Aug 17, 2001 - Eexp{itY }. A random variable Y belongs to the domain of attraction of a stable law of index 1 < α ⤠2 if and only if lnÏY (t) has the form (see [11] ...
The Rate of Convergence for Weighted Branching Processes∗ Uwe R¨osler Christian-Albrechts-Universit¨at zu Kiel Valentin Topchii Omsk Branch of Sobolev Institute of Mathematics, Omsk Vladimir Vatutin Steklov Mathematical Institute, Moscow August 17, 2001
Abstract Zn Let the martingale Wn = m n , where Zn is a weighted branching P process and m = E i Ti is the expected sum of the = factors Ti , converge to some limiting random variable W . We give conditions in terms of the factors such that W belongs to the domain of attraction or to the domain of normal attraction of an = α–stable distribution with 1 < α ≤ 2. Further the rate of convergence of Wn to W is evaluated in the sense Wn − W correctly normalized converges to a nondegenerate random variable.
1
Introduction
One of the basic facts for supercritical Galton-Watson processes Zn , n = Zn to 0, 1, . . . is the almost everywhere convergence of the martingale Wn = m n a limit W [1, 2]. Supported by grants RFBR 99-01-00012, 99-01-01130, RFBR-DFG 98-01-04132, DFG436RUS113/5/0, INTAS-99-01317 ∗
1
The random variable W (given Z0 = 1) satisfies the equation ˆ
d
W =
Z X W (j)
j=1
(1)
m
where W (j) are independent copies of W and Zˆ is = independent of W (j) and has the same distribution as the offspring Z1 of a particle. In addition, the convergence rate of Wn to W as n → ∞ is of the order −n m 2 . More precisely, for finite variance σ 2 := V arZ1 < ∞ the random variable s m2 − m Gn = m n (Wn − W ) σ2 converges in distribution to a limiting variable G (see [9] or [10] Theorem A). Here G is a mixture of normal distributions with mean 0 and the random variance W is described by the characteristic function t2 W E exp{itG} = E exp − 2 (
)
.
To formulate the up-to-date results in the case σ 2 = ∞ we recall some basic definitions and facts related with stable laws. df We denote the characteristic function of a random variable Y by ϕY (t) = E exp{itY }. A random variable Y belongs to the domain of attraction of a stable law of index 1 < α ≤ 2 if and only if ln ϕY (t) has the form (see [11], Chapter II, Section 6) ln ϕY (t) = iγt − |t|α N (t)B(t)
(2)
as t → 0. Here N (t) is a positive function which is slowly varying (in the sense of Karamata [11, 17]) as t → +0, N (t) ∼ N (−t) as t → 0, and Ã
!
πα t B(t) = c 1 − iβ tan , |t| 2 df
where γ, c, β are real constants with c > 0 and |β| ≤ 1. In case α = 2 is β = 0 required. If the limit N = lim N (t) ∈ (0, ∞) in (2) exists then Y is said to belong t→0 to the domain of normal attraction of an α–stable law. 2
In both cases for Yi , i ∈ IN being iid copies of Y there exists a monotone sequence bn such that n 1 X (Yi − EY ) bn i=1
converges in distribution to a stable law with characteristic function ϕ(t) = exp{−|t|α B(t)}. Now we return to the supercritical Galton-Watson processes and take Y = Z1 ). It is known that in general Z1 belongs to the domain of attraction of a stable law of index 1 < α ≤ 2 if and only if W belongs to the same domain (see [10] Theorems 1 and 2). If this is the case, the rate of convergence is of n 1 order (mα − m) α bm n , or, more precisely, [m ]
1
Gn = (mα − m) α
mn (Wn − W ) b[mn ]
converges in distribution to a limiting random variable G, where G is a mixture of α−stable distributions with characteristic function EϕW (t). Our aim is to prove similar results for the weighted branching processes [15, 16]. We briefly recall the definition of weighted branching processes. S n Let V = ∞ n=0 IN be the set of finite words v = (v1 , v2 , . . . , vn ) over the alphabet of natural numbers. The length n of the word v is denoted by |v|, and sometimes is called the generation. We consider V as a rooted tree with root ∅. By means of V we construct a space of elementary outcomes Ω, where an elementary outcome ω ∈ Ω is a result of assigning to each knot v ∈ V a vector t(v) = (t1 (v), t2 (v), . . .), tj (v) ∈ IR, and take the natural σ-algebra F on Ω. Now we define a probability measure P on F by assigning to each vertex v ∈ V random variables T(v) with values in RIN . The vector T(v, ω) = T(v) = (T1 (v), T2 (v), . . .), respectively, the coordinates Tj (v), j ∈ IN are called the factors. We use T = (T1 , T2 , . . .) for the coordinates, suppressing the ω and the v if possible. Notice that we allow arbitrary dependence of the factors Tj (v), j ∈ IN for fixed knot v. We skip the detailed construction of the resulting probability space (Ω, F, P since it can be done by the standard procedure. The (random) weight L(v) or length of a knot v ∈ V is recursively given by L(∅) = 1 L(vi) = L(v)Ti (v), 3
which is the weight L(v) of the mother v times a random factor Ti (v). Let df
df
Z0 = L(∅) = 1,
Zn =
X
L(v) =
|v|=n
X
L(v)
X
Tj (v)
j
|v|=n−1
be the total weight in n−th generation and let df
m = EZ1 = E
X
Tj 6= 0.
j
Zn is a martingale which converges (under One can check that Wn = m n mild conditions see Lemma 5 and Corollary 6 below) to W . Observe, that X Tj
W1 =
j
m
.
For several models of ordinary (non-weighted) branching processes Zn the problem of the relationship between the asymptotic behavior of the tail of the distribution of the number of children of a single particle Z1 and that of the distribution of the appropriate limiting variable W has been studied in [5, 6, 10, 14]. Here we restrict ourselves to the case of stable distributions with exponent 1 < α ≤ 2 and generalize the mentioned convergence results due to Heyde [10] in several directions. In the sequel we shall use the representations ln ϕW1 (t) = it − |t|α N (t)B(t),
(3)
ln ϕW (t) = it − |t|α M (t)A(t)
(4)
with the obvious meaning. These equations hold true in a neighborhood −C ≤ t ≤ C of 0. Introduce the notation df
m− (α) = E
X
|Tj |α 11Tj 0 ,
j
|Tj |α ,
df
∆m(α) = m+ (α) − m− (α).
j
Our first main result is the following statement. 4
Theorem 1 Let 1 < α ≤ 2 and assume m 6= 0, m(1) < ∞, m(α) < |m|α and Wn converges to W in L1 . i) Suppose there exist constants 0 < C1 < C2 < ∞ such that C1 < |Tj | < C2
(5)
for all Tj 6= 0. Then W belongs to the domain of attraction of a stable law of index α if and only if Z1 belongs to the domain of attraction of a stable law of the same index α. In addition, in this case N (t) ∼ M (t) as t → 0 and Ã
πα t B(t) = c 1 − iβ tan |t| 2
!
(6)
t |m|α − m(α) πα c|m|α 1 − iβ . tan A(t) = |m|α − m(α) |t| |m|α − ∆m(α) 2 Ã
!
(7)
ii) Assume there exists an α1 > α such that m(α1 ) < |m|α1 .
(8)
Then W belongs to the domain of normal attraction of a stable law of index 1 < α ≤ 2 if and only if Z1 belongs to the domain of normal attraction df of a stable law of the same index α. In addition, in this case N = lim N (t) = t→0
df
M = lim M (t) and representations (7) and (6) are true. t→0
Observe that under the conditions of Theorem 1 inequality (5) implies m(α) < ∞ for all α > 0 and, as we show below there exists an α1 > α such that relation (8) holds. There is a well known connection for a distribution belonging to the domain of attraction of an α−stable distribution and the tail behavior of the distribution ([11], Chapter II, Section 6). Using this connection it is not difficult to establish by direct calculations the validity of the following corollary being an extension and generalization of Theorem 2 in [6] which was stated for branching random walks. Corollary 2 Under the basic conditions of Theorem 1 and the validity of one of the restrictions (5) or (8) the following representations are equivalent for 1 < α < 2 : P(W1 < −x) =
p1 + o(1) L(x), xα
P(W1 > x) = 5
p2 + o(1) L(x) xα
as x → ∞ with p1 , p2 ≥ 0, p1 + p2 > 0, and P(W < −x) =
q1 + o(1) L(x), xα
P(W > x) =
q2 + o(1) L(x) xα
as x → ∞ with q1 , q2 ≥ 0, q1 + q2 > 0. Here L(x) is a slowly varying function at ∞ in the sense of Karamata if (5) is true and, in addition, lim L(x) = L ∈ (0, ∞) if (8) is valid. x→∞ Further, in this case p1 + p2 p1 − p2 + |m|α , α 2(|m| − m(α)) 2(|m|α − ∆m(α)) p1 − p2 p1 + p2 α − |m| . = |m|α 2(|m|α − m(α)) 2(|m|α − ∆m(α))
q1 = |m|α q2
Remark 3 This corollary complements a result of Liu [13] in which the case of nonnegative Tj was investigated and where (in our terms) under the assumption that there exists an χ such that E
X
Tjχ = 1,
E
X
Tjχ log+ Tj < ∞,
j
j
χ X E Tj < ∞
j
it is shown that lim xχ P(W > x) exists and is strictly positive. x→∞ P χ Observe that under our conditions if E j Tj = 1 for some χ > α, then ´χ ³P = ∞. And this is an essential difference with [13]. Indeed, in our E j Tj paper the assumption that the respective functionals of T belong to a stable distribution means that there is a large number of Tj 6= 0 each of which is bounded from above and is separated from zero, while in [13] the number of Tj 6= 0 may be finite or infinite but for a fixed N0 the tail of the distribution P of the sum Tj should be ”heavy”. j 0 in representation (3). Suppose t→0 that 1) m 6= 0, m(1) < ∞ and m(α) < |m|α ; 2) Wn converges in L1 to W ; 3) there is an α1 > α such that (8) is true; 4) the condition X
E
|Ti |α ln |Ti |α < m(α) ln m(α)
(9)
i
is valid. Then c i) there exists a sequence of constants {cn } such that n+1 cn → 1 as n → ∞ and Wn (α) → Y (10) cn n→∞ in probability to a random variable Y which is strictly positive whenever the process survives; ii) for any t ∈ (−∞, ∞) n
o
lim E exp it (mn (α)cn )−1/α (W − Wn ) n→∞ (
Ã
t c πα 1 − iβ ∗ tan = E exp −|t| N Y α |m| − m(α) |t| 2 α
where ∗ df
β =
(
1 0
m− (α) = 0, m− (α) 6= 0.
if if 7
!)
,
For the Galton-Watson process the factors Tj are 0 or 1 and the weights L(v) are 0 or 1. Since the Galton-Watson dies out almost everywhere as m ≤ 1 only the supercritical case m > 1 is interesting. This implies our condition m(α) ≡ m < |m|α . The factors are bounded below and above. Our Theorem 1 contains convergence in domains of attractions for the GaltonWatson process. It is necessary to note that for Galton-Watson processes we always have β ∗ = 1 and m− (α) = 0 while the case m− (α) > 0 has not been studied before and gives essentially new results. Observe that (irrespectively of the validity of the inequality m(α) < 1) P P 00 00 m (α) = (E j |Tj |α ) = α2 E j |Tj |α ln2 |Tj | > 0 for α ≥ 1 and, therefore, 0 m (α) is an increasing function in α. One of the key problems of the given paper is to investigate the conver0 gence rate of Wn to W . If all Tj are nonnegative then the condition m (1) < 0 is a necessary one for the convergence of Wn to W with W 6= 0 by Biggins’ criterion (see Theorem 7). To get the convergence of Wn (α) to an W (α) with W (α) 6= 0 we need 0 (again by Biggins’ criterion) m (α) < 0 that is under our conditions we always have m(α) < m = m(1) = 1 if Tj ≥ 0 for all j.
2
Auxiliary results
We say that a random variable X satisfies the fixed point equation with factors T = (T1 , T2 , . . .) if d X Tj Xj , (11) X= j
where Xj are independent copies of X and T, Xj , j ∈ IN are independent. The right-hand side is understood in the sense
n P
j=1
Tj Xj converges as n → ∞
in distribution. In our setting this is the same as the pointwise limit of the sum without the expectation. The fixed point equation (11) is equivalent to ϕX (t) = E
Y
ϕX (tTj )
j
for all t ∈ IR. Here the right-hand side is understood as the limit of n Q E ϕX (tTj ) as n tends to infinity. In our setting this will be the same j=1
8
as the pointwise limit of the product
Q j
ϕX (tTj ) and then taking expectation.
Lemma 5 Assume m(1) < ∞ and E|X| < ∞. Then the sum
P j
Tj Xj , where
Xj are independent copies of X and T is independent of the X random variables, is absolute convergent. The characteristic function
n Q
j=1
verges as n tends to ∞ to the characteristic function ments are in the sense P T almost everywhere. Proof. By the imposed strong assumptions
n P
j=1
Q j
ϕX (tTj ) con-
ϕX (tTj ). Both state-
|Tj Xj | is a Cauchy se-
quence and converges pointwise almost everywhere. Argue for m < n E
n X
|Tj Xj | ≤ E|X|E
j=m
X
j≥m
|Tj | m→∞ → 0
uniformly in n. P For fixed T = s ∈ IRIN the sum si Xi is almost everywhere absolute convergent. Therefore,
for all t ∈ IR.
n Q
j=1
i
ϕX (tsj ) converges to
Q j
ϕX (tsj ) almost everywhere q.e.d.
Corollary 6 Assume 0 < m(1) < ∞ and sup E|Wn | < ∞. Then the limiting n
random variable W satisfies the fixed point equation (11) with factors T/m. The ordinary Galton-Watson process is a special case of the weighted branching process with factors in the set {0, 1}. The sequence {Wn } is a positive martingale and converges according to Doob almost everywhere to W . A theorem by Kesten-Stigum states that EW is either 1 or 0. The equality EW = 1 is equivalent to the so-called L log L condition E(Z1 ln Z1 ) < ∞. (We use 0 ln 0 = 0.) In both cases W satisfies the fixed point equation for the factor T . The branching random walk is a special case of the weighted branching process with finitely many positive factors (see [4] and [15] for more details). Again {Wn } is a positive martingale and converges according to Doob almost everywhere to W . The limit W satisfies the fixed point equation of the form (11). The problem of L1 convergence for branching random walk is settled in the following theorem. 9
Theorem 7 [4] Wn (α) converges to some random variable called W (α) in L1 if and only if EW1 (α) ln W1 (α) < ∞ (12) and inequality (9) is true. If only the inequality (9) is satisfied then there exists a sequence of conc stants {cn } such that n+1 cn → 1 as n → ∞ and Wn (α) → Y cn n→∞ in probability to a random variable Y , which is strictly positive whenever the process survives. Remark 8 Note that under condition (12) the second part of Theorem 7 is valid for cn ≡ 1 and X = W (α). It will be convenient for us to use equation (10) in both cases.
3
Domain of attraction of an α−stable law
According to Lemma 5 and Corollary 6 W satisfies the fixed point equation d
W =
X Tj
m
j
W (j)
(13)
where W (j) are independent copies of W and T = (T1 , T2 , . . .) is independent of W (j) . Observe that equation (1) is a particular case of (13). An equivalent description via characteristic functions is ϕW (t) = E
Y
µ
ϕW t
j
Tj . m ¶
We consider here only the case m 6= 0. The fixed point equation, assuming EW = 1, rewrites d
W −1=
X Tj ³ j
m
´
W (j) − 1 +
X Tj j
In terms of characteristic functions
ϕW −1 (t) = ϕW1 −1 (t) + E exp{it(W1 − 1)} 10
Y j
m
− 1 .
ϕW −1
µ
(14)
Tj − 1 . t m ¶
(15)
If we use the fixed point equation (14) recursively then we obtain the solution d X L(v) W −1= (W1 (v) − 1), (16) |v| v∈V m P Tj (v)
df
where the random variables W1 (v) =
m
j
knots v.
are independent for different
Proposition 9 Let m(α) < ∞. Then E
X
|L(v)|α
|v|=n
L(v) (∆m(α))n |L(v)|
and the sequence df
ξn = (∆m(α))−n
X
|L(v)|α
|v|=n
L(v) |L(v)|
is a martingale with respect to the sequence of σ-algebras Fn = σ{L(v), |v| ≤ n}. Proof. The proof runs by induction. We show the induction step. E
X
|v|=n+1
|L(v)|α
X X L(v) L(v)Tj (v) |L(v)Tj (v)|α = E |L(v)| |L(v)Tj (v)| |v|=n j
= E
X
|L(v)|α
|v|=n
= ∆m(α)E
X
|v|=n
L(v) X Tj (v) |Tj (v)|α |L(v)| j |Tj (v)| |L(v)|α
L(v) = (∆m(α))n+1 . |L(v)|
The second statement of the proposition easily follows from the first one. q.e.d. We now prove our main Theorem 1. Proof of Theorem 1: In order to avoid unnecessary writing we always assume m = 1 and write Zn for Wn and Z for W . We use the representation (3) and (4). First we show that if Z belongs to the domain of attraction of an α-stable law 1 < α ≤ 2 then Z1 also possesses this property. 11
Using (15) for W = Z we have
Y ϕZ1 −1 (t) = ϕZ−1 (t) − E exp{it(Z1 − 1)} ϕZ−1 (tTj ) − 1 . j
Since the factors |Tj | are uniformly bounded from above we have for |t| sufficiently small ϕZ1 −1 (t) = ϕZ−1 (t) − E(exp{it(Z1 − 1)}(ea − 1)) with
df
a = −|t|α
X
|Tj |α A(tTj )N (tTj ).
j
Denoting df
b = −|t|α N (t)
X
|Tj |α A(tTj )
j
we obtain the identity ϕZ1 −1 (t) = ϕZ−1 (t)eEb + (ϕZ−1 (t) − 1)(1 − eEb ) + (Eea − eEb ) df
+E [(exp{it(Z1 − 1)} − 1)(ea − 1)] = Q(1) + Q(2) + Q(3) + Q(4).
(17)
We establish now a series of partial claims marked by •, which all together imply the main first statement. • Q(1) = exp{−|t|α B(t)N (t)}, where B(t) is given in (6). Denote τ = |t|α N (t). By Proposition 9 and Definition (7) we have (recall that we have selected m = 1) −τ −1 ln Q(1) = A(t) − E
X
|Tj |α A(tTj )
j
1−E = c
P j
|Tj |α
1 − m(α)
t − icβ |t|
Tj |T πα j| j tan = B(t). 1 − ∆m(α) 2
1−E
P
|Tj |α
• τ −1 Q(2) → 0. t→0 We will use |ez − 1| ≤ |z| for a complex number such that Re z < 0. 12
(18)
From (18) for z = Eb we have in view of the continuity of φZ−1 (t) at t = 0 τ −1 |Q(2)| ≤
c m(α)|ϕZ−1 (t) − 1| = o(1). 1 − m(α)
• τ −1 Q(3) → 0. t→0 It is necessary to show that Eea − exp{τ Eη} →0 τ &0 τ df
where η = − j |Tj |α A(Tj t) is a complex random variable with Re η < 0. Note that |η| does not depend on t while η depends on the sign of t ∈ IR only. More precisely, when t → −t the only changes in η are related with A: df A(−t) = A(t), |A(t)| = |A(1)| = C3 . To establish the desired convergence we use a standard argument. Fix some constant d. The left-hand side of the relation under investigation is equal to P
1 − exp{τ Eη} ea − exp{τ η} + E 11|η|
