (cid:19) . (3.3)Here e X has the conditional distribution P ( e X > t ) = P ( X > t | X > y ) and does notdepend on A and B i ; and the equivalence in the pre-last line follows from Proposition 1.1since G is strong subexponential.Since G belongs to D , there exist δ ( ε ) and x ( ε ) such that G (cid:18) ε/ b + ε x (cid:19) ≤ δ ( ε ) G ( x ) , ∀ x ≥ x ( ε ) . Since E ( X ) < ∞ , one can choose sufficiently large y = y ε,δ for any small δ > E (( c + c X ) { X > y } ) δ ( ε ) ≤ δ , and therefore so large that, for all x ≥ x ( ε ) and y ≥ y ε,δ , E (( c + c X ) { X > y } ) G (cid:18) ε/ b + ε x (cid:19) ≤ δ G ( x ) . x ≡ x ( ε, δ )such that, for any x ≥ x , P ( x < X ≤ d x ) ∼ E (( c + c X ) { X ≤ y } ) G ( x ) + P A + X X i =1 B i > x, y < X ≤ d x ! . ( δ + c + c E ( X )) G ( x ) . (3.4)This implies that, for all sufficiently large x , P ( X > x ) = ∞ X k =0 P ( xd k < X ≤ xd k +11 ) ≤ ( δ + c + c E ( X )) T d ( x ) (3.5)Therefore, we get:lim sup x →∞ P ( X > x ) /T d ( x ) ≤ ( δ + c + c E ( X )) , for any sufficiently small positive ε and δ . Letting δ tend to zero, we obtain the followingresult: lim sup x →∞ P ( X > x ) /T d ( x ) ≤ ( c + c E ( X )) . (3.6)In particular, if condition (2.5) holds, then we may tend d to 1 /b and obtain the desiredupper bound. Further, if G has an extended regularly varying distribution, then wehave this upper bound by Lemma 2.3. This implies that, if G has a regularly varyingdistribution with index α >
0, then we get the upper bound in (2.9).We next consider the lower bound for I + ( x, y ). Letting d = ( b − ε ) − > ε >
0, we have, for y < x , I + ( x, y ) ≥ P A + X X i =1 B i > x, X > d x ! ≥ P ( X > d x ) − P X X i =1 B i ≤ x, X > d x ! ≥ P ( X > d x ) − P d x X i =1 B i ≤ x, X > d x ! = P ( X > d x ) − P d x X i =1 B i ≤ ( b − ε ) d x !! . (3.7)Here the subtrahend in (3.7) decays exponentially fast, due to the Chernoff’s bound: since B i are positive, there exist universal positive constants K and α such that, for all x > P d x X i =1 B i ≤ ( b − ε ) d x ! ≤ Ke − αx . ε > δ > y = y ε,δ so large that P ( x < X ≤ d x ) ≥ (1 − δ )( c + c E ( X )) G ( x ) − Ke − αx , (3.8)for all sufficiently large x . Then, for the appropriate K > K , P ( X > x ) = ∞ X k =0 P ( xd k < X ≤ xd k +12 ) ≥ (1 − δ )( c + c E ( X )) T d ( x ) − K e − αx & (1 − δ )( c + c E ( X )) T d ( x ) , for any small δ >
0. Letting δ tend to 0, we obtain the desired lower bound. Then,under assumption (2.5), we may let d tend to 1 /b and obtain the lower bound thatcoincides with the upper bound obtained earlier. By Lemma 2.3, the statements in thelast paragraph of this theorem are legitimated. This completes the proof of the theoremin the case (i).The proof in the case (ii) is similar to the proof above, and even simpler. Since P ( X ≥ y ) >
0, for any positive y , the distributional tail of A is negligible with respect tothat of P X ( X ≤ y ) i =1 B i , of P X ( X>y ) i =1 B i (see Section 1 for the corresponding property), andof max { S n , ≤ n ≤ X ( X > y ) } . Then, clearly, (3.1) holds with c = 0. Further, (3.3) isalso valid with c = 0. Then we get (2.11) and (2.9) for c = 0, and the proof is complete.We next consider the case (iii). First, for any j = 1 , , . . . , the tail distribution of P ji =1 B i is negligible with respect to that of A (see again Section 1). Therefore, we maytake y such that P ( X ≤ y ) > I − ( x, y ) = P A ( X ≤ y ) + X ( X ≤ y ) X i =1 B i > x = y X j =0 P ( X = j ) P A + j X i =1 B i > x ! ∼ y X j =0 P ( X = j ) P ( A > x ) ∼ c P ( X ≤ y ) G ( x ) . For I + ( x, y ), we use the same arguments as in the case (i), and then (2.8) follows, with ac = 0.This completes the proof of Theorem 2.1. Proof of Theorem 2.2 . We again obtain the upper and lower bounds for I − ( x, y )like in the case (i). However, since a = ∞ , E ( X ) is infinite too, and we can not usethe last two formulas of (3.3) for getting the upper bound for I + ( x, y ) because E ( X ) isinfinite. Therefore we modify these lines as follows.12n the case (I), we get P A + X X i =1 B i > x, y < X ≤ d x ! ≤ P A + d x X i =1 B i > x, y < X ≤ d x ! ≤ P A + d x X i =1 B i > x, y < X ! = P A + d x X i =1 ( B i − b ) > x (1 − d b ) ! P ( X > y ) ≤ P ( A > (1 − d b ) x/
2) + P d x X i =1 ( B i − b ) > x (1 − d b ) / !! P ( X > y ) ≤ (cid:18) ( c + o (1)) G (cid:18) ε b + ε ) x (cid:19) + 4( b + ε ) d σ xε (cid:19) P ( X > y ) (3.9)where the inequality in the last line follows from the Chebyshev’s inequality. Since G ∈ D implies that there is a δ ( ε ) > ε > G (cid:16) ε b + ε ) x (cid:17) ≤ ( δ ( ǫ ) + o (1)) G ( x )and since the condition (I) implies that ( C + o (1)) x − ≤ G ( x ), the last line of (3.9) is notless than (cid:18) c δ ( ε ) + o (1) + 4( b + ε ) d σ ( C + o (1)) ε (cid:19) G ( x ) P ( X > y ) . Therefore,lim sup x →∞ G ( x ) P A + X X i =1 B i > x, y < X ≤ d x ! ≤ (cid:18) c δ ( ε ) + 4( b + ε ) d σ Cε (cid:19) P ( X > y ) , where P ( X > y ) in the RHS can be made arbitrarily small by taking y sufficiently large.Thus, the numerator in the ratio is ignorable with respect to G ( x ), and we can concludethat the tail asymptotics coincide with the lower bound.In the case (II), we have P A + X X i =1 B i > x, y < X ≤ d x ! ≤ P (cid:18) A + max ≤ n ≤ X S n > ε/ b + ε x, X > y (cid:19) ≤ P (cid:18) A + max n ≥ S n > ε/ b + ε x (cid:19) P ( X > y ) . (3.10)Since H I is subexponential, the integrated tail distribution of B i − ( b + ε/
2) is alsosubexponential. Since S n has a negative mean drift, we have, from Theorem 5.2 of [11]and (2.10), P (cid:18) max n ≥ S n > x (cid:19) ∼ b + ε/ H I ( x ) . b + ε/ G ( x ) . Hence, similar to (3.9), it follows from (3.10) that P A + X X i =1 B i > x, y < X ≤ d x ! . (cid:18) c + 1 b + ε/ (cid:19) G (cid:18) ε/ b + ε x (cid:19) P ( X > y ) . (3.11)13hus, this term is ignorable compared with G ( x ) by G ∈ D since P ( X > y ) can bearbitrarily small for large y , while G (cid:16) ε/ b + ε x (cid:17) /G ( x ) is bounded. This completes the prooffor the case (II). A natural continuous counterpart of (1.1) is e X = st e A + Z e X d e B ( t ) , (4.1)where e A is a non-negative random variable independent of e X , and e B ( t ) is a non-decreasingprocess with stationary independent increments which is independent of e A and e X . Thatis, e B ( · ) is a non-decreasing Levy process (subordinator).We consider a simple case that e B ( · ) is a compound Poisson process. Namely, e B ( t ) = N ( t ) X i =1 e B i , t ≥ , (4.2)where N ( t ) is the Poisson process with intensity λ >
0, and e B i for i = 1 , , . . . arenon-negative i.i.d. random variables which are independent of N ( · ). Assume that e A is anon-negative random variable independent of everything else. Then, (4.1) becomes e X = st e A + N ( e X ) X i =1 e B i . (4.3)Similar to the Markov chain { X n } defined by (2.13), we recursively define a discretetime Markov process e X n with state space R + . This model may be applied, say, to anenergy reproduction system. In this system, e A is a base production of energy, and extraenergy is reproduced according to the compound Poisson process in the time intervalwhose length equals the amount of the previous energy production.The fixed point equation (4.3) can be solved essentially in the same way as Theorem 2.1because P ( N ( e X ) > x ) ∼ P ( λ e X > x ) , x → ∞ . Assume that e a ≡ E ( e A ) and e b ≡ E ( e B ) are finite. Then, Theorem 2.1 holds true for a = e a and b = λ e b if the solution e X of (4.3) uniquely exists in distribution, where A and B arereplaced by e A and e B in the cases (i)–(iii). 14 .2 2nd order branching process with immigration In this Subsection, we introduce another extension of the model, formulate a particularresult and make short comments on its proof.Consider a branching process in which two subsequent generations produce the nextgeneration. Namely, let X n be the population of the n ’th generation, then X n = A n + X n − X i =1 B ,n,i + X n − X i =1 B ,n,i , n ≥ . (4.4)where { A n ; n ≥ } , { B ,n,i ; n ≥ , i ≥ } , { B ,n,i ; n ≥ , i ≥ } , are sequences of i.i.d non-negative integer-valued random variables, they are mutually independent, and theyare independent of X n − , X n − . We refer to { X n ; n ≥ } as a second order branchingprocess.Let a = E ( A ) and b k = E ( B k ) for k = 1 ,
2. We assume that both a and b k arefinite. Then, it is not difficult to see that the process { X n ; n ≥ } is stable if and onlyif b + b <
1. We assume this stability condition, and consider the following fixed pointequation. (cid:18) XY (cid:19) = st (cid:18) A + P Xi =1 B ,i + P Yi =1 B ,i X (cid:19) . (4.5)This fixed point equation uniquely determines the stationary distribution of X n similarlyto (1.1). Note that (4.5) is equivalent to P ( X > x, Y > y ) = P A + X X i =1 B ,i + Y X i =1 B ,i > x, X > y ! . (4.6)However, the tail asymptotics for two-dimensional distribution is generally hard tostudy. So, we restrict our attention to the tail asymptotics of the linear combination X + δY of X and Y , for a particular choice of coefficient δ .Note that P ( X > x ) = P ( Y > x ) = P A + X X i =1 B ,i + Y X i =1 B ,i > x ! . (4.7)From (4.7), one can find the expectation m ≡ E ( X ): m = a + ( b + b ) m, and therefore, under the stability assumption, m = a − ( b + b ) < ∞ . (4.8)15rom (4.5), we have, for a constant δ > X + δY = st A + X X i =1 B ,i + Y X i =1 B ,i + δX ≡ A + X X i =1 ( δ + B ,i ) + Y X i =1 B ,i . (4.9)In the Proposition below, we provide the distributional tail asymptotics for X + δY ,for a particular choice of δ , under a version of condition (2.5). Under weaker assumptions,one can obtain also upper and lower bounds. Proposition 4.1
Assume that a < ∞ and that b + b < (this is the stability condition).Assume that there is a reference distribution G such that lim x →∞ P ( A > x ) G ( x ) = c , lim x →∞ P ( B ,i > x ) G ( x ) = c , lim x →∞ P ( B ,i > x ) G ( x ) = c (4.10) for some constants c , c , c ≥ such that c + c + c > .(I). Let δ > be the solution to equation δ = b / ( b + δ ) , i.e. δ = (cid:18)q b + 4 b − b (cid:19) / . Then b + δ < .(II). Assume that condition (2.5) holds with c = b + δ . Then P ( x < X + δY ≤ x/ ( b + δ )) ∼ ( c + E X ( c + c )) G ( x ) and, therefore, P ( X + δY > x ) ∼ ( c + E X ( c + c )) T ( b + δ ) − ( x ) . Comments on the
Proof of the proposition. Statement (I) is straightforward. Toobtain the tail asymptotics, we follow the lines of the proof of Theorem 2.1, with minormodifications. Therefore we replace most of the proof by its sketch, with providing somedetails.We take the event that the right-hand side of (4.9) exceeds level x , and consider theprobabilities I − ( x, y ) and I + ( x, y ) of the intersection of this event with events { X + δY ≤ y } and { X + δY > y } , respectively. For the probability of the first intersection of events,we use again the result from Proposition 1.1, while for the second probability we consideragain the upper and lower bounds. There are slightly novel arguments in getting the upperbound only, so we give it in full. We take ε ∈ (0 , − b − δ ) and let d = ( b + δ + ε ) − ,16 = ε/ ε = ε b / ( b + δ ). We have I + ( x, y ) = P ( A + X X ( δ + B ,i ) + Y X B ,i > x, X + δY > y ) ≤ P ( X + δY > d x ) + P ( A + X X ( δ + B ,i ) + Y X B ,i > x, d ≥ X + δY > y ) ≡ P ( X + δY > d x ) + P ( x, y ) . Here P ( x, y ) = P ( A + X X ( B ,i − b − ε ) + Y X ( B ,i − b − ε ) + ( δ + b + ε ) X + ( b + ε ) Y > x,y < X + δY ≤ d x ) ≤ P ( A + M ,X + M ,Y + ( δ + b + ε ) d x > x, max( X, Y ) > y/ δ ) ≤ P ( A + M ,Z + M ,Z > εx/ δ + b + ε , Z > y/ δ )where, for n = 1 , , . . . , M ,n = max ≤ j ≤ n P j ( B ,i − b − ε ), M ,n = max ≤ j ≤ n P j ( B ,i − b − ε ), and Z = max( X, Y ). Let γ = ε/ δ + b + ε ). Then the latter probability is notsmaller than P ( A > γx ) P ( Z > y/ δ ) + X k =1 P ( Z > y/ δ ) P ( M k,Z > γx | Z > y/ δ )Like in the derivation of the upper bound for I + ( x, y ) in the proof of Theorem 2.1, we mayuse Proposition 1.1 and the property of the class D to find that each of the conditionalprobabilities is proportional to G ( x ). Therefore the upper bound to I + ( x, y ) is of order cG ( x ) where coefficient c may be made as small as one wishes, by taking y sufficientlylarge. We do not know, whether (2.8) is the only possible asymptotics for P ( X > x ) in the class
L ∩ D of heavy-tailed distributions. To formulate a more precise open problem, we lookcloser at equation (2.12). One may, in turn, represent X (2) as X (2) = A + P X (3) B i, ,then use the same representation for X (3) , etc. As a result, one can obtain the followinga.s. representation for X = X (1) : X = A + A X i =1 D ,i + A X i =1 D ,i + . . . + A n X i =1 D n − ,i + . . . where, by convention, P = 0, all random variables on the right are mutually independent, A = A and { A i } are i.i.d., D , = st B , D , = st P Bi =1 B i and, for n = 2 , , . . . , D n +1 , = st Bi =1 D n,i where all random variables in the right-hand side of each formula are mutuallyindependent.Consider a particular ”boundary” example, with P ( A > x ) = (1 + x ) − , for γ ∈ (0 , P ( B > x ) = L ( x )(1 + x ) − where L ( x ) ∼ (log x ) − − ε , ε >
0. Then a is infinite and E log max(1 , A ) is finite. Further, b = E B is finite and we can make it smaller than 1.Then one can use Theorem 7 from [7] to obtain P ( D , > x ) ∼ b P ( B > x ) and, usingthe induction argument, P ( D n, > x ) ∼ nb n − P ( B > x ), for any n = 2 , , . . . . Further,using the uniform convergence result in Theorem 2 of [13], one can get the asymptotics P A n +1 X i =1 D n,i > x ! ∼ E (cid:0) A ( A ≤ xb − n ) (cid:1) · nb n − P ( B > x ) + P ( A > xb − n ) . (5.11)We can expect the PSBJ to hold again and formulate the following conjecture : in theexample above, P ( X > x ) ∼ P ( A > x ) + ∞ X n =1 P A n +1 X D n,i > x ! , where the asymptotics for each term in the latter sum are given by (5.11). However, wedo not know how to substantiate these asymptotics. Appendix
A Proof of Lemma 2.3
By [5, Proposition 2.2.3], if c + ( G ) <
0, for any d > c + ( G ) and any γ >
1, there exists x ( γ, d ) such that, for each real c > n ≥ G ( c n x ) /G ( x ) ≤ γc dn , ∀ x ≥ x ( γ, d ) . (A.1)We choose d <
0. Since 0 < b <
1, we can choose c = c +1 ( δ ) for δ > < c +1 ( δ ) < /b and lim δ ↓ c +1 ( δ ) = 1 /b . Then, for any ε >
0, there exists N ( ε ) suchthat ∞ X n = N ( ε )+1 G ( c n +1 ( δ ) x ) /G ( x ) ≤ γ ∞ X n = N ( ε )+1 c − dn +1 ( δ ) < ε, ∀ x ≥ x ( A, − d ) . Hence, we have 1 ≤ T c +1 ( δ ) ( x ) T /b ( x ) ≤ P N ( ε ) n =0 G ( c n +1 ( δ ) x ) /G ( x ) + ε P N ( ε ) n =0 G ( b − n x ) /G ( x )= 1 + P N ( ε ) n =0 G ( b − n x ) G ( x ) (cid:16) G ( c n +1 ( δ ) x ) G ( b − n x ) − (cid:17) + ε P N ( ε ) n =0 G ( b − n x ) G ( x ) . (A.2)18ince 1 ≤ P N ( ε ) n =0 G ( b − n x ) G ( x ) ≤ N ( ε ) for all x ≥ n ≥ δ ↓ lim x →∞ G ( c n +1 ( δ ) x ) G ( b − n x ) = lim δ ↓ lim x →∞ G ( x ) G (( c +1 ( δ ) b ) − n x ) = 1by G ∈ IRV , taking the limit of (A.2) as x → ∞ then as δ ↓ ≤ lim δ ↓ lim sup x →∞ T c +1 ( δ ) ( x ) T /b ( x ) ≤ ε. Letting ε ↓
0, we have the first equality of (2.5). The second equality is similarly obtainedby choosing c = c +2 ( δ ) such that 1 < /b < c +2 ( δ ) and lim δ ↓ c +2 ( δ ) = 1 /b .The remaining parts of this lemma are obvious, so are omitted. B Proof of Lemma 2.4
Let A and { B i } be independent copies of A n and { B i,n } , that also does not depend on all { X n } n ≥ which are obtained by (2.13). Since X = 0 and X = A ≥ X ≤ X . Then X = st A + X X i =1 B i ≤ A + X X i =1 B i = st X . Thus, we have X ≤ st X . We then can choose e X , e X such that they are independent of A and { B i } , e X , ≤ e X a.s. and e X ℓ = st X ℓ for ℓ = 1 ,
2. Hence, X = st A + e X X i =1 B i ≤ A + e X X i =1 B i = st A + X X i =1 B i = st X . One can repeat this induction argument to conclude that X n ≤ st X n +1 for n ≥
3. Since(1.1) has the solution X which is unique in distribution, we have X ≤ st X = st A + X X i =1 B i Then, we can use the same induction argument as above to get X n ≤ st X . Hence, thedistribution of X n weakly converges to some proper distribution ν as n → ∞ . Denotea random variable subject to this ν by Y . Since X n +1 = st A + P X n i =1 B i implies that Y = st A + P Yi =1 B i , we have Y = st X by the uniqueness of the solution of (1.1) indistribution. Thus, X n converges to X in distribution as n → ∞ .Any stochastically non-decreasing sequence of random variables has a (possibly im-proper) weak limit, call it X . We have E X n +1 = a + b E X n = . . . = a (1 − b n +1 ) / (1 − b ).By the monotone convergence theorem, E X = lim n →∞ E X n = a/ (1 − b ) < ∞ and, inparticular, X is finite a.s. This completes the proof.19 Example of the tail distribution from the class
E RV \ RV
We provide an example of the tail distribution function g ( x ) = G ( x ) that is extendedregularly varying, but not regularly varying.Let c > < a < a . We assume that function g ( x ) has the “cycle” behaviourand define it by induction. At ”time” t = 1, we take g ( t ) = 1, and the first cycle starts.Given the n ’th cycle starts at time t n , we let u n = ct n and define g ( t ) = ( t/t n ) − a g ( t n ) forall t ∈ ( t n , u n ]. Then let t n +1 = cu n and define g ( t ) = ( t/u n ) − a g ( u n ) for all t ∈ ( u n , t n +1 ]. Acknowledgements
The authors are grateful to Charles Goldie for his invaluable comments on the extendedregular variations,
E RV and for suggesting references [6, 8].
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