Harmonic moments and large deviations for a supercritical branching process in a random environment
aa r X i v : . [ m a t h . P R ] A ug HARMONIC MOMENTS AND LARGE DEVIATIONS FOR ASUPERCRITICAL BRANCHING PROCESS IN A RANDOMENVIRONMENT
ION GRAMA, QUANSHENG LIU, AND ERIC MIQUEU
Abstract.
Let ( Z n ) be a supercritical branching process in an independent andidentically distributed random environment ξ . We study the asymptotic of theharmonic moments E [ Z − rn | Z = k ] of order r > n → ∞ . We exhibit aphase transition with the critical value r k > E p k = E m − r k , where m = P ∞ k =0 kp k with p k = P ( Z = k | ξ ) , assuming that p = 0 . Contrary to the constant environment case (the Galton-Watson case), this criticalvalue is different from that for the existence of the harmonic moments of W =lim n →∞ Z n / E ( Z n | ξ ) . The aforementioned phase transition is linked to that for therate function of the lower large deviation for Z n . As an application, we obtain alower large deviation result for Z n under weaker conditions than in previous worksand give a new expression of the rate function. We also improve an earlier resultabout the convergence rate in the central limit theorem for W − W n , and find anequivalence for the large deviation probabilities of the ratio Z n +1 /Z n . Introduction
A branching process in a random environment (BPRE) is a natural and impor-tant generalisation of the Galton-Watson process, where the reproduction law variesaccording to a random environment indexed by time. It was introduced for the firsttime in Smith and Wilkinson [24] to modelize the growth of a population submittedto an environment. For background concepts and basic results concerning a BPREwe refer to Athreya and Karlin [4, 3]. In the critical and subcritical regime thebranching process goes out and the research interest has been mostly concentratedon the survival probability and conditional limit theorems, see e.g. Afanasyev,Böinghoff, Kersting, Vatutin [1, 2], Vatutin [26], Vatutin and Zheng [27], and thereferences therein. In the supercritical case, a great deal of current research hasbeen focused on large deviations, see e.g. Bansaye and Berestycki [6], Bansaye andBöinghoff [7, 8, 9], Böinghoff and Kersting [11], Huang and Liu [17] and Nakashima[22]. In the particular case when the offspring distribution has a fractional lineargenerating function, precise asymptotics can be found in Böinghoff [10] and Kozlov[19]. An important closely linked issue is the asymptotic behavior of the harmonic
Date : March 15, 2018.2010
Mathematics Subject Classification.
Primary 60J80, 60K37, 60J05. Secondary 60J85,92D25.
Key words and phrases.
Branching processes, random environment, harmonic moments, largedeviations, phase transitions, central limit theorem. moments E [ Z − rn | Z = k ] of the process Z n starting with Z = k initial individu-als. For the Galton-Watson process which corresponds to the constant environmentcase, the question has been studied exhaustively in Ney and Vidyashankar [23]. Fora BPRE, it has only been partially treated in [17, Theorem 1.3].In the present paper, we give a complete description of the asymptotic behaviorof the harmonic moments E k [ Z − rn ] = E [ Z − rn | Z = k ] of the process Z n starting with k individuals and assuming that each individual gives birth to at least one offspring(non-extinction case). As a consequence, we improve the lower large deviation resultfor the process Z n obtained in [8, Theorem 3.1] by relaxing the hypothesis therein.In the meanwhile we give a new characterization of the rate function in the largedeviation result stated in [8]. We also improve the exponential convergence rate inthe central limit theorem for W − W n established in [17]. Furthermore, we investigatethe large deviation behavior of the ratio R n = Z n +1 Z n , i.e. the asymptotic of the largedeviation probability P ( | R n − m n | > a ) for a >
0, where m n is the expected valueof the number of children of an individual in generation n given the environment ξ .For the Galton-Watson process, the quantity R n is the Lotka-Nagaev estimator ofthe mean E Z , whose large deviation probability has been studied in [23].Let us explain briefly the findings of the paper in the special case when we startwith Z = 1 individual. Assume that P ( Z = 0) = 0 and P ( Z = 1) >
0. Define r as the solution of the equation(1.1) E m − r = γ, with(1.2) γ = P ( Z = 1) . From Theorem 2.1 we get the following asymptotic behavior of the harmonic mo-ments E [ Z − rn ] for r >
0. Assume that E m r + ε < ∞ for some ε >
0. Then, wehave(1.3) E [ Z − rn ] γ n −→ n →∞ C ( r ) if r > r , E [ Z − rn ] nγ n −→ n →∞ C ( r ) if r = r , E [ Z − rn ] (cid:16) E m − r (cid:17) n −→ n →∞ C ( r ) if r < r , where C ( r ) are positive constants for which we find integral expressions. This showsthat there are three phase transitions in the rate of convergence of the harmonicmoments for the process Z n , with the critical value r . It generalizes the result of[23] for the Galton-Watson process. For a BPRE, it completes and improves theresult of [17], where the asymptotic equivalent of the quantity E [ Z − rn ] has beenestablished in the particular case where r < r and under stronger assumptions. ARMONIC MOMENTS OF Z n FOR A BPRE 3
The proof presented here is new and straightforward compared to that for theGalton-Watson process given in [23]. Indeed, we prove (1.3) starting from thebranching property(1.4) Z n + m = Z m X i =1 Z ( m ) n,i , where conditionally on the environment ξ , for i >
1, the sequences of randomvariables { Z ( m ) n,i : n > } are i.i.d. branching processes with the shifted environment T m ( ξ , ξ , . . . ) = ( ξ m , ξ m +1 , . . . ), and are also independent of Z m . This simple idealeads to the following equation which will play a key role in our arguments:(1.5) E h Z − rn +1 i = γ n +1 + n X j =0 b j γ n − j c jr , where c r = E m − r and ( b j ) j > is an increasing and bounded sequence. Such a relationhighlights the main role played by the quantities γ and c r in the asymptotic studyof E [ Z − rn ] whose behavior depends on whether γ < c r , γ = c r or γ > c r . Note thatthe complete proof of (1.3) relies on some recent and important results establishedin [14] concerning the critical value for the existence of the harmonic moments ofthe r.v. W and the asymptotic behavior of the distribution P ( Z n = j ) as n → ∞ ,with j >
1. For the Galton-Watson process, our approach based on (1.5) is muchsimpler than that in [23].Our proof also gives an expression of the limit constants in (1.3). For the Galton-Watson process, it recovers the expressions of [23, Theorem 1] in the cases where γ > c r and γ < c r . In the critical case where r = r , the limit constant obtainedin this paper is different to that of [23, Theorem 1], which leads to an alternativeexpression of the constant and the following nice identity involving the well-knownfunctions G , Q and φ : defining G ( t ) = ∞ X k =0 t k P ( Z = k ) ,Q ( t ) = lim n →∞ γ − n G ◦ n ( t ) ,φ ( t ) = E [ e − tW ] , and denoting m = E [ Z ], γ = P ( Z = 1) and ¯ G ( t ) = G ( t ) − γt , we have(1.6) 1 γ Z ∞ ¯ G ( φ ( u )) u r − du = Z m Q ( φ ( u )) u r − du. For a BPRE, we will show a generalization of (1.6) in Proposition 2.2.As a consequence of Theorem 2.1 and of a version of the Gärtner-Ellis theorem,we obtain a lower large deviation result for Z n under conditions weaker than thosein [8, Theorem 3.1]. Assume that P ( Z = 0) = 0 and E m r + ε < ∞ for some ε > λ ) = log E e λX ION GRAMA, QUANSHENG LIU, AND ERIC MIQUEU be the log-Laplace transform of X = log m and(1.8) Λ ∗ ( x ) = sup λ { λx − Λ( λ ) } be the Fenchel-Legendre transform of Λ( · ). Then, for any θ ∈ (0 , E [ X ]), we havelim n →∞ − n log P (cid:16) Z n e θn (cid:17) = χ ∗ ( θ ) ∈ (0 , ∞ ) , (1.9)where χ ∗ ( θ ) = ( − r θ − log γ if 0 < θ < θ , Λ ∗ ( θ ) if θ θ < E [ X ] , (1.10)with(1.11) θ = Λ ′ ( − r ) ∈ (0 , E [ X ]) . Equation (1.9) improves the result of [8, Theorem 3.1(ii)] in the case when P ( Z =0) = 0, since it is assumed in [8] that E m t < ∞ for all t >
0, whereas we onlyrequire that E m r + ε < ∞ for some ε >
0. Moreover, equations (1.10) and (1.11)also give new and alternative expressions of the rate function and the critical value.In fact, it has been proved in [8] that, in the case when P ( Z = 0) = 0 and Z = 1,(1.12) lim n →∞ − n log P (cid:16) Z n e θn (cid:17) = I ( θ ) ∈ (0 , ∞ ) , with I ( θ ) = ( ρ (cid:16) − θθ ∗ (cid:17) + θθ ∗ Λ ∗ ( θ ∗ ) if 0 < θ < θ ∗ , Λ ∗ ( θ ) if θ ∗ θ < E [ X ] , (1.13)where ρ = − log γ and θ ∗ the unique solution on (0 , E [ X ]) of the equation(1.14) ρ − Λ ∗ ( θ ∗ ) θ ∗ = inf θ E [ X ] ρ − Λ ∗ ( θ ) θ . It follows directly from the relations (1.10) to (1.14) that θ = θ ∗ and χ ∗ ( θ ) = I ( θ )for all θ ∈ (0 , E [ X ]).This fact can also be shown by using simple duality argumentsbetween Λ and Λ ∗ , as will be seen in the next section.The rest of the paper is organized as follows. In Section 2 we give the precisestatements of the main theorems with applications. Section 3 is devoted to theproof of the main results, Theorems 2.1 and 2.3. The proofs for the applications aredeferred to Section 4.Throughout the paper, we denote by C an absolute constant whose value maydiffer from line to line. ARMONIC MOMENTS OF Z n FOR A BPRE 5 Main results
A BPRE ( Z n ) can be described as follows. The random environment is representedby a sequence ξ = ( ξ , ξ , ... ) of independent and identically distributed random vari-ables (i.i.d. r.v.’s) taking values in an abstract space Ξ, whose realizations determinethe probability generating functions(2.1) f ξ n ( s ) = f n ( s ) = ∞ X i =0 p i ( ξ n ) s i , s ∈ [0 , , p i ( ξ n ) > , ∞ X i =0 p i ( ξ n ) = 1 . The BPRE ( Z n ) n > is defined by the relations(2.2) Z = 1 , Z n +1 = Z n X i =1 N n,i , for n > , where the random variables N n,i ( i = 1 , , . . . ) represent the number of children ofthe i -th individual of the generation n . Conditionally on the environment ξ , ther.v.’s N n,i ( n > , i >
1) are independent of each other, and each N n,i ( i >
1) hascommon probability generating function f n . In the sequel we denote by P ξ the quenched law , i.e. the conditional probabilitywhen the environment ξ is given, and by τ the law of the environment ξ . Then P ( dx, dξ ) = P ξ ( dx ) τ ( dξ ) is the total law of the process, called annealed law . Thecorresponding quenched and annealed expectations are denoted respectively by E ξ and E . We also denote by P k and E k the corresponding annealed probability andexpectation starting with Z = k individuals, with P = P and E = E . From(2.2), it follows that the probability generating function of Z n conditionally on theenvironment ξ is given by(2.3) g n ( t ) = E ξ [ t Z n ] = f ◦ . . . ◦ f n − ( t ) , t . Since ξ , ξ , . . . are i.i.d. r.v.’s, we get that the annealed probability generatingfunction G k,n of Z n starting with Z = k individuals is given by(2.4) G k,n ( t ) = E k [ t Z n ] = E h g kn ( t ) i , t . We also define, for n > m n = m ( ξ n ) = ∞ X i =0 ip i ( ξ n ) and Π n = E ξ Z n = m ...m n − , where m n represents the average number of children of an individual of generation n when the environment ξ is given, and Π = 1 by convention. Let(2.5) W n = Z n Π n , n > , be the normalized population size. It is well known that under the quenched law P ξ , as well as under the annealed law P , the sequence ( W n ) n > is a non-negativemartingale with respect to the filtration F n = σ ( ξ, N k,i , k n − , i = 1 , . . . ) , ION GRAMA, QUANSHENG LIU, AND ERIC MIQUEU where by convention F = σ ( ξ ). Then the limit W = lim W n exists P - a.s. and E W
1. We also denote the quenched and annealed Laplace transform of W by(2.6) φ ξ ( t ) = E ξ h e − tW i and φ ( t ) = E h e − tW i , for t > , while starting with Z = 1 individual. For k >
1, while starting with Z = k individuals, we have(2.7) φ k ( t ) := E k h e − tW i = E [ φ kξ ( t )] . Another important tool in the study of a BPRE is the associated random walk S n = log Π n = n X i =1 X i , n > , where the r.v.’s X i = log m i − ( i >
1) are i.i.d. depending only on the environment ξ . For the sake of brevity, we set X = log m and µ = E X. We shall consider a supercritical BPRE where µ ∈ (0 , ∞ ), so that under the extracondition E | log(1 − p ( ξ )) | < ∞ (see [24]), the population size tends to infinity withpositive probability. For our propose, in fact we will assume in the whole paper thateach individual gives birth to at least one child, i.e.(2.8) p ( ξ ) = 0 a.s. Therefore, under the condition(2.9) E Z m log + Z < ∞ , the martingale ( W n ) converges to W in L ( P ) (see e.g. [25]) and P ( W >
0) = P ( Z n → ∞ ) = 1 . Now we can state the main result of the paper about the asymptotic of the harmonicmoments E k [ Z − rn ] of the process ( Z n ) for r >
0, starting with Z = k for k > γ k = P k ( Z = k ) = E [ p k ( ξ )]and(2.11) ¯ G k, ( t ) = G k, ( t ) − γ k t k = ∞ X j = k +1 t j P ( Z = j ) , where G k, is the generating function of Z defined in (2.4). Let r k be the solutionof the equation(2.12) γ k = E h m − r k i , with the convention that r k = + ∞ if p ( ξ ) = 0 a.s. For any r >
0, set(2.13) c r = E m − r . ARMONIC MOMENTS OF Z n FOR A BPRE 7
For any k > r >
0, let P ( r ) k be the probability measure (depending on r )defined, for any F n -measurable r.v. T , by(2.14) E ( r ) k [ T ] = E k [Π − rn T ] c nr . Set P ( r ) = P ( r )1 and E ( r ) = E ( r )1 . It is easily seen that under P ( r ) , the process ( Z n ) isstill a supercritical branching process in a random environment with P ( r ) ( Z = 0) =0, and ( W n ) is still a non-negative martingale which converges a.s. to W . Moreover,by (2.9) and the fact that m >
1, we have E ( r ) (cid:20) Z m log Z (cid:21) = E " Z m r log Z / E h m − r i E (cid:20) Z m log Z (cid:21) / E h m − r i < ∞ , which implies that(2.15) W n → W in L ( P ( r ) ) . Then we have the following equivalent for the harmonic moments of Z n . Let(2.16) A k,n ( r ) = γ nk , if r > r k ,nγ nk , if r = r k ,c nr , if r < r k . Theorem 2.1.
Let k > and assume that E m r k + ε < + ∞ for some ε > . Thenwe have (2.17) lim n →∞ E k [ Z − rn ] A k,n ( r ) = C ( k, r ) := r ) Z ∞ Q k ( e − t ) t r − dt if r > r k ,γ − k Γ( r ) E ( r ) (cid:20)Z ∞ ¯ G k, ( φ ξ ( t )) t r − dt (cid:21) if r = r k , r ) Z ∞ φ ( r ) k ( t ) t r − dt if r < r k , where C ( k, r ) ∈ (0 , ∞ ) , Γ( r ) = R ∞ x r − e − x dx is the Gamma function, and φ ( r ) k ( t ) = E ( r ) k [ e − tW ] is the Laplace transform of W under P ( r ) k . This theorem shows that there is a phase transition in the rate of convergence ofthe harmonic moments of Z n with the critical value r k > r k is generally different from the critical value a k for the existence ofthe harmonic moment of W. Indeed, as shown in [14, Theorem 2.1] (see Lemma 3.1below), the critical value a k is determined by(2.18) E h p k ( ξ ) m a k i = 1 , which is in general different from the critical value r k determined by (2.12), that is(2.19) E h p k ( ξ ) i = E h m − r k i . ION GRAMA, QUANSHENG LIU, AND ERIC MIQUEU
This is in contrast with the Galton-Watson process where r k = a k = r k , with r thesolution of the equation p m r = 1 which coincides with both equations (2.18) and(2.19). Theorem 2.1 generalizes the result of [23] for the Galton-Watson process.For a BPRE, it completes and improves Theorem 1.3 of [17], where the formula(2.17) was only proved for k = 1 and r < r , and under the following much strongerboundedness condition: there exist some constants p, c , c > c m c and c m ( p ) c a.s., where m ( p ) = P ∞ i =1 i p p i ( ξ ). Instead of this boundedness condition, here we onlyrequire the moment assumption E [ m r k + ε ] < ∞ for some ε > k = 1 initial individuals, the expression ofthe limit constant in the case when r = r (up to the constant factor Γ( r )) becomes1 γ Z ∞ ¯ G ( φ ( u )) u r − du, whereas it has been proved in [23] that the limit constant is equal to Z m Q ( φ ( u )) u r − du. Actually, the above two expressions coincide, as shown by the following result validfor a general BPRE. Let Q k ( t ) be defined by(2.20) Q k ( t ) = lim n →∞ G k,n ( t ) γ nk = ∞ X j = k q k,j t j , t ∈ [0 , , where G k,n is defined by (2.4) and the limit exitsts according to [14, Theorem 2.3](see Lemma 3.2 below). Proposition 2.2.
For k > and r = r k , we have (2.21) 1 γ k E ( r ) (cid:20)Z ∞ ¯ G k, ( φ ξ ( u )) u r − du (cid:21) = E ( r ) (cid:20)Z m Q k ( φ ξ ( u )) u r − du (cid:21) . As an application of Theorem 2.1 we get a large deviation result. Indeed, E [ Z λn ] = E [ e λ log Z n ] is the Laplace transform of log Z n . From Theorem 2.1 we obtain(2.22) lim n →∞ n log E k [ Z λn ] = χ k ( λ ) = ( log γ k if λ λ k , Λ( λ ) if λ ∈ [ λ k , . Thus using a version of the Gärtner-Ellis theorem adapted to the study of tailprobabilities (see [17, Lemma 3.1]), we get the following lower large deviation resultfor the BPRE ( Z n ). Recall that Λ( λ ) = log E e λX is the log-Laplace of X = log m ,and Λ ∗ ( · ) is the Fenchel-Legendre transform of Λ( · ) defined in (1.8). Theorem 2.3.
Let k > and r k be the solution of the equation (2.12) . Assume that E m r k + ε < ∞ for some ε > . Then, for any θ ∈ (0 , E [ X ]) , we have (2.23) lim n →∞ − n log P k (cid:16) Z n e θn (cid:17) = χ ∗ k ( θ ) ∈ (0 , ∞ ) , ARMONIC MOMENTS OF Z n FOR A BPRE 9 where χ ∗ k ( θ ) = sup λ { λθ − χ k ( λ ) } = ( − r k θ − log γ k if < θ < θ k , Λ ∗ ( θ ) if θ k θ < E [ X ] , (2.24) with (2.25) θ k = Λ ′ ( − r k ) . The value χ ∗ k ( θ ) can be interpreted geometrically as the maximum distance be-tween the graphs of the linear function l θ : λ θλ with slope θ and the function χ k : λ χ k ( λ ) defined in (2.22). Taking into account the fact that χ ( λ ) = Λ( λ )for λ ∈ [ − r k ,
0] and χ ( λ ) = log γ k for λ − r k , we can easily describe the phasetransitions of χ ∗ ( θ ) depending on the value of the slope θ of the function l θ :(1) in the case when θ ∈ ( θ k , E [ X ]), the maximum sup λ { l θ ( λ ) − χ ( λ ) } isattained for λ ∈ ( − r k ,
0) such that χ ′ ( λ θ ) = Λ ′ ( λ θ ) = θ , whose value is χ ∗ ( θ ) = Λ ∗ ( θ ) (see Fig. 1);(2) the case when θ = θ k is the critical slope for which the equation Λ ′ ( λ ) = θ k has a solution given by λ = − r k (see Fig. 2);(3) in the case when θ ∈ (0 , θ k ), the maximum sup λ { l θ ( λ ) − Λ( λ ) } is attainedfor λ = − r k , and then χ ∗ ( θ ) = − r k θ − log γ k becomes linear in θ (see Fig.3). Fig. 1,2,3: Geometrical interpretation of χ ∗ ( θ ) λl θ ( λ ) χ ∗ ( θ ) λ θ − r k – log γ k Λ( λ ) χ ( λ ) Fig. 1: θ ∈ ( θ k , E [ X ]) λl θ k ( λ ) χ ∗ ( θ k ) − r k – log γ k Λ( λ ) χ ( λ ) Fig. 2: θ = θ k λl θ ( λ ) χ ∗ ( θ ) − r k – log γ k Λ( λ ) χ ( λ ) Fig. 3: θ ∈ (0 , θ k ) Remark 2.4.
Theorem 2.3 corrects and improves the result of [8, Theorem 3.1(ii)] .Moreover it gives new and alternative expressions of the rate function and the criticalvalue. Actually it was proved in [8, Theorem 3.1(ii)] that, assuming P ( Z = 0) = 0 and E m t < ∞ for all t > , we have (2.26) lim n →∞ − n log P k (cid:16) Z n e θn (cid:17) = I k ( θ ) ∈ (0 , ∞ ) , where I k ( θ ) = ρ k (cid:18) − θθ ∗ k (cid:19) + θθ ∗ k Λ ∗ ( θ ∗ k ) if < θ θ ∗ k , Λ ∗ ( θ ) if θ ∗ k θ < E [ X ] , (2.27) with (2.28) ρ k = lim n →∞ − n log P k ( Z n = j ) and θ ∗ k the unique solution on (0 , E [ X ]) of the equation (2.29) ρ k − Λ ∗ ( θ ∗ k ) θ ∗ k = inf θ E [ X ] ρ k − Λ ∗ ( θ ) θ . It has been stated mistakenly in [9] that ρ k = − k log γ , whereas the correct state-ment is (2.30) ρ k = − log γ k , according to [14, Theorem 2.3] (see Lemma 3.2 below). With this correction, the twocritical values θ k and θ ∗ k and the two rate functions I k and χ ∗ k coincide, that is (2.31) θ k = θ ∗ k and χ ∗ k ( θ ) = I k ( θ ) for all θ ∈ (0 , E [ X ]) . Indeed, by the definition of θ ∗ k , the derivative of the function θ ρ k − Λ ∗ ( θ ) θ vanishesfor θ = θ ∗ k . Therefore, since (Λ ∗ ) ′ ( θ ) = λ θ with Λ ′ ( λ θ ) = θ , we get, for θ = θ ∗ k , (2.32) Λ ∗ ( θ ) = λ θ θ + ρ k . ARMONIC MOMENTS OF Z n FOR A BPRE 11
Using the identity Λ ∗ ( θ ) = λ θ θ − Λ( λ θ ) , we obtain Λ( λ θ ) = − ρ k , which implies that λ θ = − r k and then θ ∗ k = Λ ′ ( − r k ) = θ k .Moreover, coming back to (2.32) and using the identities Λ( − r k ) = log γ k = − ρ k and θ k = Λ ′ ( − r k ) , we get − r k θ k − Λ( − r k ) = − r k θ k + ρ k = Λ ∗ k ( θ k ) . Therefore, for any θ ∈ [0 , θ k ] , − r k θ − log γ k = − r k θ − Λ( − r k )= θθ k ( − r k θ k − Λ( − r k )) + θθ k Λ( − r k ) − Λ( − r k )= θθ k Λ ∗ ( θ k ) − − θθ k ! log γ k , so that I k ( θ ) = χ ∗ k ( θ ) , which ends the proof of (2.31) . From (2.31) and Theorem2.1, we see that (2.26) is valid assuming only P ( Z = 0) = 0 and E [ m r k + ε ] < ∞ forsome ε > . Actually when P ( Z = 0) > , as shown in [8, Theorem 3.1 (i)] , (2.26) remains valid with ρ k = lim n →∞ − n log P k ( Z n = j ) = ρ > independent of k . Similarly, one can apply Theorem 2.1 to get the decay rate for the probability P ( Z n k n ), where k n is any sub-exponential sequence in the sense that k n → ∞ and k n / exp( θn ) → θ >
0, as stated in the following corollary.
Corollary 2.5.
Let k > and assume that E [ m r k + ε ] < ∞ for some ε > . Let k n > be such that k n → ∞ and k n / exp( θn ) → for every θ > , as n → ∞ .Then (2.33) lim n →∞ n log P k ( Z n k n ) = log γ k . It was stated mistakenly in [8, Theorem 3.1(ii)] that lim n →∞ n log P k ( Z n k n ) = k log γ . To show (2.33), it suffices to note that by Markov’s inequality and Theorem2.1, we have, for any r > r k , γ nk = P k ( Z n = k ) P k ( Z n k n ) E [ Z − rn ] k rn min { γ nk k rn , γ nk k r k n n } . The above argument leads to a precise large deviation bound as stated below.
Corollary 2.6.
Let k > and assume that E [ m r k + ε ] < ∞ for some ε > . Then (2.34) P k (cid:16) Z n e θn (cid:17) inf r> E k [ Z − rn ] e θrn = e − n ( − θr k − Λ( − r k )) if < θ θ k ,ne − n ( − θ k r k − Λ( − r k )) if θ = θ k ,e − n Λ ∗ ( θ ) if θ k θ < E [ X ] . The question of the exact decay rate of P ( Z n e θn ) will be treated in a forth-coming paper.As an example, let us consider the case where the reproduction law has a fractionallinear generating function, that is when(2.35) p ( ξ ) = a and p k ( ξ ) = (1 − a )(1 − b ) b b k for all k > , for which the generating function is f ( t ) = a + (1 − a )(1 − b ) t − b t , where a ∈ [0 , and b ∈ (0 ,
1) are random variables depending on the environment ξ . This case has been examinated by several authors (see e.g. [19, 22]). In thecase where a = 0 (non-extinction), the BPRE is said to be geometric; in this case X = log m = − log(1 − b ), log γ k = log E [ e − kX ] = Λ( − k ) and r k = k . Therefore,we obtain the following explicit version of Theorem 2.1: Corollary 2.7.
Let Z n be a geometric BPRE. Assume that there exists ε > suchthat E [ e ( k + ε ) X ] < ∞ . Then (2.23) holds with χ ∗ k ( θ ) = ( − kθ − log E [ e − kX ] if < θ θ k , Λ ∗ ( θ ) if θ k θ < E [ X ] , (2.36) where (2.37) θ k = E [ Xe − kX ] / E [ e − kX ] . Corollary 2.7 recovers and completes the large deviation result in [8, Corollary3.3] for a fractional linear BPRE with a >
0, where (2.23) was obtained with I ( θ ) = ( − θ − log E [ e − X ] if 0 < θ θ ∗ , Λ ∗ ( θ ) if θ ∗ θ < E [ X ] , (2.38)and(2.39) θ ∗ = E [ Xe − X ] / E [ e − X ] . In fact the result in [8, Corollary 3.3] was stated without the hypothesis a > a = 0, this result is valid only for k = 1, as shown by Corollary 2.7 (for k >
2, the factor k is missing in [8, Corollary 3.3]).As another consequence of Theorem 2.1, we improve an earlier result about therate of convergence in the central limit theorem for W − W n . Theorem 2.8.
Assume essinf m (2) m > and E Z ε < ∞ for some ε ∈ (0 , . Thenthere exists a constant C > such that, for all k > , (2.40) sup x ∈ R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P k Π n ( W − W n ) √ Z n δ ∞ ( T n ξ ) x ! − Φ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) CA k,n ( ε/ , where Φ( x ) = √ π R x −∞ e − t / dt is the standard normal distribution function. ARMONIC MOMENTS OF Z n FOR A BPRE 13
Theorem 2.8 improves the exponential rate of convergence in [17, Theorem 1.7]. Inour approach the assumption essinf m (2) m > δ ∞ ( ξ ) of the variable W is a.s. separated from 0. This hypothesis does notseem natural and should be relaxed. One should be able to find a suitable hypothesisto ensure the existence of harmonic moments for the random variable δ ∞ ( ξ ), whichwould be enough for our objective.As another consequence of Theorem 2.1, we give some large deviation results onthe ratio(2.41) R n = Z n +1 Z n toward the conditional mean m n = P ∞ k =1 kp k ( ξ n ). Let(2.42) M n,j = j − j X i =1 N n,i be the empirical mean of m n of size j under the environment ξ , where the r.v.’s N n,i ( i = 1 , . . . , j ) are i.i.d. with generating function f n . Theorem 2.9.
Let k > . If, for some set D ⊂ R , there exist some constants C > and r > such that, for all j > , (2.43) P ( M ,j − m ∈ D ) C j r , then there exists a constant B ∈ (0 , ∞ ) such that for all n > , (2.44) P k ( R n − m n ∈ D ) B A k,n ( r ) , where A k,n ( r ) is defined in (2.16) . Similarly, if there exist some constants C > and r > such that, for all j > , (2.45) P ( M ,j − m ∈ D ) > C j r , then there exists a constant B ∈ (0 , ∞ ) such that for all n > , (2.46) P k ( R n − m n ∈ D ) > B A k,n ( r ) . This result shows that there exist some phase transitions in the rate of convergencedepending on whether the value of r is less than, equal or greater than the constant γ k . The next result gives a bound of the large deviation probability of R n − m n under a simple moment condition on Z . Theorem 2.10.
Let k > . Assume that there exists p > such that E | Z − m | p < ∞ . Then, there exists a constant C p > such that, for any a > , (2.47) P k ( | R n − m n | > a ) ( C p a − p A k,n ( p − if p ∈ (1 , ,C p a − p A k,n ( p/ if p ∈ (2 , ∞ ) . Proof of main theorems
In this section we will prove the main results of this paper, Theorems 2.1 and 2.3,and the associated result, Proposition 2.2. In Section 3.1 we present some auxiliaryresults concerning the critical value for the existence of the harmonic moments ofthe r.v. W and the asymptotic behavior of the asymptotic distriution P k ( Z n = j ) as n → ∞ , with j > k >
1. In Sections 3.2 and 3.3 we prove respectively Theorems 2.1and 2.3. The proof of Proposition 2.2 is given in Section 3.4 for a Galton-Watsonprocess and in Section 3.5 for a general BPRE.3.1.
Auxiliary results.
We recall some results to be used in the proofs. The firstone concerns the critical value for the existence of harmonic moments of the r.v. W . Lemma 3.1 ([14], Theorem 2.1) . Assume that there exists a constant p > suchthat E [ m p ] < ∞ . Then for any a ∈ (0 , p ) , E k W − a < ∞ if and only if E h p k ( ξ ) m a i < . The second result is about the asymptotic equivalent of the probability P k ( Z n = j )as n → ∞ , for any j > k > Lemma 3.2 ([14], Theorem 2.3) . Assume that P ( Z = 1) > . For any k > thefollowing assertions hold.a) For any accessible state j > k in the sense that P k ( Z l = j ) > for some l > ,we have (3.1) P k ( Z n = j ) ∼ n →∞ γ nk q k,j , where q k,k = 1 and, for j > k , q k,j ∈ (0 , + ∞ ) is the solution of the recurrencerelation (3.2) γ k q k,j = j X i = k p ( i, j ) q k,i , with q k,i = 0 for any non-accessible state i (i.e. P k ( Z l = i ) = 0 for all l > ).b) Assume that there exists ε > such that E [ m r k + ε ] < ∞ . Then, for any r > r k ,we have (3.3) ∞ X j = k j − r q k,j < ∞ . In particular, the radius of convergence of the power series (3.4) Q k ( t ) = + ∞ X j = k q k,j t j is equal to 1.c) For all t ∈ [0 , and k > , we have, (3.5) G k,n ( t ) γ nk ↑ Q k ( t ) as n → ∞ , ARMONIC MOMENTS OF Z n FOR A BPRE 15 where G k,n is the probability generating function of Z n when Z = k , defined in (2.4) .d) Q k ( t ) is the unique power series which verifies the functional equation (3.6) γ k Q k ( t ) = E [ Q k ( f ( t ))] , t ∈ [0 , , with the condition Q ( k ) k (0) = 1 . Proof of Theorem 2.1.
In this section we give a proof of the convergenceof the normalized harmonic moments E k [ Z − rn ] /A k,n ( r ) as n → ∞ , where A k,n ( r ) isdefined in (2.16).a) We first consider the case when r < r k (which corresponds to the case γ k < c r ).By the change of measure (2.14), we obtain(3.7) E k h Z − rn i = E ( r ) k [ W − rn ] c nr , with c r = E m − r . From (2.15) and [17, Lemma 2.1], it follows that the sequence( E ( r ) k [ W − rn ]) is increasing and(3.8) lim n →∞ E ( r ) k [ W − rn ] = sup n ∈ N E ( r ) k [ W − rn ] = E ( r ) k [ W − r ] . Moreover, for any r < r k , we have γ k < c r , which implies that E ( r ) [ p k ( ξ ) m r ] = γ k / E m − r <
1. So, by Lemma 3.1, we get, for any r < r k ,(3.9) E ( r ) k [ W − r ] < ∞ . Therefore, coming back to (3.7) and using (3.8) and (3.9), we obtain(3.10) E k h Z − rn i c − nr ↑ n →∞ E ( r ) k h W − r i ∈ (0 , ∞ ) . To give an integral expression of the limit constant C ( k, r ), we shall use the followingexpression for the inverse of a positive random variable X r : for any r >
0, we have(3.11) 1 X r = 1Γ( r ) Z + ∞ e − uX u r − du. Then, from (3.10), (3.11) and Fubini’s theorem, we get(3.12) E k h Z − rn i c − nr ↑ n →∞ r ) Z ∞ φ ( r ) k ( u ) u r − du, which proves (2.17) for r < r k .b) Next we consider the case when r > r k (which corresponds to the case γ k > c r ).Using parts a) and b) of Lemma 3.2 and the monotone convergence theorem, itfollows that lim n →∞ ↑ E k [ Z − rn ] γ nk = lim n →∞ ↑ ∞ X j = k +1 k − r P ( Z n = k ) γ nk = ∞ X j = k +1 k − r q k,j < ∞ . (3.13) Moreover, using (3.11) together with Fubini’s theorem and the change of variable ju = t , we obtain1Γ( r ) Z Q k ( e − u ) u r − du = ∞ X j = k q k,j j − r r ) Z ∞ e − t t r − dt = ∞ X j = k q k,j j − r . Therefore, coming back to (3.13), we get(3.14) lim n →∞ ↑ E k [ Z − rn ] γ nk = 1Γ( r ) Z Q k ( e − u ) u r − du, which proves (2.17) for r > r k .c) Now we consider the case when r = r k (which corresponds to r = γ k ). For n > m >
0, we have the following well-known branching property for Z n :(3.15) Z n + m = Z m X i =1 Z ( m ) n,i , where the r.v.’s Z ( m ) n,i ( i >
1) are independent of Z m under P ξ and P . Moreover,under P ξ , for each n >
0, the r.v.’s Z ( m ) n,i ( i >
1) are i.i.d. with the same condi-tional probability law P ξ (cid:16) Z ( m ) n,i ∈ · (cid:17) = P T m ξ ( Z n ∈ · ), where T m is the shift operatordefined by T m ( ξ , ξ , . . . ) = ( ξ m , ξ m +1 , . . . ). Intuitively, relation (3.15) shows that,conditionally on Z m = i , the annealed law of the process Z n + m is the same as thatof a new process Z n starting with i individuals. Using (3.15) with m = 1, we obtain E k h Z − rn +1 i = E k h Z − rn i P k ( Z = k ) + ∞ X i = k +1 E i h Z − rn i P k ( Z = i ) . (3.16)From (3.7), we have E i [ Z − rn ] = E ( r ) i [ W − rn ] c nr . Substituting this into (3.16) and setting b n = ∞ X i = k +1 E ( r ) i [ W − rn ] P k ( Z = i ) , we get(3.17) E k [ Z − rn +1 ] = E k [ Z − rn ] γ k + b n c nr , with γ k = P k ( Z = k ). Iterating (3.17) leads to(3.18) E k h Z − rn +1 i = γ n +1 k k − r + n X j =0 γ n − jk b j c jr . Using the fact that r = r k (which corresponds to γ k = c r ) and dividing (3.18) by γ n +1 n , we get E k h Z − rn +1 i nγ n +1 = k − r n + γ − k n n X j =0 b j . (3.19) ARMONIC MOMENTS OF Z n FOR A BPRE 17
To prove the convergence in (3.19) we need to show that lim n →∞ b n < ∞ . By (3.8)and the monotone convergence theorem, we have(3.20) b := lim n →∞ ↑ b n = ∞ X i = k +1 E ( r ) i [ W − r ] P k ( Z = i ) . Now we show that b < ∞ . Using (3.15) for m = 0 and Z = i , with i > k , and thefact that Z n,j > j i , we obtain(3.21) E i [ Z − rn ] = E h ( Z n, + . . . + Z n,i ) − r i E h ( Z n, + . . . + Z n,k ) − r i = E k [ Z − rn ] . By (3.21) and the change of measure (3.7), we get, for any i > k + 1, E ( r ) i [ W − rn ] E ( r ) k +1 [ W − rn ] . Then, as in (3.8), letting n → ∞ leads to(3.22) E ( r ) i [ W − r ] E ( r ) k +1 [ W − r ] . Now we shall prove that(3.23) E ( r ) k +1 [ W − r ] < ∞ . For this it is enough to verify the condition of Lemma 3.1 under the measure E ( r ) k defined by (2.14): indeed, for r = r k we have γ k = c r , which implies that E ( r ) [ p k +11 ( ξ ) m r ] = γ k +1 E m − r = γ k +1 γ k < . This proves (3.23). Using (3.22) and (3.23), we obtain b = ∞ X i = k +1 E ( r ) i [ W − r ] P k ( Z = i ) E ( r ) k +1 [ W − r ] ∞ X j = k +1 P k ( Z = j ) < ∞ . Therefore, coming back to (3.19), using (3.20) and Cesaro’s lemma, we get(3.24) lim n →∞ E k h Z − rn +1 i nγ n +1 k = 1 γ k ∞ X i = k +1 E ( r ) i [ W − r ] P k ( Z = i ) < ∞ , which proves (2.17) for r = r k , with C ( k, r ) = 1 γ k ∞ X i = k +1 E ( r ) i [ W − r ] P k ( Z = i ) . We now show an integral expression of the constant C ( k, r ). Recall that W admitsthe following decomposition(3.25) W = Z X j =1 W ( j ) , where the r.v.’s W ( j ) ( j = 1 , , . . . ) are independent of Z and m under P ξ and P .Moreover, conditionally on the environement ξ , the r.v.’s W ( j ) ( j = 1 , , . . . ) are i.i.d. with common law P ξ ( W ( j ) ∈ · ) = P ξ ( W ∈ · ). With these considerations, itcan be easily seen that(3.26) φ ( r ) i ( t ) = E ( r ) i [ φ ξ ( u )] = E ( r ) [ φ ξ ( u ) i ] . Therefore, using (3.11) with r = r k , together with (3.26) and Fubini’s theorem, weobtain1 γ k ∞ X i = k +1 E ( r ) i [ W − r ] P k ( Z = i ) = 1 γ k Γ( r ) ∞ X i = k +1 E ( r ) i Z ∞ e − uW u r − du P k ( Z = i )= 1 γ k Γ( r ) ∞ X i = k +1 E ( r ) Z ∞ φ iξ ( u ) u r − du P k ( Z = i )= 1 γ k Γ( r ) E ( r ) Z ∞ ∞ X i = k +1 φ iξ ( u ) P k ( Z = i ) u r − du = 1 γ k Γ( r ) E ( r ) (cid:20)Z ∞ ¯ G k, ( φ ξ ( u )) u r − du (cid:21) , (3.27)where ¯ G k, ( u ) = G k, ( u ) − γ k u k = P ∞ j = k +1 u j P ( Z = j ). Therefore, using (3.24) and(3.27), we get lim n →∞ E k h Z − rn +1 i nγ n +1 k = 1 γ k Γ( r ) E ( r ) (cid:20)Z ∞ ¯ G k, ( φ ξ ( u )) u r − du (cid:21) , which ends the proof of Theorem 2.1.3.3. Proof of Theorem 2.3.
In this section we prove Theorem 2.3. For conve-nience, let λ k = − r k . From Theorem 2.1, for any k >
1, we have(3.28) lim n →∞ n log E k [ Z λn ] = χ k ( λ ) = ( log γ k if λ λ k , Λ( λ ) if λ ∈ [ λ k , . Thus using a version of the Gärtner-Ellis theorem adapted to the study of tailprobabilities (see [17, Lemma 3.1]) and the fact that χ k ( λ ) = log γ k for all λ λ k ,we obtain, for all θ ∈ (0 , E [ X ]),(3.29) lim n →∞ − n P k ( Z n e θn ) = χ ∗ ( θ ) , with(3.30) χ ∗ k ( θ ) = sup λ { λθ − χ k ( λ ) } = max ( λ k θ − Λ( λ k ) , sup λ k λ { λθ − Λ( λ ) } ) . It is well-known (see e.g. [13, Lemma 2.2.5]) that the functionΛ ∗ ( θ ) = sup λ { λθ − Λ( λ ) } = { λ θ θ − Λ( λ θ ) } , with Λ ′ ( λ θ ) = θ, is non-increasing for θ ∈ (0 , E [ X ]). Therefore, letting(3.31) θ k = Λ ′ ( λ k ) , it follows that: ARMONIC MOMENTS OF Z n FOR A BPRE 19 (1) for any θ ∈ (0 , θ k ], λ k θ − Λ( λ k ) > λ k θ k − Λ( λ k ) = Λ ∗ ( θ k ) = sup λ k λ { λθ − Λ( λ ) } ;(2) for any θ ∈ [ θ k , µ ),Λ ∗ ( θ ) = sup λ k λ { λθ − Λ( λ ) } > Λ ∗ ( θ k ) = λ k θ k − Λ( λ k ) > λ k θ − Λ( λ k ) . With these considerations, we get from (3.30) that(3.32) χ ∗ k ( θ ) = ( λ k θ − Λ( λ k ) if θ ∈ (0 , θ k ] , Λ ∗ ( θ ) if θ ∈ [ θ k , E [ X ]) , which ends the proof of Theorem 2.3.3.4. Proof of Proposition 2.2 for the Galton-Watson case.
In this section weassume that ( Z n ) is a Galton-Watson process and prove (1.6), which is a particularbut simpler case of Proposition 2.2. Proof.
First note that for the Galton-Watson case, we have(3.33) γm r = 1 . For convenience, we shall write r = r . Using the additive property of integrationand the change of variable u = tm k for k >
0, together with Fubini’s theorem andthe fact that γm r = 1, we obtain1 γ Z ∞ ¯ G ( φ ( u )) u r − du = 1 γ ∞ X k =0 Z m k +1 m k ¯ G ( φ ( u )) u r − du = 1 γ ∞ X k =0 Z m ¯ G ( φ ( tm k ))( m r ) k t r − dt = 1 γ Z m ∞ X k =0 γ − k ¯ G ( φ ( tm k )) t r − dt. (3.34)Since ¯ G ( t ) = G ( t ) − γt and G ( φ ( t )) = φ ( tm ), we obtain, for any k > γ − k ¯ G ( φ ( tm k )) = γ − k G ( φ ( tm k )) − γ − k γφ ( tm k )= γ − k G ◦ k +1 ( φ ( t )) − γ k − G ◦ k ( φ ( t )) . (3.35)By (3.35), using a telescoping argument and the fact that lim k →∞ γ − k G ◦ k ( t ) = Q ( t ),we get ∞ X k =0 γ − k ¯ G ( φ ( tm k )) = γQ ( φ ( t )) − γφ ( t ) . (3.36)Therefore, coming back to (3.34) and using (3.36), we have(3.37) 1 γ Z ∞ ¯ G ( φ ( u )) u r − du = Z m Q ( φ ( u )) u r − du − Z m φ ( u ) u r − du. Moreover, using the change of variable u = t/m and the relations G ( φ ( t/m )) = φ ( t ) and γm r = 1, we get1 γ Z G ( φ ( u )) u r − du = m − r γ Z m φ ( t ) t r − dt = Z m φ ( t ) t r − dt. Therefore, since ¯ G ( u ) = G ( u ) − γu , we obtain1 γ Z ¯ G ( φ ( u )) u r − du = 1 γ Z G ( φ ( u )) u r − du − Z φ ( u ) u r − du = Z m φ ( u ) u r − du − Z φ ( u ) u r − du = Z m φ ( u ) u r − du. (3.38)Finally, using (3.37), (3.38) and the additive property of integration, we obtain(3.39) 1 γ Z ∞ ¯ G ( φ ( u )) u r − du = Z m Q ( φ ( u )) u r − du, which ends the proof of (1.6). (cid:3) Proof of Proposition 2.2.
Let k >
1. For convenience, let r = r k . Usingthe additive property of integration, the change of variable u = t Π kj for j > γ k E ( r ) (cid:20)Z ∞ ¯ G k, ( φ ξ ( u )) u r − du (cid:21) = 1 γ k E ( r ) ∞ X j =0 Z Π j +1 Π j ¯ G k, ( φ ξ ( u )) u r − du = 1 γ k E ( r ) ∞ X j =0 Z m j ¯ G k, ( φ ξ ( t Π j ))Π rj t r − dt = 1 γ k ∞ X j =0 E ( r ) (cid:20)Z m j ¯ G k, ( φ ξ ( t Π j ))Π rj t r − dt (cid:21) . (3.40)Recall that φ ξ ( t Π j ) = g j ( φ T j ξ ( t )), where g j ( t ) = f ◦ . . . ◦ f j − ( t ) is a random functiondepending on the environment ξ , . . . , ξ j − and T j ξ = ( ξ j , ξ j +1 , . . . ). Then, using thechange of measure (2.14), the independence of the environment sequence ( ξ i ) andFubini’s theorem, we see that(3.41) E ( r ) (cid:20)Z m j ¯ G k, ( φ ξ ( t Π j ))Π rj t r − dt (cid:21) = E ( r ) (cid:20)Z m j c − jr E T j ξ h ¯ G k, ( g j ( φ T j ξ ( t )) i t r − dt (cid:21) . Using the fact that ¯ G k, ( t ) = G k, ( t ) − γ k t k and the relations E [ G k,n ( g j ( t ))] = G k,n + j ( t ) and E [ g kj ( t )] = G k,j ( t ), we get E ( r ) (cid:20)Z m j c − jr E T j ξ h ¯ G k, ( g j ( φ T j ξ ( t )) i t r − dt (cid:21) = E ( r ) (cid:20)Z m j c − jr h G k,j +1 ( φ T j ξ ( t )) − G k,j ( φ T j ξ ( t )) i t r − dt (cid:21) . (3.42) ARMONIC MOMENTS OF Z n FOR A BPRE 21
Moreover, since the environment sequence ( ξ , ξ , . . . ) is i.i.d., we obtain, for any j > E ( r ) (cid:20)Z m j c − jr h G k,j +1 ( φ T j ξ ( t )) − G k,j ( φ T j ξ ( t )) i t r − dt (cid:21) = E ( r ) (cid:20)Z m c − jr [ G k,j +1 ( φ ξ ( t )) − G k,j ( φ ξ ( t ))] t r − dt (cid:21) . (3.43)Therefore, coming back to (3.40) and using the fact that c r = γ k (for r = r k ) togetherwith Fubini’s theorem, we obtain(3.44)1 γ k E ( r ) (cid:20)Z ∞ ¯ G k, ( φ ξ ( u )) u r − du (cid:21) = E ( r ) Z m ∞ X j =0 G k,j +1 ( φ ξ ( t )) γ j +1 k − G k,j ( φ ξ ( t )) γ jk t r − dt . Using a telescoping argument and the assertion that lim j →∞ γ jk G k,j ( t ) = Q k ( t ) forall t ∈ [0 , ∞ X j =0 " G k,j +1 ( φ ξ ( t )) γ j +1 k − G k,j ( φ ξ ( t )) γ jk = Q k ( φ ξ ( t )) − G k, ( φ ξ ( t ))= Q k ( φ ξ ( t )) − φ kξ ( t ) . (3.45)Therefore, by (3.44) and (3.45), we have1 γ k E ( r ) (cid:20)Z ∞ ¯ G k, ( φ ξ ( u )) u r − du (cid:21) = E ( r ) (cid:20)Z m Q k ( φ ξ ( t )) t r − dt (cid:21) − E ( r ) (cid:20)Z m φ kξ ( t ) t r − dt (cid:21) . (3.46)Moreover, using the identity φ ξ ( u ) = f ( φ T ξ ( t/m )), the change of variable t = u/m , the independence between ξ and T ξ , the relation γ k = c r and Fubini’stheorem, we get E ( r ) (cid:20)Z m φ kξ ( t ) t r − dt (cid:21) = E ( r ) (cid:20)Z f k ( φ T ξ ( u )) m r u r − du (cid:21) = E ( r ) (cid:20)Z E ( r ) T ξ h f k ( φ T ξ ( u )) m r i u r − du (cid:21) = E ( r ) (cid:20)Z c − r G k, ( φ T ξ ( u )) u r − du (cid:21) = γ − k E ( r ) (cid:20)Z G k, ( φ ξ ( u )) u r − du (cid:21) . (3.47)Therefore, from the identity ¯ G k, ( t ) = G k, ( t ) − γ k t k and (3.47), it follows that1 γ k E ( r ) (cid:20)Z ¯ G k, ( φ ξ ( u )) u r − du (cid:21) = 1 γ k E ( r ) (cid:20)Z G k, ( φ ξ ( t )) t r − dt (cid:21) − E ( r ) (cid:20)Z φ kξ ( t ) t r − dt (cid:21) = E ( r ) (cid:20)Z m φ kξ ( t ) t r − dt (cid:21) − E ( r ) (cid:20)Z φ kξ ( t ) t r − dt (cid:21) = E ( r ) (cid:20)Z m φ kξ ( t ) t r − dt (cid:21) . (3.48) Finally, using (3.46) and (3.48), we obtain(3.49) 1 γ k E ( r ) (cid:20)Z ∞ ¯ G k, ( φ ξ ( u )) u r − du (cid:21) = E ( r ) (cid:20)Z m Q k ( φ ξ ( u )) u r − du (cid:21) , which ends the proof of Proposition 2.2.4. Applications
In this section we present the proofs of Theorems 2.8, 2.9 and 2.10 as applicationsof Theorem 2.1. In Section 4.1 we give the rate of convergence in the central limittheorem for W − W n where we prove Theorem 2.8. In Section 4.2 we deal with thelarge deviation results for the ratio R n = Z n +1 /Z n , where we prove Theorems 2.9and 2.10.4.1. Central Limit Theorem for W − W n . In this section we prove Theorem 2.8.
Proof of Theorem 2.8.
It is well known that W admits the following decomposition:Π n ( W − W n ) = Z n X i =1 ( W ( i ) − , where under P ξ , the random variables W ( i ) ( i >
1) are independent of each otherand independent of Z n , with common distribution P ξ ( W ( i ) ∈ · ) = P T n ξ ( W ∈ · ).Let δ ∞ ( ξ ) = ∞ X n =0 n m n (2) m n − ! . The r.v. δ ∞ ( ξ ) is the variance of W under P ξ (see e.g. [16]). Notice that if c :=essinf m (2) m > , then δ ∞ ( ξ ) > c − >
0. Therefore, condition E Z ε < ∞ impliesthat, for all k >
1, it holds E k (cid:12)(cid:12)(cid:12) W − δ ∞ (cid:12)(cid:12)(cid:12) ε Cc − E k | W − | ε < ∞ (see [15]). By theBerry-Esseen theorem, we have for all x ∈ R , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ξ Π n ( W − W n ) √ Z n δ ∞ ( T n ξ ) x ! − Φ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C E T n ξ (cid:12)(cid:12)(cid:12)(cid:12) W − δ ∞ (cid:12)(cid:12)(cid:12)(cid:12) ε E ξ h Z − ε/ n i . Taking expectation with Z = k and using Theorem 2.1, we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P k Π n ( W − W n ) √ Z n δ ∞ ( T n ξ ) x ! − Φ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C E k (cid:12)(cid:12)(cid:12)(cid:12) W − δ ∞ (cid:12)(cid:12)(cid:12)(cid:12) ε E k h Z − ε/ n i CA k,n ( − ε/ . (4.1) (cid:3) ARMONIC MOMENTS OF Z n FOR A BPRE 23
Large deviation rate for R n . This section is devoted to the proof of Theo-rems 2.9 and 2.10. Recall that M n,j is defined by (2.42), where N n,i are i.i.d. withgenerating f n , given the environment ξ (see Section 2). Proof of Theorem 2.9.
Since for all n ∈ N , M n,j − m n has the same law as M ,j − m ,and is independent of Z n , we obtain P k ( R n − m n ∈ D ) = X j > k P ( M n,j − m n ∈ D ) P k ( Z n = j ) X j > k C j r P k ( Z n = j )= C E k h Z − rn i . The result (2.43) follows from Theorem 2.1, and (2.45) follows similarly. (cid:3)
Proof of Theorem 2.10.
We stat with a lemma which is a direct consequence of theMarcinkiewicz-Zygmund inequality (see [12, p. 356]).
Lemma 4.1 ([21], Lemma 1.4) . Let ( X i ) i > be a sequence of i.i.d. centered r.v.’s.Then we have for p ∈ (1 , ∞ ) , (4.2) E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 X i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ( ( B p ) p E ( | X i | p ) n, if < p , ( B p ) p E ( | X i | p ) n p/ , if p > , where B p = 2 min n k / : k ∈ N , k > p/ o is a constant depending only on p (so that B p = 2 if < p ). We shall prove Theorem 2.10 in the case when p ∈ (1 , M n,j − m n has the same law as M ,j − m and is independent of Z n , we obtain afterconditioning(4.3) P k ( | R n − m n | > a ) = ∞ X j = k P ( | M ,j − m | > a ) P k ( Z n = j ) . Using (2.42) and the fact that, under P ξ , the r.v.’s N ,i − m ( i = 1 , . . . , j ) are i.i.d.centered and with generating function f , we get from Lemma 4.1 that, for p ∈ (1 , P ξ ( | M ,j − m | > a ) a − p E ξ | M ,j − m | p (cid:18) B p a (cid:19) p j − p E ξ | Z − m | p . Taking expectation, we obtain P ( | M n,j − m n | > a ) (cid:18) B p a (cid:19) p j − p E | Z − m | p . Therefore, coming back to (4.3) and applying Theorem 2.1, we get P k ( | R n − m n | > a ) (cid:18) B p a (cid:19) p E | Z − m | p ∞ X j = k j − p P k ( Z n = j )= (cid:18) B p a (cid:19) p E | Z − m | p E k h Z − pn i = C p a − p A k,n ( p − , with C p = B p E | Z − m | p . This ends the proof of Theorem 2.10 in the case when p ∈ (1 , p > (cid:3) References [1] V. I. Afanasyev, C. Böinghoff, G. Kersting, and V. A. Vatutin. Limit theorems for weaklysubcritical branching processes in random environment.
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