Convergence in Lp and its exponential rate for a branching process in a random environment Chunmao HUANGa,b , Quansheng LIUa,b,∗
arXiv:1011.0533v1 [math.PR] 2 Nov 2010
a
LMAM, Universit´e de Bretagne-Sud, Campus de Tohannic, BP 573, 56017 Vannes, France b Universit´e Europ´eenne de Bretagne, France
Abstract We consider a supercritical branching process (Zn ) in a random environment ξ. Let W be the limit of the normalized population size Wn = Zn /E[Zn |ξ]. We first show a necessary and sufficient condition for the quenched Lp (p > 1) convergence of (Wn ), which completes the known result for the annealed Lp convergence. We then show that the convergence rate is exponential, and we find the maximal value of ρ > 1 such that ρn (W − Wn ) → 0 in Lp , in both quenched and annealed sense. Similar results are also shown for a branching process in a varying environment. Key words: branching process, varying environment, random environment, moments, exponential convergence rate, Lp convergence AMS subject classification: 60K37, 60J80
1
Introduction and main results
We consider a branching process in a random environment (BPRE). Let ξ = (ξ0 , ξ1 , ξ2 , · · · ) be a stationary and ergodic process taking values in some space Θ. Each realization of ξn corresponds to a probability distribution on N = {0, 1, 2, · · · }, denoted by p(ξn ) = {pi (ξn ) : i ∈ N}, where X X ipi (ξn ) ∈ (0, ∞). pi (ξn ) = 1 and pi (ξn ) ≥ 0, i
i
The sequence ξ = (ξn ) will be called random environment. A branching process (Zn ) in the random environment ξ is a class of branching processes in varying environment indexed by ξ. By definition, Z0 = 1,
Zn+1 =
Zn X
Xn,i
(n ≥ 0),
(1.1)
i=1
where Xn,i (i = 1, 2, · · · ) denotes the number of offspring of the ith particle in the nth generation. Given ξ, {Xn,i : n ≥ 0, i ≥ 1} is a family of (conditionally) independent random variables and each Xn,i has distribution p(ξn ) on N = {0, 1, · · · }. Let (Γ, Pξ ) be the probability space under which the process is defined when the environment ξ is fixed. As usual, Pξ is called quenched law. The total probability space can be formulated as the ∗ Corresponding author at: LMAM, Universit´e de Bretagne-Sud, Campus de Tohannic, BP 573, 56017 Vannes, France. Email addresses:
[email protected] (C. Huang),
[email protected] (Q. Liu).
1
product space (ΘN × Γ, P), where P = τ ⊗ Pξ in the sense that for all measurable and positive g, we have Z Z Z g(ξ, y)dPξ (y)dτ (ξ), gdP = ΘN
Γ
where τ is the law of the environment ξ. The total probability P is usually called annealed law. The quenched law Pξ may be considered to be the conditional probability of P given ξ. Let F0 = σ(ξ0 , ξ1 , ξ2 , · · · ) and Fn = σ(ξ0 , ξ1 , ξ2 , · · · , Xk,i , 0 ≤ k ≤ n − 1, i = 1, 2, · · · ) be the σ-field generated by Xk,i (0 ≤ k ≤ n − 1, i = 1, 2, · · · ), so that Zn are Fn -measurable. For n ≥ 0 and p ≥ 1, set X mn (p) = ip pi (ξn ), mn = mn (1), (1.2) i
and
P0 = 1,
Pn =
n−1 Y
mi (n ≥ 1).
(1.3)
i=0
p So mn (p) = Eξ Xn,i and Pn = Eξ Zn . It is well known that the normalized population size
Wn =
Zn Pn
(1.4)
is a non-negative martingale both under Pξ for every ξ and under P, hence the limit W = lim Wn n→∞
(1.5)
exists almost surely (a.s.) with EW ≤ 1 by Fatou’s lemma. Assume throughout the paper that the process is supercritical in the sense that E log m0 is well defined with E log m0 > 0. We are interested in the Lp convergence rate of Wn both in the quenched sense (under Pξ ) and in the annealed sense (under P). We first show a criterion for the quenched Lp convergence of Wn . Theorem 1.1 (Quenched Lp convergence). Let p > 1. Consider the following assertions: p Z1 < ∞; (ii) supn Eξ Wnp < ∞ a.s.; (i) E log Eξ m 0 (iii) Wn → W in Lp under Pξ for almost all ξ; (iv) 0 < Eξ W p < ∞ a.s..
Then the following implications hold: (i) ⇒ (ii) ⇔ (iii) ⇔ (iv). If additionally (ξn ) are i.i.d. and E log m0 < ∞, then all the four assertions are equivalent. Z1 Z1 p ) < ∞ if and only if E log+ Eξ | m − 1|p < ∞, where It can be easily seen that ∀p > 0, E log Eξ ( m 0 0 and hereafter we use the following usual notations:
log+ x = max(log x, 0),
log− x = max(− log x, 0),
a ∧ b = min(a, b),
a ∨ b = max(a, b).
We next give a description of the quenched Lp convergence rate. Theorem 1.2 (Exponential rate of quenched Lp convergence). Let p > 1, ρ > 1 and m = exp(E log m0 ) > 1. 2
(a) If E log Eξ
Z1 m0
p
< ∞, then
lim ρn (Eξ |W − Wn |p )1/p = 0
for ρ < min{m1−1/p , m1/2 }.
a.s.
n→∞
p∧2 p∨2 Z1 Z1 + < ∞ and E log E < ∞, then a.s. − 1 − 1 (b) If E log− Eξ m ξ m0 0 n
p 1/p
lim sup ρ (Eξ |W − Wn | ) n→∞
where ρ¯c = m1/2 =
p
=0 >0
if ρ < ρ¯c , if ρ > ρ¯c ,
exp(E log m0 ) > 1.
Remark. We mention that the theorem is valid with evident interpretation even if E log m0 = ∞ (so Z1 − p − 1 < ∞, that m = ∞). We also remark that by the monotonicity of the L norm, if E log Eξ m 0 p∧2 Z1 then E log− Eξ m − 1 < ∞. 0
Theorem 1.2(a) shows that Wn → W in Lp under Pξ at an exponential rate; Theorem 1.2(b) means that ρ¯c is the critical value of ρ > 1 for which ρn (W − Wn ) → 0 in Lp under Pξ for almost all ξ. For the classical Galton-Watson process, Theorem 1.2(a) reduces to the result of Liu (2001, [15]) that if EZ1p < ∞, then ρn (W − Wn ) → 0 in Lp for 1 < ρ < min{m1−1/p , m1/2 }, where m = EZ1 ∈ (1, ∞); Theorem 1.2(b) can be obtained by a result of Alsmeyer, Iksanov, Polotsky and R¨ osler (2009, [1]) on branching random walks. Recall that for a Galton-Watson process with m = EZ1 ∈ (1, ∞) and P(W > 0) > 0, Asmussen (1976, [2]) showed that for p ∈ (1, 2), W − Wn = o(m−n/q ) a.s. if and only if EZ1p < ∞. As an application of Theorem 1.2, we immediately obtain the following similar result for a branching process in a random environment. Corollary 1.3 (Exponential rate of a.s. convergence). Let p ∈ (1, 2) and m = exp(E log m0 ) ∈ Z1 p ) < ∞, then ∀ε > 0, (0, ∞). If E log Eξ ( m 0 n
W − Wn = o(m− q+ε )
a.s.,
(1.6)
where 1/p + 1/q = 1. 1
In fact, to see the conclusion, let ρ1 = m q+ε and take ρ satisfying ρ1 < ρ < m1/q . By Theorem 1.2(a), ρn (Eξ |W − Wn |p )1/p → 0, so that ! X X ρ1 n n ρ1 |W − Wn | ≤ ρn (Eξ |W − Wn |p )1/p < ∞ a.s.. Eξ ρ n n Therefore the series
P
n n ρ1 (W
− Wn ) converges a.s., which implies (1.6).
Corollary 1.3 has recently been shown by Huang and Liu ([12], 2010) by a truncating argument. The approach here is quite different. We now turn to the annealed Lp convergence of Wn . When the environment is i.i.d., a necessary and sufficient condition was shown by Guivarc’h and Liu (2001, [10], Theorem 3): 3
Proposition 1.4 (Annealed Lp convergence [10]). Assume that (ξn ) are i.i.d. and p > 1. Then the following assertions are equivalent: p Z1 (i) E m < ∞ and Em1−p < 1; (ii) supn EWnp < ∞; 0 0 (iii) Wn → W in Lp under P;
(iv) 0 < EW p < ∞.
We shall prove the following theorem about the rate of convergence. Theorem 1.5 (Exponential rate of annealed Lp convergence). Assume that (ξn ) are i.i.d.. Let p > 1 and ρ > 1. p Z1 (a) Assume that E m < ∞ and Em1−p < 1. Then 0 0 lim ρn (E|W − Wn |p )1/p = 0
n→∞
where ρ0 > 1 is defined by ( ρ0 =
for ρ < ρ0 ,
−1/p (Em1−p if p ∈ (1, 2), 0 ) −p/2 −1/p 1−p −1/p min{(Em0 ) , (Em0 ) } if p ≥ 2.
(b) Assume that P(W1 = 1) < 1 and that either of the following conditions is satisfied: p/2 2 −p/2 −p/2−1 Z1 (i) p ∈ (1, 2), E Eξ m < ∞, Em0 log m0 > 0 and Em0 Z1 log+ Z1 < ∞; 0 Z1 p ) < ∞. (ii) p ≥ 2 and E( m 0
Set ρc =
(
−p/2
(Em0 )−1/p if p ∈ (1, 2), −p/2 −1/p 1−p −1/p min{(Em0 ) , (Em0 ) } if p ≥ 2.
Then n
p 1/p
lim sup ρ (E|W − Wn | ) n→∞
=0 >0
if ρ < ρc , if ρ > ρc . −p/2
Remark. By the convexity of the function Em−x log m0 > 0 ensures that 0 , the condition Em0 −p/2 1−p Em0 < Em0 for p ∈ (1, 2), so that ρ0 ≤ ρc . If p0 (ξ0 ) = 0 and p1 (ξ0 ) < 1 a.s., we have m0 > 1 −p/2 −p/2−1 Z1 Z1 log+ Z1 < ∞ whenever E m a.s., so that Em0 log m0 > 0 and Em0 log+ Z1 < ∞. 0 Theorem 1.5(a) implies that Wn → W in Lp under P (annealed) at an exponential rate. Theorem 1.5(b) shows that under certain moment conditions, ρc is the critical value of ρ > 1 for the annealed Lp convergence of ρn (W − Wn ) to 0, while Theorem 1.2(b) shows that ρ¯c is the critical value for the quenched Lp convergence. Notice that by Jensen’s inequality, p p −p/2 Em0 = E exp(− log m0 ) ≥ exp(− E log m0 ), 2 2 −p/2 −1/p )
so that (Em0
≤ exp(− 21 E log m0 ). This shows that ρc ≤ ρ¯c .
The rest of this paper is organized as follows. In Section 2, we state results of the Lp convergence of the martingale Wn and its exponential rate for a branching process in a varying environment. In Section 3, we study an associated martingale Aˆn which will be used for the proof of the exponential convergence rate of Wn . Section 4 is devoted to the proofs of the results of Section 2. In Sections 5 and 6, we consider the random environment case, and give the proofs of the main results: in Section 5, we study the quenched moments of Wn and Aˆn , and prove Theorems 1.1 and 1.2; in Section 6, we consider the annealed moments of Aˆn and give the proof of Theorem 1.5. 4
2
Branching process in a varying environment
In this section we study the Lp convergence and its convergence rate for a branching process (Zn ) in a varying environment (BPVE). By definition, Z0 = 1,
Zn+1 =
Zn X
(n ≥ 0),
Xn,i
(2.1)
i=1
where Xn,i (i = 1, 2, · · · ) denotes the number of offspring of the ith particle in the nth generation, each Xn,i has distribution p(n) = {pi (n) : i ∈ N} on N = {0, 1, · · · }, where X X pi (n) ≥ 0, pi (n) = 1 and ipi (n) ∈ (0, ∞); i
i
all the random variables Xn,i (n ≥ 0, i ≥ 1) are independent of each other. Let (Ω, P) be the underlying probability space. For n ≥ 0 and p ≥ 1, set X p mn (p) = EXn,i = kp pk (n), mn = mn (1), (2.2) k
and
p X k − mn p Xn,i − 1 = m ¯ n (p) = E mn pk (n). mn
(2.3)
k
Let F0 = {∅, Ω} and Fn = σ(Xk,i : 0 ≤ k ≤ n − 1, i = 1, 2, · · · ) be the σ-field generated by Xk,i (0 ≤ k ≤ n − 1, i = 1, 2, · · · ). Like the case of BPRE, let P0 = 1,
Pn =
n−1 Y
mi (n ≥ 1).
(2.4)
i=0
Then the normalized population size Wn = Zn /Pn is a non-negative martingale with respect to the filtration Fn , and limn→∞ Wn = W a.s. for some non-negative random variable W with EW ≤ 1. It is well known that there is a non-negative but possibly infinite random variable Z∞ such that Zn → Z∞ in distribution as n → ∞. We P are interested in the supercritical case where P(Z∞ = 0) < 1, so that ∞ by ([13], Corollary 3), either n=0 (1 − µn (1)) < ∞, or limn→∞ Pn = ∞. Here we assume that limn→∞ Pn = ∞. For simplicity, let ¯n = X ¯ n,1 . ¯n,i = Xn,i , X (2.5) X mn From the definitions of Zn and Wn , we have Zn 1 X ¯ n,i − 1). (X Wn+1 − Wn = Pn
(2.6)
i=1
We are interested in the Lp convergence of the martingale Wn and its convergence rate. Firstly, we have the following theorem about the Lp convergence of Wn . Theorem 2.1 (Lp convergence of Wn for BPVE). Let (Zn ) be the BPVE defined in (2.1). P p(1/r−1) (i) Let p ∈ (1, 2). If n Pn m ¯ n (r)p/r < ∞ for some r ∈ [p, 2], then Wn → W
Conversely, if lim inf n→∞
log Pn n
in Lp .
> 0 and (2.7) holds, then 5
P
−s−p/2 m ¯ n (p) n Pn
(2.7)
< ∞ for all s > 0.
P ¯ n (p)2/p < ∞, then (2.7) holds. Conversely, if (2.7) holds, then (ii) Let p ≥ 2. If n Pn−1 m P p(1/r−1) m ¯ n (r)p/r < ∞ for all r ∈ [2, p]. n Pn
Remark 1. As Wn is a martingale, ∀p > 1, (2.7) holds if and only if supn EWnp < ∞.
P∞ −1 ¯ (2). So sup EW 2 < ∞ if and only if 2 = 1 + Remark 2. For p = 2, sup EW n n n n n n=0 Pn m P −1 m ¯ (2) < ∞, as shown by Jagers (1974, [13], Theorem 4). P n n n
P P −α ¯ n (β)γ < ∞ is equivalent to < ∞, then for α, γ > 0 and β ≥ 1, n Pn−α m Remark 3. If P P −α mn (β)γ n n P log Pn < ∞. The condition that lim inf n→∞ n > 0 ensures that n Pn−α < ∞ for any βγ n Pn mn P −α mn (β)γ P α > 0. Moreover, < ∞ implies that n Pn−α < ∞, since mn (β) ≥ 1 by Jensen’s βγ n Pn mn mβ n inequality. To estimate the exponential rate of Lp convergence of Wn , following [1], we consider the series A(ρ) =
∞ X
ρn (W − Wn ) (ρ > 1).
(2.8)
n=0
Here A(ρ) just denotes the series. The convergence of the series A(ρ) reflects the exponential rate of W − Wn . More precisely, if the series A(ρ) converges a.s. (resp. in Lp , p > 1), then ρn (W − Wn ) → 0 a.s. (resp. in Lp ). Conversely, if ρn (W − Wn ) → 0 a.s. (resp. in Lp ), then for any ρ1 ∈ (1, ρ), the series A(ρ1 ) converges a.s. (resp. in Lp ). Moreover, from the remak after Lemma 3.2, we see that the Lp convergence of A(ρ) implies its a.s. convergence. Theorem 2.2 (Lp convergence of A(ρ) for BPVE). Let (Zn ) be the BPVE defined in (2.1) and let ρ > 1. P p(1/r−1) (i) Let p ∈ (1, 2). If n ρpn Pn m ¯ n (r)p/r < ∞ for some r ∈ [p, 2], then the series A(ρ) converges in Lp .
(2.9)
P −s−p/2 m ¯ n (p) < ∞ for all Conversely, if lim inf n→∞ lognPn > 0 and (2.9) holds, then n ρpn Pn s > 0. P ¯ n (p)2/p < ∞, then (2.9) holds. Conversely, if (2.9) holds, then (ii) Let p ≥ 2. If n ρ2n Pn−1 m P pn p(1/r−1) m ¯ n (r)p/r < ∞ for all r ∈ [2, p]. n ρ Pn
By the relations between W − Wn and A(ρ), from Theorem 2.2 we immediately obtain:
Theorem 2.3 (Exponential rate of Lp convergence of Wn for BPVE). Let (Zn ) be the BPVE defined in (2.1) and let ρ > 1. P p(1/r−1) (i) Let p ∈ (1, 2). If n ρpn Pn m ¯ n (r)p/r < ∞ for some r ∈ [p, 2], then (E|W − Wn |p )1/p = o(ρ−n ).
(2.10)
P −s−p/2 Conversely, if lim inf n→∞ lognPn > 0 and (2.10) holds, then n ρ1 pn Pn m ¯ n (p) < ∞ for all ρ1 ∈ (1, ρ) and all s > 0. P ¯ n (p)2/p < ∞, then (2.10) holds. Conversely, if (2.10) holds, (ii) Let p ≥ 2. If n ρ2n Pn−1 m P p(1/r−1) then n ρ1 pn Pn m ¯ n (r)p/r < ∞ for all ρ1 ∈ (1, ρ) and r ∈ [2, p]. 6
3
The martingale {Aˆn }
To study the convergence of the series A(ρ) defined in (2.8), as in [1], we introduce an associated martingale {Aˆn }. Let ρ > 1, and define Aˆn = Aˆn (ρ) =
n X
k
ρ (Wk+1 − Wk ),
ˆ A(ρ) =
∞ X
ρn (Wn+1 − Wn ).
(3.1)
n=0
k=0
ˆ Here A(ρ) denotes the series; it will also denote the sum of the series when the series converges. It is easy to see that {(Aˆn ; Fn+1 )} forms a martingale. Let us first recall the Bukholder’s inequality for martingales before showing relations between ˆ A(ρ) and A(ρ). Lemma inequality, see [8]). Let {Sn } be a L1 martingale with S0 = 0. Let Pn3.1 (Bukholder’s Pe.g. ∞ 2 1/2 Qn = ( k=1 (Sk − Sk−1 ) ) and Q = ( n=1 (Sn − Sn−1 )2 )1/2 . Then ∀p > 1, ap k Qn kp ≤k Sn kp ≤ bp k Qn kp ,
ap k Q kp ≤ sup k Sn kp ≤ bp k Q kp , n
where ap = (p −
1)/18p3/2 ,
bp =
18p3/2 /(p
− 1)1/2 .
Lemma 3.2 ([1], Lemma 3.1). Let p > 1 and ρ > 1. The series A(ρ) converges a.s. (resp. in Lp ) if ˆ and only if the same is true for the series A(ρ). ˆ Remark. By the convergence theorems for martingales, supn E|Aˆn |p < ∞ implies that A(ρ) converges p p p ˆ a.s. and in L . Therefore the L convergence of A(ρ) is equivalent to supn E|Aˆn | < ∞. Moreover, ˆ if A(ρ) converges in Lp , then it also converges a.s.. Lemma 3.2 in fact tells us that A(ρ) converges p in L if and only if supn E|Aˆn |p < ∞, and the Lp convergence of A(ρ) implies its a.s. convergence. For reader’s convenience, we give a proof of Lemma 3.2, following the basic idea of the proof in ([1], Lemma 3.1). ˆ Proof of Lemma 3.2. For convenience, we write A and Aˆ for A(ρ) and A(ρ) respectively, and set An = An (ρ) =
n X
ρk (W − Wk ).
k=0
We shall show that An and Aˆn satisfy the following equality: An =
1 ρn+1 ρ ˆ (W − Wn+1 ) − (W − 1) An + ρ−1 ρ−1 ρ−1
7
a.s..
(3.2)
Indeed, we have a.s. An =
n X
k
ρ (W − Wk ) =
n X
lim
l→∞
k=0
k=0
l X ρ (Wj+1 − Wj ) k
j=k
j n l n X X X X (Wj+1 − Wj ) ρk lim (Wj+1 − Wj ) ρk +
=
l→∞ n X
=
j=0
j=n+1
k=0
(Wj+1 − Wj )
j=0
j X
ρk +
(Wj+1 − Wj )
j=n+1
k=0
ρn+1
∞ X
k=0
n X
ρk
k=0
ρ ˆ 1 (W − Wn+1 ) − (W − 1). An + ρ−1 ρ−1 ρ−1
=
For the a.s. convergence, if the series A converges a.s., then ρn (W − Wn ) → 0 a.s.. From (3.2) we ˆ Conversely, assume that Aˆ converges a.s.. immediately obtain the a.s. convergence of the series A. Pn+j −k k Set ak = ρ , bk = ρ (Wk+1 − Wk ) and Bj = k=n bk = Aˆn+j − Aˆn−1 for fixed n. By Abel’s lemma, n+l X
n+l X
(Wk+1 − Wk ) =
ak bk k=n n+l−1 X
k=n
(ak − ak+1 )Bk−n + an+l Bl
=
k=n n+l−1 X
(ρ−k − ρ−k−1 )(Aˆk − Aˆn−1 ) + ρ−n−l (Aˆn+l − Aˆn−1 ).
=
k=n
Therefore, n+l X |ρ (W − Wn )| ≤ ρ lim (Wk+1 − Wk ) l→∞ n
n
k=n
≤ ρn lim sup |Aˆn+j − Aˆn−1 | l→∞ 0≤j≤l
n+l X
(ρ−k − ρ−k−1 ) + ρ−n−l
k=n
≤ sup |Aˆn+j − Aˆn−1 | → 0
!
as n → ∞.
j
This leads to the a.s. convergence of A by (3.2). For the Lp convergence, if the series A converges in Lp , then we have k Am − An kp → 0 as m, n → ∞. In particular, k ρn (W − Wn ) kp =k An − An−1 kp → 0 as n → ∞. By (3.2) and applying the triangular inequality in Lp , we have ρ k Aˆm − Aˆn kp ρ−1
ρm+1 ρn+1 (W − Wm+1 ) + (W − Wn+1 ) kp ρ−1 ρ−1 ρm+1 ρn+1 ≤ k Am − An kp + k W − Wm+1 kp + k W − Wn+1 kp ρ−1 ρ−1 → 0 as m, n → ∞. =
k Am − An −
Thus Aˆ also converges in Lp . Conversely, if Aˆ converges in Lp , we have k Aˆm − Aˆn kp → 0 as 8
m, n → ∞. Again by (3.2), ρ ρm+1 ρn+1 (Aˆm − Aˆn ) + (W − Wm+1 ) − (W − Wn+1 ) kp ρ−1 ρ−1 ρ−1 ρ ρm+1 ρn+1 k Aˆm − Aˆn kp + k W − Wm+1 kp + k W − Wn+1 kp . ρ−1 ρ−1 ρ−1
k Am − An kp = k ≤
It remains to show that ρn k W −Wn kp → 0 as n → ∞. Using Bukholder’s inequality (see Lemma 3.1) first for the martingale differences {Wk+1 −Wk : k ≥ n} and then for {ρk (Wk+1 −Wk ) : n ≤ k ≤ n+l}, we obtain p/2 ∞ X ρpn E|W − Wn |p ≤ Cρpn E (Wk+1 − Wk )2 k=n !p/2 ∞ X 2k 2 ≤ C ρ (Wk+1 − Wk ) k=n
n+l X
= C lim
l→∞
ρ2k (Wk+1 − Wk )2
k=n
!p/2
≤ C sup E|Aˆn+l − Aˆn−1 |p → 0 as n → ∞, l
where C ∈ (0, ∞) is a constant that may differ from line to line. This completes the proof. The following lemma gives relations between supn E|Aˆn |p and E|Wn+1 − Wn |p that we shall use later. Lemma 3.3. Let p > 1and write ap = (p − 1)/18p3/2 , bp = 18p3/2 /(p − 1)1/2 . Then: (i) For p ∈ (1, 2) and N ≥ 1, ap N
p/2−1
N −1 X
ρ E|Wn+1 − Wn | ≤ sup E|Aˆn |p ≤ bp pn
p
n
n=0
(ii) For p = 2, sup E|Aˆn |2 = n
∞ X
∞ X
ρpn E|Wn+1 − Wn |p .
(3.3)
n=0
ρ2n E|Wn+1 − Wn |2 .
(3.4)
n=0
(iii) For p > 2, ap
∞ X
n=0
ρ E|Wn+1 − Wn | ≤ sup E|Aˆn | ≤ bp pn
p
p
n
∞ X
n=0
2n
p 2/p
ρ (E|Wn+1 − Wn | )
!p/2
.
(3.5)
Proof. (i) Let p ∈ (1, 2). Set Bn = Aˆn−1 and B0 = 0. Applying Burkholder’s inequality to the
9
martingale {Bn }, we have sup E|Aˆn | = sup E|Bn | p
p
n
n
p/2 ∞ X 2 ≤ bp E (Bn − Bn−1 ) n=1 p/2 ∞ X 2n 2 ρ (Wn+1 − Wn ) = bp E n=0
≤ bp
∞ X
ρpn E|Wn+1 − Wn |p .
n=0
The last step above is due to the concavity of xp/2 (0 < p/2 < 1). To prove the first inequality in (3.3), using the other side of Burkholder’s inequality and again the concavity of xp/2 , we obtain sup E|Aˆn |p n
p/2 ∞ X ρ2n (Wn+1 − Wn )2 ≥ ap E n=0 !p/2 N −1 X 2n 2 ≥ ap E ρ (Wn+1 − Wn ) n=0
≥ ap N
p/2−1
= ap N p/2−1
N −1 X
n=0 N −1 X
E(ρ2n (Wn+1 − Wn )2 )p/2 ρpn E|Wn+1 − Wn |p .
n=0
(ii) For p = 2, by the orthogonality of martingale, we have E|Aˆn |2 = =
n X
k=1 n X
E|Aˆk − Aˆk−1 |2 + E|Aˆ0 |2 E|ρk (Wk+1 − Wk )|2 + E|W1 − W0 |2
k=1
=
n X
E|Wk+1 − Wk |2 .
k=0
Letting n → ∞ we obtain (3.4). (iii) Let p > 2. By Burkholder’s inequality, ∞ p/2 ∞ p/2 X X 2n 2 p 2n 2 ˆ ap E ρ (Wn+1 − Wn ) ≤ sup E|An | ≤ bp E ρ (Wn+1 − Wn ) . n n=0
(3.6)
n=0
Since p/2 > 1, the triangular inequality in Lp/2 gives
p/2 2/p ∞ ∞ X X 2n 2 E ρ (Wn+1 − Wn ) ρ2n (E|Wn+1 − Wn |p )2/p . ≤ n=0
n=0
10
(3.7)
On the other hand, by the convexity of xp/2 , p/2 ∞ ∞ X X ρ2n (Wn+1 − Wn )2 E (ρ2n |Wn+1 − Wn |2 )p/2 ≥ E n=0
=
n=0 ∞ X pn
ρ E|Wn+1 − Wn |p .
(3.8)
n=0
Combing (3.7), (3.8) with (3.6), we get (3.5). For a BPRE, notice that {Wn } is a martingale both under Pξ (for every ξ) and under P, and so is {Aˆn }. Thus Lemmas 3.2 and 3.3 hold for both expectations Eξ and E.
4
Moments and Lp convergence rate for BPVE; Proofs of Theorems 2.1 and 2.2
In this section, we shall prove the results introduced in Section 2. Lemma 4.1. Let p > 1, n ≥ 0 and write ap = (p − 1)/18p3/2 , bp = 18p3/2 /(p − 1)1/2 . Then: (i) For p ∈ (1, 2) and r ∈ [p, 2], ¯ n − 1|p ≤ E|Wn+1 − Wn |p ≤ bp Pnp(1/r−1) (E|X ¯ n − 1|r )p/r . ap Pn−p/2 EWnp/2 E|X (ii) For p = 2,
(4.1)
¯ n − 1|2 . E|Wn+1 − Wn |2 = Pn−1 E|X
(4.2)
¯n − 1|r )p/r ≤ E|Wn+1 − Wn |p ≤ bp Pn−p/2 EWnp/2 E|X ¯ n − 1|p . ap Pnp(1/r−1) (E|X
(4.3)
(iii) For p > 2 and r ∈ [2, p],
Proof. We first prove (ii). By (2.6), E|Wn+1 − Wn |2 = =
=
2 Z n 1 X ¯n,i − 1) ( X E Pn2 1 E Pn2
i=1 Z n X
¯ n,i − 1) (X
Zn X ¯n,j − 1) (X j=1
i=1
Zn X X 1 ¯n,i − 1)2 + ¯n,i − 1)(X ¯n,j − 1) E (X (X Pn2 i=1
i6=j
Z
=
n 1 X ¯ n,i − 1)2 = Pn−1 E|X ¯ n − 1|2 . (X E Pn2
i=1
We then prove (i) and (iii). Let p > 1. Fix n ≥ 0 and let S0 = 0,
Sk = Pn−1
k X ¯ n,i − 1)1{Z (X i=1
11
n ≥i}
.
Let G0 = Fn and Gk = σ(Fn , Xn,i , 1 ≤ i ≤ k). It is not difficult to verify that {Sk } forms a martingale with respectPto Gk and {Sk } is uniformly integrable, so that supk E|Sk |p = E|S|p , where n ¯ S = limk→∞ Sk = Pn−1 Z i=1 (Xn,i − 1) = Wn+1 − Wn . By Burkholder’s inequality, p/2 p/2 ∞ ∞ X X 2 p 2 ap E (Sk − Sk−1 ) ≤ E|S| ≤ bp E (Sk − Sk−1 ) , k=1
k=1
which means that p/2 p/2 Zn Zn 1 X 1 X 2 2 p ¯ ¯ (Xn,i − 1) ≤ E|Wn+1 − Wn | ≤ bp E 2 (Xn,i − 1) . ap E 2 Pn Pn i=1
(4.4)
i=1
For p ∈ (1, 2) and r ∈ [p, 2], by the concavity of xr/2 , xp/r and xp/2 , we have p/2 2r · rp Zn Zn 1 X 1 X ¯n,i − 1)2 ¯n,i − 1)2 (X = E 2 (X E 2 Pn Pn i=1 i=1 !p/r Zn X −r r ¯ n,i − 1| ≤ E P |X n
≤
i=1 p(1/r−1) ¯n Pn (E|X
− 1|r )p/r ,
(4.5)
and p/2 Zn Zn 1 X X 2 ¯n,i − 1|p ¯ n,i − 1) E 2 |X (X ≥ Pn−p EZnp/2−1 Pn i=1
=
i=1 −p/2 p/2 ¯n Pn EWn E|X
− 1|p .
(4.6)
Combing (4.5),(4.6) with (4.4), we obtain (4.1). For p > 2 and r ∈ [2, p], since xr/2 , xp/r and xp/2 are convex, (4.5) holds with ” ≤ ” replaced by ” ≥ ”, while (4.6) holds with ” ≥ ” replaced by ” ≤ ”. Remark. The second inequality in (4.1) and (4.3) can also be obtained similarly as ([15], Proposition 1.3). Lemma 4.2. Let p ∈ (1, 2) and s > 0. If η = η(s) := and bp = 18p3/2 /(p − 1)1/2 , we have sup E|Wn − 1|p ≥ ap η p/2−1 n
and sup E|Aˆn |p ≥ ap η p/2−1 n
∞ X
P
n
Pn−s < ∞, then writting ap = (p−1)/18p3/2
Pns(p/2−1) E|Wn+1 − Wn |p ,
(4.7)
ρpn Pns(p/2−1) E|Wn+1 − Wn |p .
(4.8)
n=0 ∞ X
n=0
12
Proof. Applying Burkholder’s inequality to the martingale {Wn − 1}, and then using Jensen’s inequality, we get p/2 ∞ X ≥ ap E (Wn+1 − Wn )2
sup E|Wn − 1|p n
n=0
!p/2 ∞ X 1 s 2 = ap E (ηPn |Wn+1 − Wn | ) ηPns n=0
∞ X 1 (ηPns |Wn+1 − Wn |2 )p/2 ≥ ap E s ηP n n=0
= ap η
p/2−1
∞ X
Pns(p/2−1) E|Wn+1 − Wn |p .
n=0
So (4.7) is proved. The proof of (4.8) is similar.
Proof of Theorem 2.1. (i) Let p ∈ (1, 2). Obviously, the condition supn EWnp < ∞ is equivalent to supn E|Wn − 1|p < ∞. By Burkholder’s inequality and Lemma 4.1, for r ∈ [p, 2], p
sup E|Wn − 1| n
p/2 ∞ X 2 ≤ CE (Wn+1 − Wn ) n=0
≤ C
∞ X
E|Wn+1 − Wn |p
n=0
≤ C
X
¯ n − 1|r )p/r < ∞. Pnp(1/r−1) (E|X
n
P p(1/r−1) m ¯ n (r)p/r < ∞ for some r ∈ [p, 2]. Thus supn EWnp < ∞ if n Pn 2s Conversely, assume that supn EWnp < ∞. For s > 0, let s′ = 2−p > 0. It is easy to see that P −s′ 1/n ′ lim inf n→∞ Pn > 0 implies that η = η(s ) = n Pn < ∞. By Lemmas 4.2 and 4.1, sup E|Wn − 1|p ≥ Cη p/2−1 n
≥ Cη p/2−1
∞ X
n=0 ∞ X
′
Pns (p/2−1) E|Wn+1 − Wn |p
¯ n − 1|p Pn−s−p/2 EWnp/2 E|X
n=0
≥ Cη
p/2−1
inf EWnp/2 n
∞ X
Pn−s−p/2 m ¯ n (p).
n=0
P −s−p/2 Thus n Pn m ¯ n (p) < ∞, ∀s > 0. (ii) For p = 2, by the orthogonality of martingale and (4.2), EWn2 = 1 +
n−1 X
E|Wk+1 − Wk |2 = 1 +
k=0
n−1 X k=0
13
¯ k − 1|2 . Pk−1 E|X
Letting n → ∞, we obtain sup EWn2 = 1 + n
∞ X
¯ n − 1|2 = 1 + Pn−1 E|X
P
Pn−1 m ¯ n (2).
n=0
n=0
supn EWn2
∞ X
−1 ¯ (2) n n Pn m
Thus < ∞ if and only if < ∞. P −1 p 2/p = ¯ we consider the case where p > 2. Firstly we assume that n Pn (E|Xn − 1| ) P Now −1 2/p ¯ n (p) < ∞. We want to prove that for every integer b ≥ 1, n Pn m X ¯n − 1|p )2/p < ∞, ∀p ∈ (2b , 2b+1 ]. sup EWnp < ∞ if Pn−1 (E|X (4.9) n
n
Using Burkholder’s inequality, the triangular inequality in Lp/2 , together with (4.3), we get p
sup E|Wn − 1|
≤ CE|
n
∞ X
(Wn+1 − Wn )2 |p/2
n=0 ∞ X
≤ C
(E|Wn+1 − Wn |p )2/p
n=0 ∞ X
≤ C
!p/2
¯ n − 1|p )2/p (Pn−p/2 EWnp/2 E|X
n=0 ∞ X
≤ C sup EWnp/2 n
!p/2
¯ n − 1|p )2/p Pn−1 (E|X
n=0
!p/2
.
(4.10)
We shall prove (4.9) by induction on b. For b = 1, we consider p ∈ (2, 22 ], so that p/2 ∈ (1, 2]. By H¨older’s inequality, X X ¯ n − 1|2 ≤ ¯ n − 1|p )2/p < ∞. Pn−1 E|X Pn−1 (E|X n
n
Hence supn EWn2 < ∞, so that
p/2 supn EWn
p
sup E|Wn − 1| ≤ C n
< ∞. By (4.10), ∞ X
sup EWnp/2 n
¯n Pn−1 (E|X
p 2/p
− 1| )
n=0
!p/2
< ∞.
So (4.9) holds for b = 1. Now assume that (4.9) holds for p ∈ (2b , 2b+1 ] for some integer b ≥ 1. For p ∈ (2b+1 , 2b+2 ], we have p/2 ∈ (2b , 2b+1 ]. By H¨older’s inequality, X X ¯ n − 1|p/2 )4/p ≤ ¯ n − 1|p )2/p < ∞. Pn−1 (E|X Pn−1 (E|X n
n
p/2
Using (4.9) for p/2, we obtain supn EWn < ∞, so that supn EWnp < ∞ from (4.10). Therefore (4.9) still holds for p ∈ (2b+1 , 2b+2 ], which implies that (4.9) holds for all integers b ≥ 1. Conversely, assume that supn EWnp < ∞. By Burkholder’s inequality and Lemma 4.1, for all r ∈ [2, p], sup E|Wn − 1|p ≥ CE| n
≥ C ≥ C
∞ X
(Wn+1 − Wn )2 |p/2
n=0 ∞ X
E|Wn+1 − Wn |p
n=0 ∞ X
¯ n − 1|r )p/r . Pnp(1/r−1) (E|X
n=0
14
Thus
P
p(1/r−1) m ¯ n (r)p/r n Pn
=
P
p(1/r−1)
n
Pn
¯ n − 1|r )p/r < ∞. (E|X
Proposition 4.1 (Moments of Aˆn for BPVE). P p(1/r−1) (i) Let p ∈ (1, 2). If n ρpn Pn m ¯ n (r)p/r < ∞ for some r ∈ [p, 2], then sup E|Aˆn |p < ∞. n
(4.11)
P 1/n −s−p/2 Conversely, if lim inf n→∞ Pn > 0 and (4.11) holds, then n ρpn Pn m ¯ n (p) < ∞ for any s > 0. P ¯ n (p)2/p < ∞, then (4.11) holds. Conversely, if (4.11) holds, (ii) Let p ≥ 2. If n ρ2n Pn−1 m P pn p(1/r−1) then for any r ∈ [2, p], n ρ Pn m ¯ n (r)p/r < ∞. P 2n −1 ¯ (2). Remark. For p = 2, supn E|Aˆn |2 = ∞ n n=0 ρ Pn m Proof of Proposition 4.1. (i) By Lemmas 3.3 and 4.1, for r ∈ [p, 2], sup E|Aˆn |p ≤ C n
≤ C
∞ X
n=0 ∞ X
ρpn E|Wn+1 − Wn |p ¯n − 1|r )p/r . ρpn Pnp(1/r−1) (E|X
n=0
P p(1/r−1) m ¯ n (r)p/r < ∞ for some r ∈ [p, 2]. Conversely, assume Hence supn E|Aˆn |p < ∞ if n ρpn Pn 2s p ′ > 0. Since η = η(s′ ) < ∞, by (4.8) and Lemma that supn E|Aˆn | < ∞. For any s > 0, let s = 2−p 4.1, ∞ X ρpn Pn−s−p/2 m ¯ n (p). sup E|Aˆn |p ≥ Cη p/2−1 inf EWnp/2 n
n
n=0
−s−p/2
m ¯ n (p) < ∞, ∀s > 0. Thus n epan Pn (ii) For p = 2, by (3.4) and (4.2 ), P
sup E|Aˆn |2 = n
∞ X
¯ n − 1|2 = ρ2n Pn−1 E|X
∞ X
ρ2n Pn−1 m ¯ n (2).
n=0
n=0
P Thus supn E|Aˆn |2 < ∞ if and only if P n ρ2n Pn−1 m ¯ n (2) < ∞. ¯ n − 1|p )2/p (= P ρ2n Pn−1 m Let p > 2. We first assume that n ρ2n Pn−1 (E|X ¯ n (p)2/p ) < ∞. By n (3.5 ) and (4.3), sup E|Aˆn |p ≤ C n
∞ X
ρ2n (E|Wn+1 − Wn |p )2/p
n=0
≤ C
∞ X
!p/2
¯ n − 1|p )2/p ρ2n Pn−1 (EWnp/2 )2/p (E|X
n=0
≤ C sup EWnp/2 n
∞ X
¯ n − 1|p )2/p ρ2n Pn−1 (E|X
n=0
15
!p/2
!p/2
< ∞,
(4.12)
p/2
provided that supn EWn
< ∞. Since X X Pn−1 m ¯ n (p)2/p ≤ ρ2n Pn−1 m ¯ n (p)2/p < ∞, n
n
p/2
we have supn EWnp < ∞ by Theorem 2.1, so that supn EWn < ∞. It follows from (4.12) that supn E|Aˆn |p < ∞. Conversely, assume that supn E|Aˆn |p < ∞. Notice that ∀r ∈ [2, p], sup E|Aˆn |p ≥ C n
This implies that
P
∞ X
¯n − 1|r )p/r . ρpn Pnp(1/r−1) (E|X
n=0
pn p(1/r−1) m ¯ n (r)p/r n ρ Pn
< ∞, ∀r ∈ [2, p].
Proof of Theorem 2.2. By Lemma 3.2 and the remark following it, the assertion supn E|Aˆn |p < ∞ ˆ which is also equivalent to the Lp convergence of A. So is equivalent to the Lp convergence of A, Theorem 2.2 is just a consequence of Proposition 4.1.
5
Quenched moments and quenched Lp convergence rate for BPRE; Proofs of Theorems 1.1 and 1.2
Let us return to a BPRE (Zn ). Notice that for each fixed ξ, (Zn ) is a BPVE. So all the results for BPVE can be directly applied to BPRE by considering the quenched law Pξ and the corresponding expectation Eξ . The following lemma will be used to prove our theorems for BPRE. Lemma 5.1. Let (αn , βn )n≥0 be a stationary and ergodic sequence of non-negative random variables. If E log α0 < 0 and E log+ β0 < ∞, then ∞ X
α0 · · · αn−1 βn < ∞
a.s..
(5.1)
n=0
Conversely, we have: (a) if (αn , βn )n≥0 are i.i.d. and E log α0 ∈ (−∞, 0), then (5.1) implies that E log+ β0 < ∞; (b) if E| log β0 | < ∞, then (5.1) implies that E log α0 ≤ 0. Proof. The sufficiency is a direct consequence of the ergodic theorem and Cauthy’s test for the convergence of series, remarking that if E log α0 < 0 and E log max(β0 , 1) < ∞, then lim sup n→∞
1 log(α0 · · · αn−1 max(βn , 1)) < 0. n
For the necessity, part (a) was shown in the proof of ([9], Theorem 4.1). For part (b), again by Cauchy’s test, if (5.1) holds, then lim sup(α0 · · · αn−1 βn )1/n ≤ 1 n→∞
a.s.,
which is equivalent to n−1
lim sup( n→∞
1X 1 log αi + log βn ) ≤ 0 a.s.. n n i=0
16
By the ergodic theorem, n−1
1X log αi = E log α0 n→∞ n lim
a.s.,
i=0
and 1 lim log βn = lim n→∞ n→∞ n Hence E log α0 ≤ 0.
n
n−1
i=0
i=0
1X 1X log βi − log βi n n
!
= E log β0 − E log β0 = 0
a.s..
Proof of Theorem 1.1. The implications ”(ii) ⇒ (iii) ⇒ (iv)” are evident. We first prove that (iv) implies (ii). Notice that for n ≥ 1, Zn 1 X W = W (n, i) Pn
a.s.,
(5.2)
i=1
where under Pξ , (W (n, i))i≥1 are independent of each other and independent of Zn , with distribution Pξ (W (n, i) ∈ ·) = PT n ξ (W ∈ ·). Taking conditional expectation at both sides of (5.2), we see that Eξ W = ET n ξ W
a.s..
Therefore, by the ergodicity, Eξ W = c a.s. for some constant c ∈ [0, ∞]. As Eξ W p > 0 a.s., we have c > 0. Again by (5.2) and Jensen’s inequality, !!p Zn 1 X p = cp Wnp a.s., W (n, i) Fn Eξ (W |Fn ) ≥ Eξ Pn i=1
so that
Eξ Wnp ≤ c−p Eξ W p
a.s.,
∀n ≥ 1.
Therefore, supn Eξ Wnp ≤ c−p Eξ W p < ∞ a.s. (so that c = 1 as then Wn → W in Lp under Pξ ). We next prove that (i) implies (ii). Notice that E log Eξ
Z1 m0
p
Z1 − < ∞ is equivalent to E log+ Eξ | m 0
1|p < ∞. By Theorem 2.1, to prove that supn Eξ Wnp < ∞ a.s., it suffices to show that X Pn1−p m ¯ n (p) < ∞ a.s. if p ∈ (1, 2), n
and
X
Pn−1 m ¯ n (p)2/p < ∞ a.s. if p ≥ 2.
n
Z1 By Lemma 5.1, since E log m0 > 0 and E log+ m ¯ 0 (p) = E log+ Eξ | m − 1|p < ∞, the two series above 0 converge a.s.. We finally prove that (ii) implies (i) when the environment is i.i.d.. Assume that (ξn )n≥0 are i.id, E log m0 < ∞ and supn Eξ Wnp < ∞ a.s.. Again by Theorem 2.1, X Pn−s−p/2 m ¯ n (p) < ∞ a.s., ∀s > 0, if p ∈ (1, 2), n
and
X
Pn1−p m ¯ n (p) < ∞ a.s. if p ≥ 2.
n
Z1 − 1|p < ∞, so that As (ξn )n≥0 are i.i.d. and E log m0 ∈ (0, ∞), by Lemma 5.1, E log+ Eξ | m 0 p Z1 E log Eξ m < ∞. 0
17
Proposition 5.1 (Quenched moments of Aˆn ). Let ρ > 1 and m = exp(E log m0 ) > 1. Z1 (i) Let p ∈ (1, 2). If E log+ Eξ | m − 1|r < ∞ and ρ < m1−1/r for some r ∈ [p, 2], then 0
sup Eξ |Aˆn |p < ∞ n
a.s..
(5.3)
Z1 p < ∞ and (5.3) holds, then ρ ≤ m1/2 . − 1| Conversely, if E log Eξ | m 0
Z1 (ii) Let p ≥ 2. If E log+ Eξ | m − 1|p < ∞ and ρ < m1/2 , then (5.3) holds. Conversely, if 0 Z1 − 1|r < ∞ for some r ∈ [2, p] and (5.3) holds, then ρ ≤ m1−1/r . E log Eξ | m 0
Z1 Proof. (i) Let p ∈ (1, 2). Suppose that E log+ Eξ | m − 1|r < ∞ and ρ < m1−1/r for some r ∈ [p, 2]. P pn p(1/r−1) 0 p/r Then by Lemma 5.1, the series n ρ Pn m ¯ n (r) < ∞ a.s.. Thus supn Eξ |Aˆn |p < ∞ a.s. by Proposition 4.1. Z1 p Conversely, suppose that E log Eξ | m0 − 1| < ∞ and supn Eξ |Aˆn |p < ∞ a.s.. By Proposition P −s−p/2 4.1, we have ∀s > 0, n ρpn Pn m ¯ n (p) < ∞ a.s.. Hence by Lemma 5.1, ρ ≤ m1/2+s/p . Letting 1/2 s → 0, we get ρ ≤ m . Z1 − 1|p < ∞ and ρ < m1/2 . Then by Lemma 5.1, the (ii) Let p ≥ 2. Suppose that E log+ Eξ | m 0 P 2n −1 ¯ n (p)2/p < ∞ a.s., which implies that supn Eξ |Aˆn |p < ∞ a.s. by Proposition 4.1. series n ρ Pn m Z1 Conversely, suppose that E log Eξ | m − 1|r < ∞ for some r ∈ [2, p] and supn Eξ |Aˆn |p < ∞ a.s.. 0 P p(1/r−1) Proposition 4.1 shows that n ρpn Pn m ¯ n (r)p/r < ∞ a.s., which implies that ρ ≤ m1−1/r by Lemma 5.1.
By the relations between Aˆn and A(ρ) (defined in (2.8)) shown in Lemma 3.2, together with Proposition 5.1, we immediately obtain: Theorem 5.2 (Quenched Lp convergence of A(ρ)). Let ρ > 1 and m = exp(E log m0 ) > 1. r Z1 (i) Let p ∈ (1, 2). If E log Eξ m < ∞ and ρ < m1−1/r for some r ∈ [p, 2], then 0 the series A(ρ) converges in Lp under Pξ for almost all ξ. p Z1 − 1 Conversely, if E log Eξ m < ∞ and (5.4) holds, then ρ ≤ m1/2 . 0
(5.4)
p Z1 (ii) Let p ≥ 2. If E log Eξ m < ∞ and ρ < m1/2 , then (5.4) holds. Conversely, if 0 r Z1 − 1 < ∞ for some r ∈ [2, p] and (5.4) holds, then ρ ≤ m1−1/r . E log Eξ m 0
In the determinist case, similar results were shown by Alsmeyer et al. (2009, [1]) for branching random walks. Notice that the quenched Lp convergence of A(ρ) implies that ρn (W − Wn ) → 0 in Lp under Pξ . Conversely, ρn (W − Wn ) → 0 in Lp under Pξ implies the quenched Lp convergence of A(ρ1 ) for any ρ1 ∈ (1, ρ). So we can obtain from Theorem 5.2 the following criteria for the quenched Lp convergence rate of Wn : Theorem 5.3 (Exponential rate of quenched Lp convergence of Wn ). Let ρ > 1 and m = exp(E log m0 ) > 1. 18
(i) Let p ∈ (1, 2). If E log Eξ
Z1 Conversely, if E log Eξ m 0
Z1 m0
r
< ∞ and ρ < m1−1/r for some r ∈ [p, 2], then
(Eξ |W − Wn |p )1/p = o(ρ−n ) a.s.. p − 1 < ∞ and (5.5) holds, then ρ ≤ m1/2 .
(5.5)
p Z1 (ii) Let p ≥ 2. If E log Eξ m < ∞ and ρ < m1/2 , then (5.5) holds. Conversely, if 0 r Z1 − 1 < ∞ for some r ∈ [2, p] and (5.5) holds, then ρ ≤ m1−1/r . E log Eξ m 0
Proof of Theorem 1.2. The assertion (a) is a direct consequence of Theorem 5.3(i) with r = p for p ∈ (1, 2) and Theorem 5.3(ii) for p ≥ 2. p∨2 p Z1 Z1 + For the assertion (b), notice that the condition E log+ Eξ m − 1 − 1 < ∞ ensures that E log E < ξ m0 0 2 Z1 − 1 < ∞. If ρ < m1/2 , applying Theorem 5.3(i) with r = 2 for p ∈ (1, 2) and ∞ and E log+ Eξ m 0 Theorem 5.3(ii) for p ≥ 2, we have lim ρn (Eξ |W − Wn |p )1/p = 0
n→∞
a.s..
Now consider the case where ρ > m1/2 . Denote D = {ξ : lim ρn (Eξ |W − Wn |p )1/p = 0}. n→∞
First, we show that P(D) = 0 or 1. By the ergodicity, it suffices to show that T −1 D = D a.s.. By (5.2), Z1 1 X W (1, i) a.s.. W = m0 i=1
Similarly, we can write Wn as
Z1 1 X Wn = Wn−1 (1, i) m0
a.s.,
(5.6)
i=0
n (k,i) where Wn (k, i) = mk Z···m with Zn (k, i) denoting the branching process starting with the ith k+n−1 particle in the kth generation. Under Pξ , the sequence (Wn (k, i))i≥1 are independent of each other and independent of Zk , and have a common conditional distribution Pξ (Wn (k, i) ∈ ·) = PT k ξ (Wn ∈ ·). Therefore, Z1 1 X (W (1, i) − Wn−1 (1, i)) a.s.. (5.7) W − Wn = m0
i=1
By (5.7) and the convexity of
xp ,
we have
Eξ |W − Wn |p ≤
1 Eξ mp0
Z1 X
!p
|W (1, i) − Wn−1 (1, i)|
i=1
Z
1 X 1 p−1 ≤ |W (1, i) − Wn−1 (1, i)|p p Eξ Z1 m0 i=1 Z1 p = Eξ ET ξ |W − Wn−1 |p . m0
19
(5.8)
Therefore for almost all ξ, if T ξ ∈ D, then ξ ∈ D. So we have proved that T −1 D ⊂ D a.s.. On the other hand, notice that by Theorem 1.1, Eξ W = 1 a.s.. Using (5.7) and Burkholder’s inequality, we get p
Eξ |W − Wn |
≥
C Eξ mp0
Z1 X (W (1, i) − Wn−1 (1, i))2 i=1
!p/2
Z
≥
1 X C |W (1, i) − Wn−1 (1, i)|p p Eξ 1{Z1 ≥1} m0
i=1
= C
1 − p0 (ξ0 ) ET ξ |W − Wn−1 |p mp0
a.s..
(5.9)
Notice that p0 (ξ0 ) < 1 since m0 ∈ (0, ∞). It follows from (5.9) that for almost all ξ, if ξ ∈ D, then T ξ ∈ D. Hence D ⊂ T −1 D a.s.. So we have proved that T −1 D = D a.s.. For ρ > m1/2 , assume that P(D) = 1, so that limn→∞ ρn (Eξ |W −Wn |p )1/p = 0 a.s.. Notice that the p∧2 p 2 Z1 Z1 Z1 − − − condition E log Eξ m0 − 1 < ∞ ensures that E log Eξ m0 − 1 < ∞ and E log Eξ m0 − 1 < p 2 Z1 Z1 ∞. So we have E log Eξ m0 − 1 < ∞ and E log Eξ m0 − 1 < ∞. Applying Theorem 5.3(i) for p ∈ (1, 2) and Theorem 5.3(ii) with r = 2 for p ≥ 2, we get ρ ≤ m1/2 . This contradicts the condition that ρ > m1/2 . Thus P(D) = 0, which implies that n p 1/p P lim sup ρ (Eξ |W − Wn | ) > 0 = P(D c ) = 1. n→∞
So the proof is finished.
6
Annealed moments and annealed Lp convergence rate for BPRE; Proof of Theorem 1.5
In this section, we consider a branching process in an i.i.d. environment: we assume that (ξn )n≥0 are i.i.d.. We also assume that P(W1 = 1) < 1, (6.1) which avoids the trivial case where Wn = 1 a.s.. Let us study the annealed moments of Aˆn at first. We shall distinguish three cases: (i) p ∈ (1, 2); (ii) p = 2; (iii) p > 2. Our approach is inspired by ideas from [1], especially for the case where p > 2. r p/r Z1 Proposition 6.1 (Annealed moments of Aˆn for p ∈ (1, 2)). Let p ∈ (1, 2) and ρ > 1. If E Eξ m < 0 p(1/r−1) 1/p )
∞ and ρ(Em0
< 1 for some r ∈ [p, 2], then
sup E|Aˆn |p < ∞.
(6.2)
n
Z1 p ) < ∞ and ρ(Ems0 )−1/2s < 1 for all s > 0, so that ρ ≤ Conversely, if (6.2) holds, then E( m 0 −p/2
exp( 21 E log m0 ); if additionally Em0 1.
−p/2−1
log m0 > 0 and Em0
20
−p/2 1/p )
Z1 log+ Z1 < ∞, then ρ(Em0
0. For α > 0, H¨older’s inequality gives The assumption P(W1 = 1) < 1 ensures that E|X EWnα = EWnα Pn−α Pnα ≤ (EWnαp1 Pn−αp1 )1/p1 (EPnαq1 )1/q1 ,
(6.5)
sp where p1 , q1 > 1 and 1/p1 + 1/q1 = 1. For s > 0, take α = p+2s , p1 = 1 + p/2s and q1 = 1 + 2s/p. Then (6.5) becomes (EWnα )p1 ≤ EWnp/2 Pn−p/2 (Ems0 )pn/2s . (6.6)
Combing (6.6) with (6.4), we get sup E|Aˆn |p ≥ CN p/2−1 n
N −1 X
ρpn (Ems0 )−pn/2s (EWnα )p1
n=0
≥
C(inf EWnα)p1 N p/2−1 n
N −1 X
ρp (Ems0 )−p/2s
n=0
n
.
Hence supn E|Aˆn |p < ∞ implies that ρ(Ems0 )−1/2s < 1 for all s > 0, so that log ρ < all s > 0. Notice that (Ems0 )1/s is increasing as s increases. We have log ρ ≤ inf
s>0
1 1 1 1 log Ems0 = lim log(Ems0 ) = E log m0 , s→0 2s 2 s 2 21
1 2s
log Ems0 for
so that ρ ≤ exp( 21 E log m0 ). −p/2
−p/2−1
If additionally Em0 log m0 > 0 and Em0 Z1 log+ Z1 < ∞, we introduce a new BPRE. Denote the distribution of ξ0 by τ0 . Define a new distribution τ˜0 as τ˜0 (dx) =
m(x)−p/2 τ0 (dx) −p/2
,
Em0
P where m(x) = E[Z1 |ξ0 = x] = ∞ k=0 kpk (x). Consider the new BPRE whose environment distribution ⊗N ⊗N is τ˜ = τ˜0 instead of τ = τ0 . The corresponding probability and expectation are denoted by ˜ = Pξ ⊗ τ˜ and E, ˜ respectively. Then P ˜ p/2 (Em−p/2 )n . EPn−p/2 Wnp/2 = EW n 0
(6.7)
Combing (6.7) with (6.4), we obtain ˜ p/2 N p/2−1 sup E|Aˆn |p ≥ C inf EW n n
n
Notice that
N −1 X
−p/2 n
ρp Em0
n=0
.
˜ log m0 = Em−p/2 log m0 > 0, E 0
and
˜ Z1 log+ Z1 = Em−p/2−1 Z1 log+ Z1 < ∞. E 0 m0 p/2
˜ i.e. P(W ˜ ˜ n Hence W is non-degenerate under P, > 0) > 0 (cf. Tanny, 1988, [18]), so that inf n EW −p/2 1/p p/2 p ˜ ˆ EW > 0. Therefore, supn E|An | < ∞ implies that ρ(Em0 ) < 1.
=
Proposition 6.2 (Annealed moment of Aˆn for p = 2). Let ρ > 1. Then supn E|Aˆn |2 < ∞ if and Z1 2 1/2 < 1. ) < ∞ and ρ(Em−1 only if E( m 0 ) 0 Proof. By Lemmas 3.3 and 4.1, sup E|Aˆn |2 = n
=
∞ X
n=0 ∞ X
ρ2n E|Wn+1 − Wn |2 ¯n − 1|2 ) ρ2n E(Pn−1 Eξ |X
n=0
¯ 0 − 1|2 = E|X
∞ X
ρ2 Em−1 0
n=0
n
.
(6.8)
We get our conclusion immediately from (6.8). Finally, we consider the case where p > 2. We need two lemmas below. Denote un (s, r) = EPn−s Wnr
(s ∈ R, r > 1).
(6.9)
Lemma 6.1. For r > 2, un (s, r) satisfies the following recursive formula: 1
1
1
1
1
r r−1 un−1 (s, r − 1) r−1 . un (s, r) r−1 ≤ (Em01−r−s ) r−1 un−1 (s, r) r−1 + (Em−s 0 W1 )
22
(6.10)
(0)
(k)
Proof. Denote ϕn (t) = ET k ξ eitWn and ϕn (t) = ϕn (t) = Eξ eitWn . By (5.6), we get the functional equation t Z1 (1) ) a.s., ϕn (s) = Eξ ϕn−1 ( m0 By differentiations, this yields Z1 −1 ′ t Z1 t (1) (1) ′ a.s.. (6.11) ) ϕn−1 ( ) ϕn (t) = Eξ ϕn−1 ( m0 m0 m0 Let Vn be a random variable whose distribution is determined by Eξ g(Vn ) = Eξ Wn g(Wn ) (k)
for all bounded and measurable function g, and Vn random variable independent of
(1) Vn−1
(k)
with Pξ (Vn
∈ ·) = PT k ξ (Vn ∈ ·). Let Mn be a
under Pξ , whose distribution is determined by
Z1 −1 Z1 1 X Eξ g(Mn ) = Eξ g( Wn−1 (1, i)), m0 m0 i=0
for all bounded and measurable function g. (The probability space (Γ, Pξ ) can be taken large enough (k) to define the random variables Vn , Vn and Mn .) The Fourier transform of Vn is Eξ eitVn = Eξ Wn eitWn = −iϕ′n (t). So (6.11) implies that (1)
it( m1 Vn−1 +Mn )
Eξ eitVn = Eξ e
0
a.s.,
which is equivalent to the distributional equation d
Vn =
1 (1) V + Mn m0 n−1
under Pξ . Therefore, un (s, r) = EPn−s Wnr = EPn−s Eξ Wnr = EPn−s Eξ Vnr−1 r−1 1 (1) −s = EPn Eξ V + Mn m0 n−1 − s r−1 s − r−1 (1) = E Pn r−1 m−1 . V + P M n n 0 n−1
By the triangular inequality in Lr−1 , un (s, r)
1 r−1
1 1 (1) r−1 r−1 −s 1−r ≤ EPn m0 Vn−1 + EPn−s Mnr−1 r−1 .
(6.12)
We now calculate the two expectations of the right hand side. We have (1) r−1 r−1 EPn−s m01−r Vn−1 = EPn−s m1−r 0 ET ξ Vn−1 r−1 = Em01−r−s EPn−s Vn−1
= Em01−r−s un−1 (s, r), 23
(6.13)
and EPn−s Mnr−1 = EPn−s Eξ Mnr−1 Z1 = EPn−s Eξ m0
Z1 −1 1 X Wn−1 (1, i) m0
i=0 r−1 −s −r r EPn m0 Eξ Z1 ET ξ Wn−1 r Z1 −s r−1 EPn−1 Wn−1 Em−s 0 m0 Z1 r Em−s un−1 (s, r − 1). 0 m0
≤ = =
!r−1
(6.14)
So (6.10) is a combination of (6.12), (6.13) and (6.14). Remark. In particular, un (0, r) = EWnr . By Lemma 6.1, we can obtain the recursive formula for EWnr : 1 1 1 1 r−1 r−1 ) (r > 2). (EWnr ) r−1 ≤ (Em0r−1 ) r−1 + (EW1r ) r−1 (EWn−1
−s Lemma 6.2. Let s ∈ R and r ∈ (b, b+1], where b ≥ 1 is an integer. If Em−s 0 < ∞ and Em0 ∞, then n un (s, r) = O(n1+(b−1)r−(b−1)b/2 (max{ max Em0i−r−s, Em−s 0 }) ).
Z1 m0
r
2). Let p > 2 and ρ > 1. Then supn E|Aˆn |p < ∞ (Annealed p −p/2 Z1 if and only if E m < ∞ and ρ max{(Em01−p )1/p , (Em0 )1/p } < 1. 0
25
Proof. Suppose that E
Z1 m0
p
−p/2 1/p ) }
1/p , (Em < ∞ and ρ max{(Em1−p 0 ) 0 ∞ X
sup E|Aˆn |p ≤ C n
ρ2n (E|Wn+1 − Wn |p )2/p
n=0
!p/2
< 1. By Lemma 3.3,
.
To prove supn E|Aˆn |p < ∞, it suffices to show that ∞ X
ρ2n (E|Wn+1 − Wn |p )2/p < ∞.
n=0
By Lemma 4.1, ¯ n − 1|p E|Wn+1 − Wn |p ≤ CEPn−p/2 Eξ Wnp/2 Eξ |X ¯ 0 − 1|p = EPn−p/2 Wnp/2 E|X = Cun (p/2, p/2). −p/2 Z1 p/2 ( m0 )
Notice that Em0
−p/2
Z1 p < ∞, since E( m ) < ∞, and Em0 0
< 1 < ∞. The remark after
i−p 1≤i≤b+1 Em0 , Notice that Emx0
O(nγ (max{max
−p/2
Lemma 6.2 shows that un (p/2, p/2) = Em0 })n ) for p/2 ∈ (b + 1, b + 2] with γ = 1 for b = 0 and γ = bp/2 for b ≥ 1. is log convex. Therefore we have −p/2 −p/2 max{ max Emi−p }≤ sup {Emx0 } = max{Em01−p , Em0 }. 0 , Em0 1≤i≤b+1
Thus
∞ X
1−p≤x≤−p/2
ρ2n (E|Wn+1 − Wn |p )2/p ≤ C
∞ X
−p/2)2/p
2/p ρ2n n2γ/p (max{(Em1−p , (Em0 0 )
})n .
n=0
n=0
−p/2
1/p , (Em The series in the right side of the above inequality is finite if and only if ρ max{(Em1−p )1/p }< 0 ) 0 1. p Z1 Z1 −1|p = E|Aˆ0 |p < < ∞, since E| m Conversely, assume that supn E|Aˆn |p < ∞. Obviously, E m 0 0 ∞. By Lemma 3.3 and Lemma 4.1, we have ∀r ∈ [2, p],
sup E|Aˆn |p ≥ C n
∞ X
ρpn E|Wn+1 − Wn |p
n=0
≥ C
∞ X
¯ n − 1|r )p/r ρpn EPnp(1/r−1) (Eξ |X
n=0
= C
∞ X
p(1/r−1) n
ρpn (Em0
¯ 0 − 1|r )p/r . ) E(Eξ |X
n=0 p(1/r−1)
1/p , )1/p < 1 holds for all r ∈ [2, p]. Taking r = p, 2, we get ρ max{(Em1−p Thus ρ(Em0 0 ) −p/2 (Em0 )1/p } < 1.
ˆ Again, by relations between A(ρ) and A(ρ) (see Lemma 3.2), it can be seen that the annealed p ˆ moments of An imply the L convergence of A(ρ). More precisely, we have the following theorem about the Lp convergence of A(ρ). Theorem 6.4 (Annealed Lp convergence of A(ρ)). Let ρ > 1. 26
r p/r p(1/r−1) 1/p Z1 < ∞ and ρ(Em0 ) < 1 for some r ∈ [p, 2], then (i) Let p ∈ (1, 2). If E Eξ m 0
the series A(ρ) converges in Lp under P. (6.18) p Z1 Conversely, if (6.18) holds, then E m < ∞ and ρ ≤ exp( 21 E log m0 ); if additionally 0 −p/2
Em0
−p/2−1
log m0 > 0 and Em0
−p/2 1/p )
Z1 log+ Z1 < ∞, then ρ(Em0
< 1.
−p/2 1/p ) }
1/p , (Em (ii) Let p ≥ 2. Then (6.18) holds if and only if ρ max{(Em1−p 0 ) 0
< 1.
p/2 2 −p/2 Z1 Remark. For p ∈ (1, 2), under the conditions that E Eξ m < ∞, Em0 log m0 > 0 and 0 −p/2−1
Em0
−p/2 1/p )
Z1 log+ Z1 < ∞, A(ρ) converges in Lp under P if and only if ρ(Em0
< 1.
Proof of Theorem 6.4. (i) is a consequence of Proposition 6.1, while (ii) comes from Proposition 6.2 (for p = 2) and Proposition 6.3 (for p > 2). Similar to the quenched case, the annealed Lp convergence of A(ρ) implies that ρn (W − Wn ) → 0 in Lp under P. Conversely, ρn (W − Wn ) → 0 in Lp under P implies the annealed Lp convergence of A(ρ1 ) for any ρ1 ∈ (1, ρ). So Theorem 6.5 below is a consequence of Theorem 6.4. Theorem 6.5 (Exponential rate of annealed Lp convergence of Wn ). Let ρ > 1. r p/r p(1/r−1) 1/p Z1 < ∞ and ρ(Em0 ) < 1 for some r ∈ [p, 2], then (i) Let p ∈ (1, 2). If E Eξ m 0 (E|W − Wn |p )1/p = o(ρ−n ).
(6.19) −p/2
Conversely, if (6.19) holds, then ρ ≤ exp( 12 E log m0 ); if additionally Em0 −p/2−1 Em0 Z1 log+ Z1
< ∞, then
−p/2 ρ(Em0 )1/p
≤ 1. −p/2 1/p ) }
Z1 p (ii) Let p ≥ 2. If E( m ) < ∞ and ρ max{(Em01−p )1/p , (Em0 0
Conversely, if (6.19) holds, then
log m0 > 0 and
1/p , (Em−p/2 )1/p } ρ max{(Em1−p 0 ) 0
< 1, then (6.19) holds.
≤ 1.
p Note that ρpn E|W − Wn |p → 0 implies that ∀ρ1 ∈ (1, ρ), ρpn 1 Eξ |W − Wn | → 0 a.s. by BorelCantelli’s lemma and Markov’s inequality. So under the conditions of Theorem 6.5, we can also obtain (5.5). However, by Jensen’s inequality, it can be seen that the conditions of Theorem 6.5 are stronger than those of Theorem 5.3.
The proof of Theorem 1.5 is now easy: Proof of Theorem 1.5. Theorem 1.5 is a direct consequence of Theorem 6.5: taking r = p in Theorem 6.5 gives (a), and taking r = 2 yields (b).
27
References [1] G. Alsmeyer, A. Iksanov, S. Polotsky, U. R¨ osler, Exponential rate of Lp convergence of instrinsic martingales in supercritical branching random walks. Theory Stoch. Process 15 (2009), 1-18. [2] S. Asmussen, Convergence rates for branching processes. Ann. Proba. 4 (1976), 139-146. [3] S. Asmussen, H. Hering, Branching processes, Birkh¨auser, Boston, 1983. [4] K. B. Athreya, S. Karlin, On branching processes in random environments I & II. Ann. Math. Statist. 42 (1971), 1499-1520 & 1843-1858. [5] K. B. Athreya, P. E. Ney, Branching Processes. Springer, Berlin, 1972. [6] J. D. Biggins, The central limit theorem for the supercritical branching random walk, and related results. Stoch. Proc. Appl. 34 (1990), 255-274. [7] J. D. Biggins, J.C. D’Souza, The Supercritical Galton-Watson process in varying environments. Stoch. Proc. Appl. 42 (1992), 39-47. [8] Y. S. Chow, H. Teicher, Probability theory: Independence, Interchangeability and Martingales. Springer-Verlag, New York, 1988. [9] A. K. Grintsevichyus, On the continuity of the distribution of a sum of dependent variable connected with indepentdent walks on lines. Theory Prob. Appl. 19 (1974), 163-168. [10] Y. Guivarc’h, Q. Liu, Propri´et´es asymptotiques des processus de branchement en environnement al´eatoire. C. R. Acad. Sci. Paris, Ser I. 332 (2001), 339-344. [11] C. Huang, Q. Liu, Moments, moderate and large deviations for a branching process in a random environment. Preprint (2010). [12] C. Huang, Q. Liu, Convergence rates for a branching process in a random environment. Preprint (2010). [13] P. Jagers, Galton-Watson processes in varying environments. J. Appl. Prob. 11 (1974), 174-178. [14] Q. Liu, On generalized multiplicascades. Stoc. Proc. Appl. 86 (2000), 263-286. [15] Q. Liu, Local dimensions of the branching measure on a Galton-Watson tree. Ann. Inst. H. Poincar´e. Probabilit´es et Statistiques 37 (2001), 195-222. [16] W. L. Smith, W. Wilkinson, On branching processes in random environments. Ann. Math. Statist. 40 (1969), 814-827. [17] D. Tanny, Limit theorems for branching processes in a random environment. Ann. Proba. 5 (1977), 100-116. [18] D. Tanny, A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means. Stoch. Proc. Appl. 28 (1988), 123-139.
28
