Critical Galton-Watson processes: The maximum of total progenies within a large window

0.34 MB·English

by  Klaus Fleischmann

Critical Galton-Watson processes:
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0 The maximum of total progenies
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n within a large window
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Klaus Fleischmann∗, Vladimir A.Vatutin+,

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and Vitali Wachtel∗
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1991 Mathematics Subject Classification 60J80, 60F17
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X
Key words and phrases Branching of index one plus alpha, limit theorem, con-
r ditional invariance principle, tail asymptotics, moving window, maximal total
a
progeny,lower deviation probabilities
Runninghead Totalprogenyinawindow
Correspondingauthor V.A.Vatutin
WIASpreprintNo.1091ofJanuary13,2006; ISSN 0946-8633; win21.tex
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∗
SupportedbytheDFG
+
Supported by the program “Contemporary problems of theoretical mathematics” of RAS
andthegrantsINTAS03-51-5018andRFBR05-01-00035
Totalprogenyinawindow 1
Abstract
ConsideracriticalGalton-Watsonprocess Z ={Zn : n=0,1,...}
of index 1+α, α ∈ (0,1]. Let Sk(j) denote the sum of the Zn
with n inthewindow [k,...,k+j), and Mm(j) themaximumof
the Sk(j) with k movingin [0,m−j]. Wedescribetheasymptotic
behavioroftheexpectation EMm(j) ifthewindowwidth j =jm is
suchthat j/m convergesin [0,1] as m↑∞. Thiswill beachieved
via establishing the asymptotic behavior of the tail probabilities of
M∞(j).
Contents
1 Introductionand statement ofresults 1
2 Auxiliarytools 3
2.1 Basicpropertiesofcriticalprocessesofindex1+α. . . . . . . . 3
2.2 Basicpropertiesofcriticalprocesses . . . . . . . . . . . . . . . . 6
2.3 Aconditionalinvarianceprinciple . . . . . . . . . . . . . . . . . 11
2.4 OnthelimitingprocessX∗ . . . . . . . . . . . . . . . . . . . . . 14
3 Proofofthe main results 18
3.1 ProofofTheorem2 . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 ProofofTheorem1(a) . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 ProofofTheorem1(b) . . . . . . . . . . . . . . . . . . . . . . . 24
1 Introduction and statement of results
Let Z = {Z : n ≥ 0} denote a Galton-Watson process. As a rule, we
n
start with a single ancestor: Z = 1. It will be convenient to write ξ for
0
the intrinsic number of offspring Z . We always assume that Z is critical,
1
that is Eξ = 1. If not stated otherwise, we consider the case of branching
of index 1+α for some 0 < α ≤ 1. With this we mean that the related
offspring generating function f satisfies
f(s) := Esξ = s+(1−s)1+αL(1−s), 0≤s≤1, (1)
where x 7→ L(x) is a function slowly varying as x ↓ 0. For k ≥ 0 and
1≤j ≤m<∞, set
k+j−1
S (j) := Z and M (j) := max S (j). (2)
k l m k
0≤k≤m−j
l=k
X
Extendthesenotationsbymonotoneconvergenceto m=∞ oreven j =∞,
and put
M(j):=M (j), 1≤j ≤∞. (3)
∞
2 K.Fleischmann,V.A.VatutinandV.Wachtel
Since any criticalGalton-Watsonprocess dies a.s. in finite time, M(j) is a
proper random variable for any j. In particular, M(∞) coincides with the
total number S (∞)=Z +Z +··· of individuals of Z.
0 0 1
The main purpose of this note is to study the asymptotic behavior of
the expectation EM (j) when j might depend on m such that j/m →
m
η ∈[0,1] as m↑∞.
To getafeeling, letus firstdiscuss twospecialcases. If j =m, we have
m−1
EM (m) = ES (m) = E Z = m. (4)
m 0 l
l=0
X
Ontheotherhand,thecase j =1 reducestotheinvestigationoftheasymp-
totic behavior of the expectation of M (1) =: M = max Z as
m m 0≤k≤m−1 k
m ↑ ∞. The last issue has a rather long history. First Weiner [Wei84]
demonstratedthatifthecriticalprocesshasafinitevariance[whichrequires
α = 1 in our case (1)], then there exist constants 0 < c ≤ c¯ < ∞ such
that c ≤ EM /logm ≤ c¯ for all m. Then K¨ammerle and Schuh [KS86]
m
and Pakes [Pak87] have found explicit bounds for c from below and for c¯
from above. Finally, Athreya [Ath88] established (still under the condition
Varξ <∞) that
EM (1) = EM ∼ logm as m↑∞. (5)
m m
In Borovkov and Vatutin [BV96] the validity of (5) was proved under con-
dition (1). Moreover, in Vatutin and Topchii [VT97] and Bondarenko and
Topchii [BT01] asymptotics (5) was established under much weaker condi-
tions than (1), for instance, in [BT01] under Eξlogβ(1+ξ) < ∞ for any
β >0.
Comparing the difference of orders at the right-hand sides of (4) and
(5) leads to the following natural question: What can be said about the
behaviorofEM (j)whenthewidth j ofthemovingwindowwithinwhich
m
the total population size is calculated, may vary anyhow with m. For this
purpose, we restrict our attention to processes satisfying (1). Here is our
main result.
Theorem 1 (Expected maximal total progeny) Assume that j ≥1
m
satisfies j /m→η ∈[0,1] as m↑∞.
m
(a) If η =0, then
m
EM (j ) ∼ j log as m↑∞. (6)
m m m
j
m
(cid:16) (cid:17)
(b) If 0<η ≤1, then
EM (j ) ∼ j ϕ(η) as m↑∞, (7)
m m m
where ϕ is explicitly given in formula (177) below. In particular,
1
ϕ(η) ∼ log as η ↓0. (8)
η
Totalprogenyinawindow 3
Note that (8) yields a continuous transition between the cases (a) and
(b).
We will deduce Theorem 1 via studying finer properties of M(j) =
M (j). In fact, we will establish the following asymptotic representation
∞
for tail probabilities of M(j). As usual, we write Q(n) for the survival
probability P(Z >0).
n
Theorem 2 (Tail of maximal total progeny) Assumethat j ≥1 sat-
n
isfies j / Q(j )n →y ∈[0,∞] as n↑∞.
n n
(a) If y(cid:0)=∞, t(cid:1)hen
P M(jn)≥n ∼ P M(∞)≥n ∼ n−1+1α ℓ(n) as n↑∞, (9)
(cid:0) (cid:1) (cid:0) (cid:1)
where ℓ is a function slowly varying at infinity.
(b) If 0<y <∞, then
P M(j )≥n ∼ Q(j )ψ(y) as n↑∞, (10)
n n
where ψ is expl(cid:0)icitly given (cid:1)in formula (139) below.
(c) Finally, if y =0, then
αj
P M(j )≥n ∼ P M(1)≥nj−1 ∼ n as n↑∞. (11)
n n n
(cid:0) (cid:1) (cid:0) (cid:1)
The rest of the paper is organized as follows. In the next two subsec-
tions, we state some (partially known)properties ofcriticalGalton-Watson
processes, preparing for the proof of parts (a) and (c) of Theorem 2, given
inSubsection3.1. This is followedin2.3 by aconditionalinvarianceprinci-
ple for critical Galton-Watson processes of index 1+α, see Proposition13,
needed for the proof of Theorem 2(b) (also given in 3.1). Properties of the
limit process X∗ arising in the mentioned invariance principle, are stud-
ied in 2.4 and applied as Proposition 14 in the proof of Theorem 1(b) in
Subsection 3.3.
2 Auxiliary tools
2.1 Basic properties of critical processes of index 1+α
Westartwithsomefurthernotationalconventions. Ifsymbols L and ℓ [as
in(1)and(9),respectively]haveanindex,theyalsodenotefunctionsslowly
varyingatzeroorinfinity,respectively. Inthiscase,theindexmightreferto
thefirstplaceofitsoccurrence,forinstance, ℓ foroccurringinLemma#.
#
Furthermore,theletter c willalwaysdenotea(positiveandfinite)constant,
which might change from place to place, except it has an index, which also
might refer to the place of first occurrence. We will also use the following
4 K.Fleischmann,V.A.VatutinandV.Wachtel
convention: If a mathematical expression (as Z ) is defined only for an
n
integer (here n), but we write a non-negative number in it instead (as Z ),
x
then actually we mean the integer part of that number (here Z ).
[x]
Now we collect some basic properties of critical processes under our
assumption (1). The first lemma is taken from Vatutin [Vat81, Lemma 1].
Lemma 3 (Asymptotics of f′) For f from (1) we have
1−f′(s) ∼ (1+α)(1−s)αL(1−s) as s↑1. (12)
The next lemma is due to Slack [Sla68].
Lemma 4 (Asymptotics of survival probability) As n↑∞,
αQα(n)L Q(n) ∼ 1/n, (13)
implying (cid:0) (cid:1)
Q(n) ∼ n−1/αℓ4(n) (14)
for a function ℓ4 (slowly varying at infinity).
Set f (s) := s, and f (s) := f(f (s)), n ≥ 1, for the iterations of
0 n n−1
f. The following lemma can be considered as a local limit statement.
Lemma 5 (A local limit statement) As n↑∞,
n−1
dn := f′ fk(0) ∼ n−1−1/αℓ5(n). (15)
k=1
Y (cid:0) (cid:1)
Proof It follows from Lemmas 3 and 4 that
1+α
1−f′ 1−Q(k) = 1+δ(k) , (16)
αk
(cid:0) (cid:1) (cid:0) (cid:1)
where δ(k)→0 as k ↑∞. Hence,
n−1 n−1
1+α 1
d = exp logf′ f (0) = exp − 1+δ (k)
n k 1
α k
hkX=1 (cid:0) (cid:1)i h Xk=1 (cid:0) (cid:1)i
n−1
1+α δ (k)
∼ n−1+αα e−1+ααγexp − 1 as n↑∞, (17)
α k
h kX=1 i
where γ is Euler’s constant, and also δ (k) → 0 as k ↑ ∞. According to
1
Seneta [Sen76, Theorem 1.2 in Section 1.5], the function
n−1
1+α δ (k)
1
n 7→ exp − (18)
α k
h kX=1 i
is slowly varying at infinite. Combining this with (17) proves the lemma.(cid:3)
The next statement might also be known from the literature. Recall
that M(∞)=S (∞).
0
Totalprogenyinawindow 5
Lemma 6 (Maximal total population) As n↑∞,
P M(∞)≥n = P S0(∞)≥n ∼ n−1+1α ℓ6(n). (19)
Proof As wel(cid:0)l-known (se(cid:1)e, for in(cid:0)stance, Har(cid:1)ris [Har63, formula (1.13.3)]),
h(s):=EsS0(∞) solves the equation
h(s) = sf h(s) , 0≤s≤1. (20)
By assumption (1) we have (cid:0) (cid:1)
1+α
h(s) = sh(s) + 1−h(s) L 1−h(s) , (21)
giving in view of h(1)=1, (cid:0) (cid:1) (cid:0) (cid:1)
1+α
1−h(s) L 1−h(s) ∼ (1−s) as s↑1. (22)
Hence (cf. [Se(cid:0)n76, Sect(cid:1)ion 1.5(cid:0)]), (cid:1)
1
1−h(s) ∼ (1−s)1+α L(23)(1−s) as s↑1, (23)
and
1−h(s) = ∞ P S (∞)>n sn ∼ L(23)(1−s) as s↑1, (24)
1−s 0 (1−s)1+αα
n=0
X (cid:0) (cid:1)
implying (see, for instance, Feller [Fel71, Section XIII]),
1
P S0(∞)≥n ∼ Γ α n−1+1α L(23)(1/n) =: n−1+1α ℓ6(n) (25)
1+α
(cid:0) (cid:1)
as n↑∞. This finishes th(cid:0)e pro(cid:1)of. (cid:3)
Lemma 7 (Some tail asymptotics) The following statements hold.
(a) As n↑∞, if j ≥1 satisfies j / Q(j )n →∞, then
n n n
Q(j )(cid:0) (cid:1)
n
→ 0. (26)
P M(∞)≥n
(cid:0) (cid:1)
(b) As j ↑∞,
1
α1+α
P M(∞)≥j/Q(j) ∼ Q(j). (27)
Γ α
1+α
(cid:0) (cid:1)
(cid:0) (cid:1)
Proof (a) Recalling notation h introduced in the beginning of the proof
of Lemma 6, set b := 1−h(1−1/x), x ∈ [1,∞). As x ↑ ∞, it follows
x
from (22) that
b1+αL(b ) ∼ 1/x, (28)
x x
6 K.Fleischmann,V.A.VatutinandV.Wachtel
and from (23) that
bx ∼ x−1+1αL(23)(1/x). (29)
By our assumption, j →∞, hence, by Lemma 4,
n
Q(j )
Q1+α(j )L Q(j ) ∼ n as n↑∞. (30)
n n
αj
n
(cid:0) (cid:1)
Thus, combined with (28),
Q1+α(j )L(Q(j )) nQ(j )
n n n
∼ → 0 as n↑∞. (31)
b1n+αL(bn) αjn
Note thatthe function s7→(1−s)1+αL(1−s)=f(s)−s is monotone(its
derivative f′(s)−1 is negative for s ∈ [0,1) by criticality). Applying this
to 1−s = Q(j ) and 1−s = b , it follows from (31) and properties of
n n
slowly varying functions that
Q(j )/b →0 as nj−1Q(j )→0. (32)
n n n n
On the other hand, from (25) and (29) it follows that
1
P M(∞)≥n ∼ b as n↑∞. (33)
Γ α n
1+α
(cid:0) (cid:1)
Combining this with (32) proves part(cid:0)(a) o(cid:1)f the lemma.
(b) Observe that by (29) and (33),
1
P M(∞)≥j/Q(j) ∼ b as j ↑∞, (34)
Γ α j/Q(j)
1+α
(cid:0) (cid:1)
and that (cid:0) (cid:1)
Q(j)
b1+α L b ∼ as j ↑∞. (35)
j/Q(j) j/Q(j) j
This,combinedwith(30)an(cid:0)dprop(cid:1)ertiesofslowlyvaryingfunctions,implies
1
bj/Q(j) ∼α1+α Q(j) as j ↑∞. (36)
Substituting (36) into (34) finishes the proof. (cid:3)
2.2 Basic properties of critical processes
For a while, we now discuss general critical Galton-Watson processes [i.e.
we drop restriction (1)]. For R≥2, put
B :=E ξ(ξ−1); ξ ≤R and R :=min{R≥2: B >0}<∞, (37)
R 0 R
and set (cid:8) (cid:9)
∞
1−f(s)
F(s) := = P(ξ >j)sj, 0≤s<1. (38)
1−s
j=0
X
Totalprogenyinawindow 7
Lemma 8 (Truncated variance) Thereexistsapositiveconstant c8 such
that for any critical Galton-Watson process,
BR ≤ 2 jP(ξ >j) ≤ c8R 1−F(1−1/R) , R≥2. (39)
1≤j≤R
X (cid:0) (cid:1)
Proof The first inequality in (39) essentially follows by integration by
parts. For j satisfying 1 ≤ j ≤ R and x ∈ (0,1), we have the following
elementary inequality:
1−(1−x)j ≥ jx(1−x)j−1 ≥ jx(1−x)R, (40)
which can be rewritten as
j ≤ x−1(1−x)−R 1−(1−x)j . (41)
Choosing x = 1/R and using criticality(cid:0) ∞ P(ξ >(cid:1) j) = 1, we get from
j=0
the first inequality in (39),
P
1 −R 1 j
B ≤ 2R 1− 1− 1− P(ξ >j)
R
R R
(cid:16) (cid:17) 0≤Xj≤R(cid:18) (cid:16) (cid:17) (cid:19)
1 −R 1 j
= 2R 1− 1− P(ξ >j)− 1− P(ξ >j)
R R
(cid:16) (cid:17) (cid:18) jX>R 0≤Xj≤R(cid:16) (cid:17) (cid:19)
1 −R 1 j
≤ 2R 1− 1− 1− P(ξ >j)
R R
(cid:16) (cid:17) (cid:18) Xj≥0(cid:16) (cid:17) (cid:19)
≤ cR 1−F(1−1/R) , (42)
as desired. (cid:0) (cid:1) (cid:3)
The next statement is a particular case of Nagaev and Wachtel [NW05,
Theorem 3].
Lemma 9 (A tail estimate) For m≥0, k ≥1, y >0, and R≥2,
0
1 1 k −1
P(M ≥k) ≤ y + 1+ −1
m+1 (cid:0) 0 R(cid:1)(cid:20)(cid:16) 1/y0+(e2+ey0R)mBR/2(cid:17) (cid:21)
+ mP(ξ >R). (43)
If the variance of ξ, for the moment denoted by B is finite and
∞,
positive, then by Doob’s inequality,
mB +1 mB
∞ ∞
P(M ≥k) ≤ ≤ (1+1/B ) . (44)
m+1 k2 ∞ k2
Estimate(43)allowsustoderiveananalogousboundwithoutimposingthe
finiteness of Varξ :
8 K.Fleischmann,V.A.VatutinandV.Wachtel
Lemma 10 (A further tail estimate) There existfiniteconstants c(45)
and c10 such that
mB
k
P(Mm+1 ≥k) ≤ c(45) k2 + mP(ξ >k/2) (45)
for all k,m≥1 satisfying k/(mBk)>c10.
Weseethat,for k sufficientlylarge,thefirsttermattherighthandside
of (45) coincides with (44) concerning the truncated variance B (except
k
the choice of the constant). The second term compensates the truncation.
Proof of Lemma 10 In view of Lemma 8, B /k →0 as k ↑ ∞. Hence,
k
thereisaconstant c(46) ≥e suchthatfor k,m≥1 with k/(mBk)>c(46),
2 k 3 k
y := y (k,m) := log − loglog
0 0
k mB k mB
k k
1 k 2 k
= log log−3 > 0. (46)
k mB mB
(cid:18)(cid:16) k(cid:17) (cid:16) k(cid:17)(cid:19)
Hence,letting R=k/2≥2 in(43)andobservingthatB isnon-decreasing
R
in R, we get from Lemma 9 and our choice of R,
2 1 k −1
P(M ≥k) ≤ y + 1+ −1
m+1 (cid:0) 0 k(cid:1)(cid:20)(cid:16) 1/y0+(e2+ey0k/2)mBk/2(cid:17) (cid:21)
+ mP(ξ >k/2). (47)
From the estimates
2 k 2 k
y ≤ log ≤ log (48)
0
k mB k B
k R0
being validforallk ≥R , itfollowsthat y =y (k,m)↓0 ask ↑∞, and,
0 0 0
in addition, there exists a constant c(49) such that for k,m≥1 satisfying
k/(mBk)≥c(49),
k k
e2+ey0k/2 mB /2 = e2+ log−3/2 mB /2
k k
mB mB
(cid:18) k (cid:16) k(cid:17)(cid:19)
(cid:0) (cid:1) k 1
≤ 2klog−3/2 ≤ . (49)
mB 4y
k 0
(cid:16) (cid:17)
Hence, for these k,m,
5
1/y +(e2+ey0k/2)mB /2 ≤ . (50)
0 k
4y
0
Clearly, for sufficiently small y >0,
0
1 k 4y k
0
1+ ≥ 1+
1/y0+(e2+ey0k/2)mBk/2 5
(cid:16) (cid:17) (cid:16) (cid:17)
4y 4ky y
0 0 0
= exp klog 1+ ≥exp 1− . (51)
5 5 2
(cid:20) (cid:16) (cid:17)(cid:21) (cid:20) (cid:16) (cid:17)(cid:21)
Totalprogenyinawindow 9
By the definition of y0 there exists c(52) such that
y 5 6 2 k
0
1− ≥ and y > log (52)
0
2 6 7 k mB
k
for k/(mBk)>c(52). Thus, we get the bound
1 k k 8/7
1+ ≥ (53)
1/y0+(e2+ey0k/2)mBk/2 mBk
(cid:16) (cid:17) (cid:16) (cid:17)
for k/(mBk) ≥ c(53) := max(c(46),c(49),c(52)). Moreover, if k/(mBk) >
c10 :=max(c(53),2), then
1 k −1 mB 8/7
k
1+ −1 ≤2 . (54)
(cid:18)(cid:16) 1/y0+(e2+ey0k/2)mBk/2(cid:17) (cid:19) (cid:16) k (cid:17)
Combining (46) – (54) gives, for k/(mBk)>c10,
2 k mB 8/7
k
P(M ≥k) ≤ 2+log + mP(ξ >k/2). (55)
m+1
k mB k
(cid:18) (cid:16) k(cid:17)(cid:19)(cid:16) (cid:17)
The boundedness of the function x7→x−1/7logx for x≥2 completes the
proof of the lemma. (cid:3)
Now we return to critical processes of index 1+α.
Lemma 11 (A moment estimate) Under condition (1), for β ∈(1,1+
α), there is a constant c11 =c11(β) such that
EZmβ ≤ c11Q1−β(m), m≥1. (56)
Proof According to assumption (1), for 0≤s<1,
∞ ∞
1−F(s) f(s)−s L(1−s)
= = sl P(ξ >i) = . (57)
1−s (1−s)2 (1−s)1−α
l=0 i=l+1
X X
Therefore, by Lemma 8, for all sufficiently large k,
Bk ≤ c8k1−αL(1/k). (58)
Onthe other hand, formula (57) and a Tauberiantheorem(cf. [Fel71, The-
orem 13.5.5]) imply
k−1 ∞
1
P(ξ >i) ∼ k1−αL(1/k) as k ↑∞. (59)
Γ(α)
l=0i=l+1
X X
Hence, for sufficiently large k,
∞
2
P(ξ >i) ≤ k−αL(1/k) (60)
Γ(α)
i=k
X