aa r X i v : . [ m a t h . P R ] A ug How many families survive for a long time ∗ Vatutin V.A. † , Dyakonova E.E. ‡ Abstract
A critical branching process { Z k , k = 0 , , , ... } in a random environ-ment generated by a sequence of independent and identically distributedrandom reproduction laws is considered. Let Z p,n be the number of par-ticles at time p ≤ n having a positive offspring number at time n . Atheorem is proved describing the limiting behavior, as n → ∞ of thedistribution of a properly scaled process log Z p,n under the assumptions Z n > p ≪ n . AMS Subject Classification:
Key words branching processes, random environment, reduced processes,limit theorems
We consider branching processes in random environment specified by sequencesof independent identically distributed random laws. Denote by ∆ the space ofproper probability measures on N = { , , , ... } = 0 ∪ N + . Let Q be a randomvariable taking values in ∆. An infinite sequenceΠ = ( Q , Q , . . . ) (1)of i.i.d. copies of Q is said to form a random environment . In the sequel wemake no difference between the laws Q = ( Q ( { } ) , Q ( { } ) , ..., Q ( { k } ) , ... ) , Q n = ( Q n ( { } ) , Q n ( { } ) , ..., Q n ( { k } ) , ... )and the generating functions f ( s ) = f ( s ; Q ) := ∞ X k =0 Q ( { k } ) s k , f n ( s ) = f n ( s ; Q ) := ∞ X k =0 Q n ( { k } ) s k , n ∈ N + . (2) ∗ This work is supported by the RSF under a grant 14-50-00005. † Department of Discrete Mathematics, Steklov Mathematical Institute, 8, Gubkin str.,119991, Moscow, Russia; e-mail: [email protected] ‡ Department of Discrete Mathematics, Steklov Mathematical Institute, 8, Gubkin str.,119991, Moscow, Russia; e-mail: [email protected] N -valued random variables Z : = ( Z , Z , . . . ) is called a branch-ing process in the random environment Π, if Z is independent of Π and E (cid:2) s Z n n | Z , Z , ..., Z n − , Q , Q , . . . , Q n (cid:3) = ( f n ( s )) Z n − , n ∈ N + . Thus, Z n is the n th generation size of the population and Q n is the dis-tribution of the number of children of an individual at generation n −
1. Weassume that Z = 1 for convenience and denote the corresponding probabilitymeasure on the underlying probability space by P . (If we refer to other prob-ability spaces, then we use notation P , E and L for the respective probabilitymeasures, expectations and laws.)As it turns out the properties of Z are mainly determined by its associatedrandom walk S := { S n , n ≥ } . This random walk has initial state S = 0 andincrements X n = S n − S n − , n ≥ , defined as X n := log f ′ n (1) , which are i.i.d. copies of the logarithmic mean offspring number X := log f ′ (1) . Following [1] we call the process Z critical if and only if the random walk S is oscillating, that is,lim sup n →∞ S n = ∞ and lim inf n →∞ S n = −∞ with probability 1.It is shown in [1] that the extinction moment of the critical branching processin a random environment is finite with probability 1. Moreover, it is establishedin [1] that if lim n →∞ P ( S n >
0) =: ρ ∈ (0 , , (3)then, as n → ∞ (and some mild additional assumptions to be specified lateron) P ( Z n > ∼ θ P (min ( S , S , ..., S n ) ≥
0) = θn ρ − l ( n ) , (4)where l ( n ) is a slowly varying function and θ is a known positive constant whoseexplicit expression is given by formula (23) below.Let A := { < α < | β | < }∪{ < α < | β | < }∪{ α = 1 , β = 0 }∪{ α = 2 , β = 0 } be a subset in R . For ( α, β ) ∈ A and a random variable X write X ∈ D ( α, β )if the distribution of X belongs to the domain of attraction of a stable law withcharacteristic function H α,β ( t ) := exp (cid:26) − c | t | α (cid:18) iβ t | t | tan πα (cid:19)(cid:27) , c > , (5)2nd, in addition, E X = 0 if this moment exists.Denote by { c n , n ∈ N + } a sequence of positive integers specified by the re-lation c n := inf (cid:8) u ≥ G ( u ) ≤ n − (cid:9) , where G ( u ) := 1 u Z u − u x P ( X ∈ dx ) . It is known (see, for instance, [8, Ch. XVII, § X ∈ D ( α, β )the function G ( u ) is regularly varying with index ( − α ), and, therefore, c n = n /α l ( n ) , n ∈ N + , (6)for some function l ( n ) , slowly varying at infinity. In addition, the scaled se-quence { S n /c n , n ≥ } converges in distribution, as n → ∞ , to the stable lawgiven by (5).Observe that if X ∈ D ( α, β ) , then the quantity ρ in (3) is calculated by theformula (see, for instance, [22]) ρ = , if α = 1 or 2 , + πα arctan (cid:0) β tan πα (cid:1) , otherwise . (7)In particular, ρ ∈ (0 , M n := max ( S , ..., S n ) , L k,n := min k ≤ j ≤ n S j , L n := L ,n = min ( S , S , ..., S n )and introduce two functions V ( x ) : = 1 + ∞ X k =1 P ( − S k ≤ x, M k < , x ≥ ,U ( x ) : = 1 + ∞ X k =1 P ( − S k > x, L k ≥ , x ≤ , and 0 elsewhere. In particular, V (0) = U (0) = 1.The fundamental properties of V, U are the identities E [ V ( x + X ); x + X ≥
0] = V ( x ) , x ≥ , E [ U ( x + X ); x + X <
0] = U ( x ) , x ≤ , (8)which hold for any oscillating random walk.It follows from (8) that V and U give rise to probability measures P + x and P − x being important for subsequent arguments. The construction procedures ofthese probability measures are explained for P + x in [1] and for P − x in [2] and[15] in detail. We only recall that if T , T , . . . is a sequence of random variablesand the random walk S = ( S n , n ≥
0) with S = x are both adapted to some3ltration F = ( F n , n ≥ n and a bounded and measurablefunction g n : R n +1 → R the measures above are specified by the equalities E + x [ g n ( T , . . . , T n )] : = 1 V ( x ) E x [ g n ( T , . . . , T n ) V ( S n ); L n ≥ , E − x [ g n ( T , . . . , T n )] : = 1 U ( x ) E x [ g n ( T , . . . , T n ) U ( S n ); M n < . Observe that under the measure P + x the sequence S , S , . . . is a Markov processwith state space R := [0 , ∞ ) and transition probabilities P + ( x, dy ) := 1 V ( x ) P { x + X ∈ dy } V ( y ) , x ≥ , y ≥ . It is the random walk conditioned never to enter R − := ( −∞ , , while underthe measure P − x the process S , S , . . . becomes a Markov chain with state space R − and transition kernel P − ( x, dy ) := 1 U ( x ) P { x + X ∈ dy } U ( y ) , x ≤ , y < . (9)Note that P − ( x, [0 , ∞ )) = 0, thus the Markov process never enters [0 , ∞ ) again.It may, however, start from the boundary x = 0.We now describe in brief a construction of L´evy processes conditioned tostay nonnegative following basically the definitions given in [5] and [6].Let Ω := D ([0 , ∞ ) , R ) be the space of real-valued c`adl`ag paths on the realhalf-line [0 , ∞ ) and let B := { B t , t ≥ } be the coordinate process defined bythe equality B t ( ω ) = ω t for ω ∈ Ω. In the sequel we consider also the spacesΩ U := D ([0 , U ] , R ) , U > . We endow the spaces Ω and Ω U with Skorokhod topology and denote by F = {F t , t ≥ } and by F U = {F t , t ∈ [0 , U ] } (with some misuse of notation)the natural filtrations of the processes B and B U = { B t , t ∈ [0 , U ] } .Let P x be the law on Ω of an α − stable L´evy process B , α ∈ (0 ,
2] startedat x and let P = P . Denote by ρ := P ( B ≥
0) the positivity parameter of theprocess B (in fact, this quantity is the same as in (7) for a random walk whoseincrements X i ∈ D ( α, β )). We now introduce an analogue of the measure P + for L´evy processes. Namely, following [4] we specify for all t > A ∈ F t the law P + x on Ω of the L´evy process starting at point x > P + x ( A ) := 1 x α (1 − ρ ) E x (cid:20) B α (1 − ρ ) t I {A} I (cid:26) inf ≤ u ≤ t B u ≥ (cid:27)(cid:21) , where I {C} is the indicator of the event C .Thus, P + x is an h − transform of the L´evy process killed when it first entersthe negative half-line. The corresponding positive invariant function is H ( x ) = x α (1 − ρ ) in this case. 4his definition has no sense for x = 0. However, it is shown in [5] thatone can construct a law P + = P +0 and a c`adl`ag Markov process with the samesemigroup as ( B , { P + x , x > } ) in such a way that P + ( B = 0) = 1. Moreover, P + x = ⇒ P + , as x ↓ ⇒ stands for the weak convergence inthe respective space of c`adl`ag functions endowed with the Skorokhod topology.Let P ( m ) be the law on Ω of the meander of length 1 associated with ( B , P ) , i.e. P ( m ) ( · ) := lim x ↓ P x (cid:18) · (cid:12)(cid:12)(cid:12) inf ≤ u ≤ B u ≥ (cid:19) . (10)Thus, the law P ( m ) may be viewed as the law of the L´evy process ( B , P ) condi-tioned to stay nonnegative on the time-interval (0 ,
1] while the law P + specifiedearlier corresponds to the law of the L´evy process conditioned to stay nonneg-ative on the whole time interval (0 , ∞ ).As shown in [5], P ( m ) and P + are absolutely continuous with respect to eachother: for every event A ∈ F P + ( A ) = C E ( m ) h I {A} B α (1 − ρ )1 i , (11)where (see, for instance, formulas (3.5)-(3.6) and (3.11) in [6]) C := lim n →∞ V ( c n ) P ( L n ≥
0) = (cid:16) E ( m ) h B α (1 − ρ )1 i(cid:17) − ∈ (0 , ∞ ) , (12)and E ( m ) is the expectation with respect to P ( m ) . In fact, one may extend theabsolute continuity given in (11) to an arbitrary interval [0 , U ] be considering therespective space Ω U instead of Ω and conditioning by the event inf ≤ u ≤ U B u ≥ ζ ( a ) := P ∞ k = a k Q ( { k } )( f ′ (1)) , a ∈ N . In what follows we say that1)
Condition A is valid if X ∈ D ( α, β ) ;2) Condition A is valid if E (cid:0) log + ζ ( a ) (cid:1) α + ε < ∞ for some ε > a ∈ N ;3) Condition A is valid if Conditions A A p = p ( n ) tends to infinity as n → ∞ in such a way thatlim n →∞ n − p = lim n →∞ n − p ( n ) = 0 . (13)5ranching processes in random environment meeting Condition A have beeninvestigated in a recent paper [19]. The paper includes a Yaglom-type functionallimit theorem describing the asymptotic properties of the process (cid:8) Z [ tp ] , ≤ t < ∞ (cid:9) given Z n > Z in more detail and provea conditional limit theorem for the so-called reduced process { Z p,n , ≤ p ≤ n } , where Z p,n is the number of particles in the process at time p ∈ [0 , n ], each ofwhich has a nonempty offspring at time n . Our main result looks as follows. Theorem 1
If Condition A is valid, then for any x ≥ n →∞ P (cid:18) log Z p,n c p ≥ x | Z n > , Z = 1 (cid:19) = E + "(cid:18) − xB (cid:19) α (1 − ρ ) I { B ≥ x } = P + (cid:18) inf ≤ v< ∞ B v ≥ x (cid:19) . (14)Here and in what follows E + is the expectation with respect to P + . Thisresult complements papers [3] and [13] where it was established (under theassumptions E [ X ] = 0 , σ = E (cid:2) X (cid:3) ∈ (0 , ∞ ) and some additional technicalconditions) that, as n → ∞L (cid:18)(cid:26) log Z [ nt ] ,n σ √ n , ≤ t ≤ (cid:27) (cid:12)(cid:12)(cid:12) Z n > , Z = 1 (cid:19) = ⇒ L ( m ) (cid:18) inf t ≤ v ≤ B v , ≤ t ≤ (cid:19) = L (cid:18) inf t ≤ v ≤ B + v (cid:19) , where B + := { B + v , ≤≤ v ≤ } is the standard Brownian meander.The study of the reduced processes has a rather long history. Reducedprocesses for ordinary Galton-Watson branching processes where introducedby Fleischmann and Prehn [9], who discussed the subcritical case. Criticaland supercritical reduced Galton-Watson processes have been investigated byZubkov [23] and Fleischmann and Siegmund-Schultze [10]. The first results forreduced branching processes in random environment were obtained by Borovkovand Vatutin [3] and Fleischmann and Vatutin [11]. They considered (under theannealed approach) the case when the support of the measure P is concentratedonly on the set of fractional-linear generating functions. Vatutin proved in [13] alimit theorem for the critical reduced processes under the annealed approach andgeneral reproduction laws of particles. Papers [14], [16],[17] and [18] consider theproperties of the critical reduced branching processes in random environmentunder the quenched approach (see also survey [20]). To prove the main results of the paper we need to know the asymptotic be-havior of the function V ( x ) as x → ∞ . The following lemma gives the desiredrepresentation. 6 emma 2 (compare with Lemma 13 in [21] and Corollary 8 in [7]) If X ∈D ( α, β ) , then there exists a function l ( x ) slowly varying at infinity such that V ( x ) ∼ x α (1 − ρ ) l ( x ) as x → ∞ . In the sequel we use the symbols
K, K , K , ... to denote different constants.They are not necessarily the same in different formulas.Our next result is a combination (with a slight reformulation) of Lemma 2.1in [1] and Corollaries 3 and 8 in [7]: Lemma 3 If X ∈ D ( α, β ) , then (compare with (4)), as n → ∞ P ( L n ≥ − r ) ∼ V ( r ) P ( L n ≥ ∼ V ( r ) n ρ − l ( n ) (15) uniformly in ≤ r ≪ c n , and there exists a constants K > such that P ( L n ≥ − r ) ≤ K V ( r ) P ( L n ≥ , r ≥ , n ≥ . (16)For U > pU ≤ n let Q p,nU : = (cid:26) S [ pu ] c p , ≤ u ≤ U (cid:27) , Q p,n := Q p,n ∞ , S p,nU : = (cid:26) S [ pU ]+[( n − pU ) t ] c n , ≤ t ≤ (cid:27) , S n := S p,n . Let φ : Ω → R be a bounded uniformly continuous functional and { ε n , n ∈ N + } be a sequence of positive numbers vanishing as n → ∞ . Lemma 4 (see [19]) If Condition A is valid then E [ φ ( S n ) | L n ≥ − x ] → E ( m ) (cid:2) φ ( B ) (cid:3) as n → ∞ uniformly in ≤ x ≤ ε n c n . Lemma 5 (see [19]) If Conditions A and (13) are valid, then for any r ≥ L (cid:16) Q p,n (cid:12)(cid:12)(cid:12) L n ≥ − r (cid:17) = ⇒ L + ( B ) as n → ∞ . For x ≥ D ( x ) : = C E ( m ) h ( B − x ) α (1 − ρ ) I { B ≥ x } i = E + "(cid:18) − xB (cid:19) α (1 − ρ ) I { B ≥ x } . (17)7 emma 6 For any x > P + (cid:18) inf ≤ u< ∞ B u ≥ x (cid:19) = D ( x ) . Proof . According to Theorem 5 in [4] for any pair 0 < x ≤ y P + y (cid:18) inf ≤ u< ∞ B u ≥ x (cid:19) = (cid:18) − xy (cid:19) α (1 − ρ ) I { x ≤ y } . Hence, P + (cid:18) inf ≤ u< ∞ B u ≥ x (cid:19) = Z ∞ x P + ( B ∈ dy ) P + y (cid:18) inf ≤ u< ∞ B u ≥ x (cid:19) = Z ∞ x P + ( B ∈ dy ) (cid:18) − xy (cid:19) α (1 − ρ ) = D ( x ) , as desired.In the sequel we agree to write a n ≪ b n if lim n →∞ a n /b n = 0 . In prticular,lim n ≫ p →∞ means that the limit of the respective expression is claculated as p, n → ∞ in such a way that pn − → Lemma 7
If Condition A is valid then, for any x ≥ n ≫ p →∞ P ( L p,n ≥ xc p | L n ≥ − r ) = D ( x ) . (18) Proof.
We select
N > x and write0 ≤ P ( L p,n ≥ xc p ; L n ≥ − r ) − P ( S p ≤ N c p , L p,n ≥ xc p ; L n ≥ − r ) ≤ P ( S p > N c p ; L n ≥ − r ) . (19)It follows from Lemma 5 that for any ε > N = N ( ε ) such thatfor all N ≥ N P ( S p > N c p ; L n ≥ − r ) ≤ ε P ( L n ≥ − r ) . (20)To proceed further we denote by S ∗ := ( S ∗ = 0 , S ∗ , ..., S ∗ n , ... ) an indepen-dent probabilistic copy of the random walk S and let L ∗ k := min ( S ∗ , S ∗ , ..., S ∗ k ) .Then, for N > x ≥ P ( S p ≤ N c p , L p,n ≥ xc p ; L n ≥ − r )= Z Nc p xc p P ( S p ∈ dy ; L p ≥ − r ) P (cid:0) L ∗ n − p ≥ xc p − y (cid:1) = Z Nx P ( S p ∈ c p dz ; L p ≥ − r ) P (cid:0) L ∗ n − p ≥ ( x − z ) c p (cid:1) . p ≪ n, we deduce by (15), (12) and properties of regularly varying func-tions that if n → ∞ then for any ε > P (cid:0) L ∗ n − p ≥ ( x − z ) c p (cid:1) ∼ V (( z − x ) c p ) P ( L n ≥ ∼ ( z − x ) α (1 − ρ ) V ( c p ) P ( L n ≥ z − x ) α (1 − ρ ) V ( c p ) P ( L p ≥ P ( L n ≥ P ( L p ≥ ∼ ( z − x ) α (1 − ρ ) C P ( L n ≥ − r ) P ( L p ≥ − r )uniformly in 0 ≤ x ≤ ze ε ≤ N . Hence we conclude that given condition (13) wehave as n → ∞ P ( S p ≤ N c p ; L p,n ≥ xc p ; L n ≥ − r ) ∼ C P ( L n ≥ − r ) Z Nx ( z − x ) α (1 − ρ ) P ( S p ∈ c p dz | L p ≥ − r ) ∼ C P ( L n ≥ − r ) Z Nx ( z − x ) α (1 − ρ ) P ( m ) ( B ∈ dz )= P ( L n ≥ − r ) Z Nx (cid:16) − xz (cid:17) α (1 − ρ ) P + ( B ∈ dz ) . (21)Using (20) and (21) to evaluate (19) and letting N to infinity we get (18).For convenience we introduce the notation τ n := min { j : S j = L n } , A u.s. := { Z n > n ≥ } and recall that by Corollary 1.2 in [1], (4) and (12) P ( Z n > ∼ θ P ( L n ≥ ∼ θl ( n ) n ρ − ∼ θC V ( c n ) (22)as n → ∞ , where θ := ∞ X k =0 E [ P + Z k ( A u.s. ) ; τ k = k ] . (23)Let ˆ L k,n := min ≤ j ≤ n − k ( S k + j − S k )and let ˜ F k be the σ − algebra generated by the tuple { Z , Z , ..., Z k ; Q , Q , ..., Q k } (see (1)).For further references we formulate two statements borrowed from [1]. Lemma 8 (see Lemma 2.5 in [1]) Assume Condition A . Let Y , Y , ... be a uni-formly bounded sequence of random variables adapted to the filtration ˜ F = n ˜ F k , k ∈ N o ,which converges P + -a.s. to some random variable Y ∞ . Then, as n → ∞ E [ Y n | L n ≥ → E + [ Y ∞ ] . emma 9 (see Lemma 4.1 in [1]) Assume Condition A1 and let l ∈ N . Sup-pose that ζ , ζ , ... is a uniformly bounded sequence of real-valued random vari-ables, which, for every k ≥ meets the equality E h ζ n ; Z k + l > , ˆ L k,n ≥ | ˜ F k i = P ( L n ≥
0) ( ζ k, ∞ + o (1)) P -a.s.as n → ∞ with random variables ζ , ∞ = ζ , ∞ ( l ) , ζ , ∞ = ζ , ∞ ( l ) , .... Then E [ ζ n ; Z τ n + l >
0] = P ( L n ≥ ∞ X k =0 E [ ζ k, ∞ ; τ k = k ] + o (1) ! as n → ∞ , where the right-hand side series is absolutely convergent. For q ≤ p ≤ n and u > m ( u ; n ) = m ( u ; n, p, q ) := min { q + [ u ( p − q )] , n } and set X q,p := (cid:8) X q,pu = e − S m ( u ; n ) Z m ( u ; n ) , ≤ u < ∞ (cid:9) . The next statement is an evident corollary of Theorem 1.3 in [1].
Corollary 10
Assume Condition A . Let ( q , p ) , ( q , p ) , ... be a sequence ofpairs of positive integers such that q n ≪ p n ≪ n and q n → ∞ as n → ∞ . Then L ( X q n ,p n | Z n > , Z = 1) = ⇒ L ( W u , ≤ u < ∞ ) , where P ( W u = W, ≤ u < ∞ ) = 1 for some random variable W such that P (0 < W < ∞ ) = 1 . We conclude this section by recalling asymptotic properties of the distri-bution of the number of particles in a critical branching process in randomenvironment at moment p ≪ n given Z n > . Theorem 11 (see [19]) If Condition A is valid, then, as n → ∞ lim n ≫ p →∞ P (cid:18) log Z p c p ≤ z (cid:12)(cid:12)(cid:12) Z n > , Z = 1 (cid:19) = P + ( B ≤ z ) for any z > . Reduced processes
The proof of Theorem 1 will be divided into several steps which we formulateas lemmas.For f n ( s ) , n = 1 , , ..., specified by (2) set f p,n ( s ) := f p +1 ( f p +2 ( . . . ( f n ( s )) . . . )) , ≤ p ≤ n − , f n,n ( s ) ≡ . We label Z p particles of the p th generation by positive numbers 1 , , ..., Z p in an arbitrary but fixed way and denote by Z ( i ) n ( p ) , i = 1 , , . . . , Z p , ≤ p ≤ n, the offspring size at moment n of the population generated by the i th particleof the p th generation.For fixed positive x and N introduce the events A p,n ( x ) := { ln( e + Z p,n ) ≥ xc p } ,B p,n ( N ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z p X i =1 (cid:16) I n Z ( i ) n ( p ) > o − (1 − f p,n (0)) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > q N Z p (1 − f p,n (0)) , and C p,n := (cid:8) Z p (1 − f p,n (0) < e √ c p (cid:9) and use the notation ¯ C p,n for the event complementary to C p,n .Finally, we use for brevity the notation P j ( • ) := P ( •| Z = j ) with thenatural agreement that P ( • ) := P ( •| Z = 1). In particular, P ( Z n >
0) = P ( Z n > | Z = 1). Lemma 12
If the conditions of Theorem 1 are valid then, for any j ∈ N + lim N →∞ lim n ≫ p →∞ P j ( A p,n ( x ) B p,n ( N ); L n ≥ P ( Z n >
0) = 0 (24) and lim n ≫ p →∞ P j ( A p,n ( x ) C p,n ; L n ≥ P ( Z n >
0) = 0 . (25) Proof.
First we establish the validity of (24). To this aim we temporaryintroduce the notation P ( F ) ( • ), E ( F ) [ • ] and D ( F ) [ • ] for the probability, expec-tation and variance calculated for the fixed σ -algebra F = F p,n generated by therandom variables { Z , Z , ..., Z p } and random probability generating functions f ( s ) , f ( s ) , . . . , f n ( s ).First we note that, for 1 ≤ i ≤ Z p E ( F ) h I n Z ( i ) n ( p ) > oi = 1 − f p,n (0) , D ( F ) h I n Z ( i ) n ( p ) > oi = (1 − f p,n (0)) f p,n (0) . (26)11esides, Z p,n = Z p X i =1 I n Z ( i ) n ( p ) > o . Using these relations and applying Chebyshev’s inequality to evaluate the prob-ability under the expectation sign we obtain for sufficiently large n P j ( A p,n ( x ) B p,n ( N ); L n ≥ ≤ P j ( Z p > , B p,n ( N ); L n ≥ E h P ( F ) j ( B p,n ); Z p > , L n ≥ i ≤ E j " I { Z p > , L n ≥ } N Z p (1 − f p,n (0) D ( F ) " Zp X i =1 I n Z ( i ) n ( p ) > o ≤ N − P ( L n ≥ . This estimate along with (22) implies (24).To establish (25) observe that P j ( A p,n ( x ) C p,n ; L n ≥
0) = E h P ( F ) j ( A p,n ( x ) C p,n ); L n ≥ i ≤ x c p E j h I { C p,n ; L n ≥ } E ( F ) [ ln ( e + Z p,n ) ] i . (27)Since (ln ( e + x )) ′′ = 2( e + x ) (1 − ln( e + x )) , the function ln ( e + x ) is concave on the set x > . This fact allows us to applyJensen’s inequality to the internal expectation in the right-hand side of (27) andto obtain the estimate E ( F ) (cid:2) ln ( e + Z p,n ) (cid:3) ≤ ln ( e + E ( F ) [ Z p,n ]) = ln ( e + Z p (1 − f p,n (0))) . By the first equality in (26) we find that, for all sufficiently large n, P j ( A p,n ( x ) C p,n ; L n ≥ ≤ x c p E j (cid:2) I { C p,n ; L n ≥ } ln ( e + Z p (1 − f p,n (0))) (cid:3) ≤ x c p ln (cid:0) e + e √ c p (cid:1) P ( L n ≥ ≤ K x c p P ( L n ≥ . These estimates imply (25).The lemma is proved.
Lemma 13
If Condition A is valid, then, for any j = 1 , , ... and x ≥ n ≫ p →∞ P j ( L p,n ≥ xc p | L n ≥ , Z n >
0) = D ( x ) . roof . Clearly, P j ( L p,n ≥ xc p ; L n ≥ , Z n >
0) = E h I ( p, n ; x ) (cid:16) − f j ,n (0) (cid:17)i , (28)where I ( p, n ; x ) := I { L p,n ≥ xc p ; L n ≥ } . We now select γ > l < p and write the right-hand side of (28) as follows: E h I ( p, n ; x ) (cid:16) − f j ,n (0) (cid:17)i = G ( m, p, n ; x, γ ) + G ( p, n ; x, γ )+ G ( m, p, n ; x, γ ) , where G ( l, p, n ; x, γ ) := E h I ( p, n ; x ) ( f j ,n (0) − f j ,l (0)) I { L nγ ≥ } i ,G ( p, n ; x, γ ) := E h I ( p, n ; x ) (1 − f j ,n (0)) ( I { L n ≥ } − I { L nγ ≥ } ) i ,G ( l, p, n ; x, γ ) := E h I ( p, n ; x ) (1 − f j ,l (0)) I { L nγ ≥ } i . By (16) we have G ( l, p, n ; x, γ ) ≤ E h ( f j ,n (0) − f j ,l (0)) I { L nγ ≥ } i = E h ( f j ,n (0) − f j ,l (0)) I { L n ≥ } P (cid:16) L ∗ n ( γ − ≥ − S n | S n (cid:17)i ≤ K P (cid:0) L n ( γ − ≥ (cid:1) E h ( f j ,n (0) − f j ,l (0)) I { L n ≥ } V ( S n ) i = K P (cid:0) L n ( γ − ≥ (cid:1) E + h f j ,n (0) − f j ,l (0) i ≤ K ( γ − ρ − P ( L n ≥ E + h f j ,n (0) − f j ,l (0) i , (29)where we have used (15) to justify the last inequality. Since f ,t (0) → f , ∞ (0) ∈ (0 , P + - a.s. as t → ∞ , letting in (29) first n → ∞ and then l → ∞ , we getin account of (22) lim n →∞ G ( l, p, n ; x, γ ) P ( L n ≥
0) = 0 . Further, G ( p, n ; x, γ ) ≤ P ( L n ≥ − P ( L nγ ≥ ≤ K (cid:16) − γ − (1 − ρ ) (cid:17) P ( L n ≥ γ ↓ lim n →∞ G ( p, n ; x, γ ) P ( L n ≥
0) = 0 . By (16) we conclude that, given p ≪ nG ( l, p, n ; x, γ ) = E h (1 − f j ,l (0)) I { L p ≥ } P (cid:0) L ∗ γn − p ≥ xc p − S p | S p (cid:1)i ∼ P ( L nγ ≥ E h (1 − f j ,l (0)) I { L p ≥ } V ( S p − xc p ) i n → ∞ . Using Lemma 2 and properties of regularly varying functions weget, as p → ∞ E h (1 − f j ,l (0)) I { L p ≥ } V ( S p − xc p ) i = E (cid:20) (1 − f j ,l (0)) I { L p ≥ } V ( S p − xc p ) V ( c p ) × V ( c p ) I (cid:26) S p c p ≥ x (cid:27)(cid:21) ∼ E " (1 − f j ,l (0)) I { L p ≥ } (cid:18) S p c p − x (cid:19) α (1 − ρ ) V ( c p ) I (cid:26) S p c p ≥ x (cid:27) . Further, E " (1 − f j ,l (0)) I { L p ≥ } (cid:18) S p c p − x (cid:19) α (1 − ρ ) V ( c p ) I (cid:26) S p c p ≥ x (cid:27) = E h (1 − f j ,l (0)) I { L l ≥ } V ( c p ) P (cid:0) L ∗ p − l ≥ − S l | S l (cid:1) ×× E "(cid:18) S ∗ p − l + S l c p − x (cid:19) α (1 − ρ ) I (cid:26) S ∗ p − l + S l c p ≥ x (cid:27) | L ∗ p − l ≥ − S l . It is not difficult to conclude by Lemma 4 and (17) that, for any fixed l lim p →∞ E "(cid:18) S ∗ p − l + S l c p − x (cid:19) α (1 − ρ ) I (cid:26) S ∗ p − l + S l c p ≥ x (cid:27) | L ∗ p − l ≥ − S l = C − D ( x ) . Besides, for 0 ≤ x ≪ c p V ( c p ) P (cid:0) L ∗ p − l ≥ − x (cid:1) ∼ V ( c p ) P (cid:0) L ∗ p − l ≥ (cid:1) V ( x ) ∼ C V ( x )as p → ∞ , leading after evident transformations to E h (1 − f j ,l (0)) I { L p ≥ } V ( S p − xc p ) i ∼ D ( x ) E h (1 − f j ,l (0)) I { L l ≥ } V ( S l ) i = E + h − f j ,l (0) i D ( x ) . Hence we obtainlim γ ↓ lim l →∞ lim n ≫ p →∞ G ( l, p, n ; x, γ ) P ( L n ≥
0) = E + h − f j , ∞ (0) i D ( x ) . To complete the proof of the lemma it remains to note that P ( L n ≥ , Z n > , Z = j ) = E h(cid:16) − f j ,n (0) (cid:17) | L n ≥ i P ( L n ≥
0) (30)and that lim n →∞ E h(cid:16) − f j ,n (0) (cid:17) | L n ≥ i = E + h − f j , ∞ (0) i according to Lemma 8. 14he lemma is proved.Set η l := f ′′ l (1)( f ′ l (1)) , l ∈ N + , and let J + ( p, r ) : = r − X l = p η l e S p − S l + e S p − S r , J − ( p, r ) := r − X l = p η l e S r − S l , ˆ J − (0 , r ) : = r − X l =0 η l +1 e S l +1 . It is know (see, for instance, Lemma 2.7 in [1]) that if Conditions A A J + (0 , ∞ ) < ∞ P + - a.s.and, for any y > n →∞ P (cid:0) J + (0 , n ) < y | L n ≥ (cid:1) = P + (cid:0) J + (0 , ∞ ) < y (cid:1) . (31)In addition (compare with Lemma 2.7 in [1] or see Lemma 6 in [16]), if Condi-tions A A P − defined in (9)ˆ J − (0 , ∞ ) = ∞ X l =0 η l +1 e S l +1 < ∞ P − - a.s.,and for any y ≥ n →∞ P (cid:16) ˆ J − (0 , n ) > y | M n ≤ (cid:17) = P − (cid:16) ˆ J − (0 , ∞ ) > y (cid:17) . (32)Set τ p,n := min n p ≤ j ≤ n : S j − S p = ˆ L p,n o . Lemma 14
If Condition A is valid, then, for any j = 1 , , ... lim y →∞ lim n ≫ p →∞ P j (cid:0) J + ( τ p,n , n ) > y | L n ≥ , Z n > (cid:1) = 0 , (33)lim y →∞ lim n ≫ p →∞ P j (cid:0) J − ( p, τ p,n ) > y | L n ≥ , Z n > (cid:1) = 0 . (34) Proof . We write P j (cid:0) J + ( τ p,n , n ) > y ; L n ≥ , Z n > (cid:1) = E h I (cid:8) J + ( τ p,n , n ) > y (cid:9) (cid:16) − f j ,n (0) (cid:17) ; L n ≥ i ≤ E h I (cid:8) J + ( τ p,n , n ) > y (cid:9) (cid:16) − f j ,p (0) (cid:17) ; L n ≥ i = Z ∞ E h(cid:16) − f j ,p (0) (cid:17) ; S p ∈ c p dz, L p ≥ i P (cid:0) J + ( τ n − p , n − p ) > y ; L n − p ≥ − c p z (cid:1) . P (cid:0) J + ( τ n − p , n − p ) > y ; L n − p ≥ − c p z (cid:1) = n − p X k =0 P (cid:0) J + ( k, n − p ) > y ; τ n − p = k, L n − p ≥ − c p z (cid:1) = n − p X k =0 P ( M k < S k ≥ − c p z ) P (cid:0) J + (0 , n − p − k ) > y ; L n − p − k ≥ (cid:1) . In view of (31) for any ε > y and N = N ( y , ε ) such that, for all y ≥ y and n − p − k ≥ N P (cid:0) J + (0 , n − p − k ) > y, L n − p − k ≥ (cid:1) ≤ P (cid:0) J + (0 , n − p − k ) > y , L n − p − k ≥ (cid:1) ≤ P + (cid:0) J + (0 , ∞ ) > y (cid:1) P ( L n − p − k ≥ ≤ ε P ( L n − p − k ≥ . On the other hand, for each fixed N one can find a sufficiently large y ≥ y such that P (cid:0) J + (0 , j ) > y, L j ≥ (cid:1) ≤ ε P ( L j ≥ y ≥ y . These estimates and (30) imply P (cid:0) J + ( τ n − p , n − p ) > y ; L n − p ≥ − c p z (cid:1) ≤ ε n − p X k =0 P ( M k < S k ≥ − c p z ) P ( L n − p − k ≥
0) = ε P ( L n − p ≥ − c p z ) . Thus, Z ∞ E h(cid:16) − f j ,p (0) (cid:17) ; S p ∈ c p dz, L p ≥ i P (cid:0) J + ( τ n − p , n − p ) > y ; L n − p ≥ − c p z (cid:1) ≤ ε Z ∞ E h(cid:16) − f j ,p (0) (cid:17) ; S p ∈ c p dz, L p ≥ i P ( L n − p ≥ − c p z )= ε E h(cid:16) − f j ,p (0) (cid:17) ; L n ≥ i ≤ ε P ( L n ≥ ≤ εK P j ( L n ≥ , Z n > . This proves (33), since ε > P j (cid:0) J − ( p, τ p,n ) > y ; L n ≥ , Z n > (cid:1) ≤ E h(cid:16) − f j ,p (0) (cid:17) ; J − ( p, τ p,n ) > y ; L n ≥ i = Z N E h(cid:16) − f j ,p (0) (cid:17) ; S p ∈ c p dz, L p ≥ i P (cid:0) J − (0 , τ n − p ) > y ; L n − p ≥ − c p z (cid:1) + E h(cid:16) − f j ,p (0) (cid:17) ; S p > N c p , J − ( p, τ p,n ) > y ; L n ≥ i .
16y Lemma 5 with r = 0 for any ε > N such that the inequality E h(cid:16) − f j ,p (0) (cid:17) ; S p > N c p , J − ( p, τ p,n ) > y ; L n ≥ i ≤ P ( S p > N c p , L n ≥ ≤ ε P ( L n ≥
0) (35)is valid for all N ≥ N . Further, P (cid:0) J − (0 , τ ,n − p ) > y ; L n − p ≥ − c p z (cid:1) = n − p − X k =0 P (cid:0) J − (0 , k ) > y ; L n − p ≥ − c p z ; τ n − p = k (cid:1) = n − p − X k =0 P (cid:0) J − (0 , k ) > y ; L k ≥ − c p z ; τ k = k (cid:1) P ( L n − p − k ≥ n − p − X k =0 P (cid:16) ˆ J − (0 , k ) > y ; M k ≤ , S k ≥ − c p z (cid:17) P ( L n − p − k ≥ , where at the last step we have used the duality principle for random walks. Inview of (32) for any ε > y = y ( ε ) such that P (cid:16) ˆ J − (0 , k ) > y ; M k ≤ , S k ≥ − c p z (cid:17) ≤ P (cid:16) ˆ J − (0 , k ) > y ; M k ≤ (cid:17) ≤ ε P ( M k ≤ y ≥ y . On the other hand, one can show (compare with a similarstatement in [6] for random walk conditioned to stay positive) that there existsa proper distribution G ( · ) with G ( z ) ∈ (0 ,
1) for all z >
R > P ( S pR ≥ − c p z | M pR ≤ → − G ( zR /α )as p → ∞ uniformly in z ∈ [0 , N ]. This leads to the following chain of estimatesbeing valid for y ≥ y , a large but fixed T > k ≤ pT : P (cid:16) ˆ J − (0 , k ) > y ; M k ≤ , S k ≥ − c p z (cid:17) ≤ P (cid:16) ˆ J − (0 , k ) > y ; M k ≤ (cid:17) ≤ ε P ( M k ≤ ≤ εK P ( M k ≤ , S k ≥ − c p z ) . This, in account of (15) allows us to proceed with one more chain of estimates pT X k =0 P (cid:0) J − (0 , k ) > y ; L k ≥ − c p z ; τ k = k (cid:1) P ( L n − p − k ≥ ≤ εK pT X k =0 P ( M k ≤ , S k ≥ − c p z ) P ( L n − p − k ≥ ≤ εK P ( L n − p ≥ − c p z ) ≤ εK V ( c p z ) P ( L n − p ≥ ≤ εK z α (1 − ρ ) V ( c p ) P ( L n ≥
0) (36)17eing valid for all z ∈ [0 , N ].To consider the case k ≥ T p we use Theorem 4 of [21] according to which P ( M k ≤ , S k ≥ − c p z ) ≤ K V ( c p z ) zc p kc k if zc p ≤ εc k . Using this bound we get n − X k = pT P (cid:0) J − (0 , k ) > y ; L k ≥ − c p z ; τ k = k (cid:1) P ( L n − p − k ≥ ≤ K V ( c p z ) zc p n − X k = pT kc k P ( L n − p − k ≥ . By (6), (15) and properties of regularly varying functions we conclude that n − X k = pT kc k P ( L n − p − k ≥
0) = n/ p X k = pT kc k P ( L n − p − k ≥
0) + n − X k = n/ p +1 kc k P ( L n − p − k ≥ ≤ K P ( L n ≥ n/ p X k = pT kc k + K nc n n/ X j =0 P ( L j ≥ ≤ K c pT P ( L n ≥
0) + K nc n n P ( L n ≥ ≤ K c pT P ( L n ≥ . As a result we get n − X k = pT P (cid:0) J − (0 , k ) > y ; L k ≥ − c p z ; τ k = k (cid:1) P ( L n − p − k ≥ ≤ K V ( c p z ) zc p K c pT P ( L n ≥ ≤ K z α (1 − ρ )+1 c p c pT V ( c p ) P ( L n ≥ ≤ K z α (1 − ρ )+1 T /α V ( c p ) P ( L n ≥ . (37)Combining (36) and (37) we see that P (cid:0) J − (0 , τ ,n − p ) > y ; L n − p ≥ − c p z (cid:1) ≤ K z α (1 − ρ ) (cid:16) zT /α + ε (cid:17) V ( c p ) P ( L n ≥ . N ≤ εT /α we get in account of (12) Z N E h(cid:16) − f j ,p (0) (cid:17) ; S p ∈ c p dz, L p ≥ i P (cid:0) J − (0 , τ n − p ) > y ; L n − p ≥ − c p z (cid:1) ≤ εK P ( L n ≥ V ( c p ) ×× Z N E h(cid:16) − f j ,p (0) (cid:17) ; S p ∈ c p dz, L p ≥ i z α (1 − ρ ) (cid:16) zT /α + ε (cid:17) ≤ εK P ( L n ≥ V ( c p ) Z N z α (1 − ρ ) P ( S p ∈ c p dz, L p ≥ ≤ εK P ( L n ≥ V ( c p ) P ( L p ≥ Z N z α (1 − ρ ) P ( S p ∈ c p dz | L p ≥ ≤ εK C P ( L n ≥ P + ( B ≤ N ) ≤ εK P ( L n ≥ . This estimate combined with (35) proves (34).
Lemma 15
If Condition A is valid, then, for any j = 1 , , ... and x ≥ n ≫ p →∞ P j ( A p,n ( x ) | L n ≥ , Z n >
0) = D ( x ) . Proof . It follows from Lemma 12 and the inequalities P j ( A p,n ( x ) ¯ B p,n ( N ) ¯ C p,n ; L n ≥ ≤ P j ( A p,n ( x ); L n ≥ ≤ P j ( A p,n ( x ) ¯ B p,n ( N ) ¯ C p,n ; L n ≥ , Z n > P j ( A p,n ( x ) C p,n ; L n ≥
0) + P j ( A p,n ( x ) B p,n ( N ); L n ≥ N →∞ lim n ≫ p →∞ P j ( A p,n ( x ) ¯ B p,n ( N ) ¯ C p,n ; L n ≥ , Z n > P j ( Z n > , L n ≥
0) = D ( x ) . Using the equality Z p,n = Z p (1 − f p,n (0)) + Z p X l =1 (cid:16) I n Z ( l ) n ( p ) > o − (1 − f p,n (0)) (cid:17) , the estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z p X l =1 (cid:16) I n Z ( l ) n ( p ) > o − (1 − f p,n (0)) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ q N Z p (1 − f p,n (0) , being valid on the set ¯ B p,n ( N ) , and recalling (24), (25) and the fact that Z p (1 − f p,n (0))) ≥ e √ c p
19n the set ¯ C p,n ∩{ Z p > } , we conclude that for any δ > n = n ( δ ) such that for n ≥ n P j ( A p,n ( x ) ¯ B p,n ( N ) ¯ C p,n , L n ≥ , Z n > ≤ P j (cid:0) ln( e + Z p (1 − f p,n (0))(1 + δ )) ≥ xc p ; ¯ C p,n ; L n ≥ , Z n > (cid:1) + α ( p, n ) ≤ P j (ln Z p + ln(1 − f p,n (0)) + ln(1 + 2 δ ) ≥ xc p ; L n ≥ , Z n >
0) + α ( p, n ) , where lim n ≫ p →∞ | α ( p, n ) | P j ( Z n >
0) = 0 . In view of1 − f p,n (0) ≤ f ′ p (1) (1 − f p +1 ,n (0)) ≤ min p ≤ j ≤ n Y p ≤ i ≤ j f ′ i (1) = e ˆ L p,n , we get P j (ln Z p + ln(1 − f p,n (0)) + ln(1 + 2 δ ) ≥ xc p ; L n ≥ , Z n > ≤ P j (cid:18) c p ln Z p e S p + 1 c p L p,n + ln(1 + 2 δ ) ≥ x ; L n ≥ , Z n > (cid:19) . Using the equivalences P j ( L n ≥ , Z n > ∼ P j ( Z n > | L n ≥ P ( L n ≥ ∼ θ E + (cid:16) − f j , ∞ (0) (cid:17) P ( Z n > n → ∞ , and following the proof of Theorem 1.1 in [1], we obtain P j ( A p,n ( x ) ¯ B p,n ( N ) ¯ C p,n | L n ≥ , Z n > ≤ P j (cid:18) c p ln Z p e S p + 1 c p L p,n + ln(1 + 2 δ ) ≥ x | L n ≥ , Z n > (cid:19) . This inequality, Corollary 10 and Lemma 13 yieldlim N →∞ lim sup n ≫ p →∞ P j (cid:0) A p,n ( x ) ¯ B p,n ( N ) ¯ C p,n | L n ≥ , Z n > (cid:1) ≤ lim δ ↓ lim n ≫ p →∞ P j (cid:18) c p L p,n + ln(1 + 3 δ ) ≥ x | L n ≥ , Z n > (cid:19) = D ( x ) . To get a similar estimate from below observe that according to relations (2.2)and (2.3) of [12]1 − f p,n (0) ≥ n − X l = p η l e − ( S l − S p ) + e − ( S n − S p ) − = e ˆ L p,n n − X l = p η l e − ( S l − ˆ L p,n ) + e − ( S n − ˆ L p,n ) − = e ˆ L p,n (cid:0) J − ( p, τ p,n ) + J + ( τ p,n , n ) (cid:1) − . − f p,n (0)) ≥ ˆ L p,n − log (cid:0) J − ( p, τ p,n ) + J + ( τ p,n , n ) (cid:1) . According to Lemma 14lim y →∞ lim n ≫ p →∞ P j (cid:0) log (cid:0) J − ( p, τ p,n ) + J + ( τ p,n , n ) (cid:1) > y | L n ≥ , Z n > (cid:1) = 0 . Hence for any δ > p and n P j (ln Z p + ln(1 − f p,n (0)) + ln(1 − δ ) ≥ xc p ; L n ≥ , Z n > ≥ P j (cid:18) c p ln Z p e S p + 1 c p L p,n + ln(1 − δ ) ≥ x ; L n ≥ , Z n > (cid:19) leading by Lemma 13 to the following estimate from below:lim inf n ≫ p →∞ P j (cid:0) A p,n ( x ) ¯ B p,n ( N ) ¯ C p,n | L n ≥ , Z n > (cid:1) ≥ lim δ ↓ lim n ≫ p →∞ P j (cid:18) c p L p,n + ln(1 − δ ) ≥ x | L n ≥ , Z n > (cid:19) = D ( x ) . Lemma 15 is proved.
Proof of Theorem 1.
First we note that to check the validity of (14) it issufficent to investigate the asymptotic behavior of the probability of the event A p,n ( x ). We use Lemma 9 to this aim. For z, p, n ∈ N with p ≤ n set ψ ( z, p, n ) := P z ( A p,n ( x ) , L n ≥ . Clearly, ψ (0 , p, n ) = 0. We know by Lemmas 15 and 8 that if n ≫ p = p ( n ) → ∞ then ψ ( z, p, n ) ∼ P z ( L n ≥ , Z n > D ( x )= P ( L n ≥ E (cid:2) − f z ,n (0) | L n ≥ (cid:3) D ( x ) ∼ P ( L n ≥ E + (cid:2) − f z , ∞ (0) (cid:3) D ( x )= P ( L n ≥ P + z ( A u.s ) D ( x ) . (38)In addition, for k ≤ p ≤ n E h I { A p,n ( x ) } , L k,n ≥ | ˜ F k i = ψ ( Z k , p − k, n − k ) . (39)Relations (38) and (39) show that we may apply Lemma 9 to ζ n := I { A p,n ( x ) } , ζ k, ∞ := P + Z k ( A u.s ) D ( x )and l = 0 to conclude that P ( A p,n ( x )) = P ( A p,n ( x ); Z n > ∼ D ( x ) θ P ( L n ≥ ∼ D ( x ) P ( Z n > . This completes the proof of (14) . eferences [1] Afanasyev V.I., Geiger J., Kersting G., Vatutin V.A. Criticality for branch-ing processes in random environment. – Ann. Probab. , (2005), 645–673.[2] Afanasyev V.I., Boeinghoff Ch., Kersting G., Vatutin V.A. Limit theoremsfor weakly subcritical branching processes in random environment. – J.Theoret. Probab. , (2012), p. 703–732.[3] Borovkov K.A., Vatutin V.A. Reduced critical branching processes in ran-dom environment.– Stoch. Proc. Appl., 71 (1997), p. 225–240.[4] Chaumont L. Conditionings and path decompositions for L´evy processes. Stochastic Process. Appl., (1996), 39–54.[5] Chaumont L. Excursion normalisee, meandre at pont pour les processus deL´evy stables. Bull. Sci. Math. , (1997), 5, 377-403.[6] Caravenna F., and Chaumont L. Invariance principles for random walksconditioned to stay positive. - Ann. Inst. H. Poincare, Probab. Statist. , (2008), 170-190.[7] Doney R.A. Local behavior of first passage probabilities. – Probab. TheoryRelat. Fields , (2012), 559–588.[8] Feller W. An Introduction to Probability Theory and its Applications.
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