Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities
Let $\mathcal{M}_{(n)},\quad n = 1, 2,...,$ be the supercritical branching random walk in which the family sizes may be infinite with positive probability. Assume that a natural martingale related to $\mathcal{M}_{(n)},$ converges almost surely and in the mean to a random variable $W$. For a large s...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509565914710016 |
|---|---|
| author | Iksanov, O. M. Іксанов, О. М. |
| author_facet | Iksanov, O. M. Іксанов, О. М. |
| author_sort | Iksanov, O. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:55:25Z |
| description | Let $\mathcal{M}_{(n)},\quad n = 1, 2,...,$ be the supercritical branching random walk in which the family sizes may be infinite with positive probability. Assume that a natural martingale related to $\mathcal{M}_{(n)},$ converges almost surely and in the mean to a random variable $W$. For a large subclass of nonnegative and concave functions $f$ , we provide a criterion for the finiteness of $\mathbb{E}W f(W)$. The main assertions of the present paper generalize some results obtained recently in Kuhlbusch’s Ph.D. thesis as well as previously known results for the Galton-Watson processes. In the process of the proof, we study the existence of the $f$-moments of perpetuities. |
| first_indexed | 2026-03-24T02:43:08Z |
| format | Article |
| fulltext |
UDC 519.21
A. M. Iksanov (Kiev Nat. Taras Shevchenko Univ.),
U. Rösler (Univ. Kiel, Germany)
SOME MOMENT RESULTS
ABOUT THE LIMIT OF A MARTINGALE
RELATED TO THE SUPERCRITICAL BRANCHING
RANDOM WALK AND PERPETUITIES
DEQKI REZUL\TATY PRO MOMENTY HRANYCI
MARTYNHALA, POV’QZANOHO Z NADKRYTYÇNYM
HILLQSTYM VYPADKOVYM BLUKANNQM,
TA ROZV’QZKIV DEQKYX STOXASTYÇNYX
RIZNYCEVYX RIVNQN\
Let M(n), n = 1, 2, . . . , be the supercritical branching random walk in which the family sizes may be infinite
with positive probability. Assume that a natural martingale related to M(n) converges almost surely and in mean
to a random variable W . For a large subclass of nonnegative and concave functions f, we provide a criterion
for the finiteness of EWf(W ). The main assertions of the present paper generalize some results obtained
recently in Kuhlbusch’s Ph.D. thesis as well as previously known results for the Galton – Watson processes. In
the process of the proof, we study the existence of the f -moments of perpetuities.
Nexaj M(n), n = 1, 2, . . . , — nadkrytyçne vypadkove blukannq, u qkomu rozmir rodyny moΩe buty
neskinçennym z dodatnog jmovirnistg. Prypustymo, wo standartnyj martynhal, pov’qzanyj z M(n)
,
zbiha[t\sq majΩe napevno i v seredn\omu do vypadkovo] velyçynyW . Dlq velykoho pidklasu nevid’[mnyx
ta vhnutyx funkcij f navedeno kryterij skinçennosti EWf(W ). Osnovni tverdΩennq roboty uza-
hal\nggt\ deqki rezul\taty, otrymani v dysertaci] Kul\bußa, a takoΩ rezul\taty, vidomi dlq proce-
siv Hal\tona – Vatsona. U procesi dovedennq doslidΩu[t\sq isnuvannq f -momentiv rozv’qzkiv deqkyx
stoxastyçnyx riznycevyx rivnqn\.
1. Introduction and results. Assume that an initial ancestor of some population is placed
at the origin of the real line. She produces offspring who form the first generation of the
population. Each individual of the first generation in her turn gives birth to children too.
All children of the individuals of the first generation constitute the second generation and
so on. A point process M with points {Ai, i = 1,M(R)} controls the location of the
population over the real line in such a way. For i = 1, 2, . . . , the displacements of the
individuals of the i-th generation relative to positions of their mothers (they reside in the
i − 1-th generation) are given by independent copies of M. The sequence of the point
processes M(n), n = 1, 2, . . . , which define positions of the n-th generation individuals
is called the branching random walk (the BRW, in short). Many references related to the
BRW can be found in [1 – 3].
In the sequel, for n = 1, 2, . . . ,Fn denotes a σ-field containing all information about
the first n generations. The position of the individual u is denoted by Au; the symbol
|u| = n means that the individual u resides in the n-th generation; the symbol
∑
|u|=n
denotes the summation over all individuals of the n-th generation.
Set K := M(R) and q := P{K < ∞} ∈ [0, 1]. In this paper, we only consider the
supercritical BRW. Therefore, if q = 1, we additionally assume that EK > 1. Recall that
the supercriticality ensures the survival of the population with a positive probability.
Define the function
m(y) := E
K∑
i=1
eyAi ∈ (0,∞], y > 0.
Assume that m(γ) <∞ for some γ > 0 and set
W (γ)
n := m(γ)−n
∑
|u|=n
eγAu , n = 1, 2, . . . .
c© A. M. IKSANOV, U. RÖSLER, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4 451
452 A. M. IKSANOV, U. RÖSLER
The sequence (W (γ)
n ,Fn), n = 1, 2, . . . , is a nonnegative martingale (Kingman [4] and
Biggins [5] were the first to study such a martingale). Since γ and Fn will be the same
from line to line, in what follows the martingale is denoted just by Wn. This martingale
converges either almost surely to zero or almost surely and in mean to a random variable
W which is positive with positive probability (throughout the text we use words “positive”
and “increasing” in a strict sense). Put Yi := eγAi/m(γ). The (probability) distribution
of the W satisfies the equality
W
d=
K∑
i=1
YiW
(i),
where, given F1, W (1),W (2), . . . are conditionally independent copies of the W .
The papers [5 – 7] provide conditions for the martingale convergence in mean. How-
ever, all these authors required more or less restrictive additional assumptions. Our Propo-
sition 1 can be read from Theorem 2 [3], where the criterion of the above mentioned
convergence has been obtained (but in other terms).
The equality
E
K∑
i=1
Yit(Yi) = Et(Z), (1)
which is assumed to hold for bounded Borel functions t, defines the distribution of a
random variable Z. Notice that
P{Z = 0} = 0.
In the sequel, we additionally always assume that
P{Z = 1} < 1, P{W1 = 1} < 1.
As soon as the distribution of Z was defined, we can permit for (1) to hold for any Borel
function t. In that case, we assume that if the left-hand side is infinite or does not exist,
the same is true for the right-hand side.
Let Tn, n = 0, 1, . . . , be the random walk starting at zero with a step distributed like
V := − logZ. Define the function
AZ(y) :=
y∫
0
P{V > x}dx, y > 0.
Relevant properties of this function can be found in [8].
Proposition 1. The martingale Wn converges in mean if and only if
lim
n→∞
Tn = +∞ a.s.;
∫
(1,∞)
x log x
AZ(x)
dP{W1 ≤ x} <∞, (2)
or equivalently if and only if either
i) EV ∈ (0,∞) and EW1 log+W1 <∞, or
ii) EV = ∞ and
∫
(1,∞)
x log x
AZ(x)
dP{W1 ≤ x} <∞, or
iii) EV does not exist and E
(
log+ Z
AZ(log+ Z)
)
<∞, and
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
SOME MOMENT RESULTS ABOUT THE LIMIT OF A MARTINGALE RELATED ... 453∫
(1,∞)
x log x
AZ(x)
dP{W1 ≤ x} <∞.
Remark 1. In any case, the classical x log+ x condition together with the condition
lim
n→∞
Tn = +∞ a.s. are sufficient for the mean convergence of the martingale. A quite
remarkable fact is that when EV is infinite or does not exist, the x log+ x condition is no
longer necessary. Thus we come to a bit discouraging conclusion: the weaker moment
restriction is imposed on V , the weaker moment condition may be put on W1.
As soon as the problem of existence of somewhere positive W is settled, it is natural
to want to investigate moments ofW . Following this principle, in this paper we will study
f -moments of W . Consequently, the description of appropriate functions f will be given
next.
Throughout the text, we assume that one of the following two assumptions is in force.
Assumption A. Function f > 0 is nondecreasing and concave on [0,∞), lim
x→∞
f(x) =
= ∞. For fixed B, d ≥ 0 a new function g is defined by
g(x) := B +
x∫
d
(f(y)/y)dy for x ≥ d; g(x) = 0 for x < d.
Assumption B. Function f is nondecreasing and concave on [0,∞), lim
x→∞
f(x) = ∞
and f(0) = 0. Additionally, there exists p > 0 such that
f(xy) ≤ pf(x)f(y) for all x, y > 0. (3)
In this paper, ψ(x), x ≥ 0, is called a submultiplicative function if ψ(x) is finite,
positive, and Borel measurable and
ψ(0) = 1 and ψ(x+ y) ≤ ψ(x)ψ(y).
Recall that for a submultiplicative function ψ, there exists a limit
lim
x→∞
logψ(x)
x
∈ [0,∞).
Inequality (3) implies that
h(x+ y) := f(ex+y) ≤ ph(x)h(y) for all x, y > 0.
As pointed out by Sgibnev on [9, p. 85], the latter implies that there exists a nondecreasing
submultiplicative function ψ such that
p1ψ(x) ≤ h(x) ≤ p2ψ(x) for some positive constants p1, p2. (4)
Therefore, we can define a constant r ∈ [0, 1] by
r := lim
x→∞
log h(x)
x
.
In what follows, F � G means that
0 < lim inf
x→∞
F (x)
G(x)
≤ lim sup
x→∞
F (x)
G(x)
<∞.
We are now ready to present our first main result.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
454 A. M. IKSANOV, U. RÖSLER
Theorem 1. Let f satisfy Assumption A or B.
a) If M(−∞,−γ−1 logm(γ)) = 0 a.s., assume that Assumption A holds. If the
integral in (2) converges, then
EW1g(W1) <∞ ⇒ EWf(W ) <∞;
if EV ∈ (0,∞), then
EW1g(W1) <∞ ⇐ EWf(W ) <∞.
In particular, if EV ∈ (0,∞) and EW1 log+W1 < ∞, then both implications hold
and we have the equivalence. If f � g, then we have the equivalence under the weaker
assumption that the integral in (2) converges.
b) If M(−∞,−γ−1 logm(γ)) ≥ 1 with positive probability, assume that Assump-
tion B holds. If EV ∈ (0,+∞], then
b1) if r > 0, then
EW1f(W1) <∞, EZr < 1 ⇒ EWf(W ) <∞;
b2) if r = 0, then
EW1f(W1) <∞ ⇒ EWf(W ) <∞.
If EV ∈ (0,+∞) in both cases r > 0 and r = 0, the converse implications hold, and we
in fact have the equivalence.
Remark 2. In case f(x) = xa, a ∈ (0, 1] Theorem 1 reduces to the well-known
equivalence
EW a+1 <∞ ⇔ EW a+1
1 <∞, EZa < 1
(see, for example, Proposition 4 [3]).
There are many results in the spirit of Theorem 1 related to the Galton – Watson
process (see[10, 11] for recent developments). In the context of the branching random
walk, our Theorem 1 generalizes a statement in Section 4 [12], Corollary 10 [13], The-
orems 4.4.1 and 4.5.1 [14]. The best previously known results like our Theorem 1 were
recently obtained in Kuhlbusch’s Ph.D. thesis [14]. In Section 2, we partially compare
our results to Kuhlbusch’s ones.
The technique developed in this work is an extension of the approach proposed in
[11] for the case of the Galton – Watson processes and in [3]. It should be noted that
independently and at the same time a similar technique has also been used in [15] in the
context of branching diffusions. Our method of proof consists in comparing (under the
appropriate change of measure proposed in [7]) the random variable W with a so called
perpetuity. Keeping this in mind we find it useful to study the existence of the f -moments
of perpetuities.
Let (Q1,M1), (Q2,M2), . . . be independent copies of a random vector (Q,M). Let
Z0 be a random variable which is independent of (Q,M).
Set Π0 := 1 and Πn := M1M2 . . .Mn, n = 1, 2, . . ., X := − log |M |, AM (y) :=
:=
∫ y
0
P{X > x}dx, y > 0. The following proposition is a selection from Theo-
rem 2.1 [8].
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
SOME MOMENT RESULTS ABOUT THE LIMIT OF A MARTINGALE RELATED ... 455
Proposition 2. The following assertions are equivalent:
lim
n→∞
Πn = 0 a.s.,
∞∫
1
(
log q
AM (log q)
)
dP{|Q| ≤ q} <∞, (5)
∞∑
n=1
|Πn−1Qn| <∞ a.s.
Each of these ensures
lim
n→∞
(
n∑
k=1
Πn−1Qn + ΠnZ0
)
= Z∞ a.s.,
where
Z∞ :=
∞∑
n=1
Πn−1Qn.
In the literature, there exist several results about the existence of moments (or the tail
behaviour) of the random variable Z∞ called a perpetuity. We only mention two of them:
a) E|M |p < 1 and E|Q|p < ∞ ⇔ E|Z∞| < ∞, p > 0 (in [16] this has been shown
in case M,Q ≥ 0, in [17] the implication ⇒ has been proved in case p > 1);
b) if P{|M | ≤ 1} = 1 and Eeε|Q| < ∞ for some ε > 0, then Eeρ|Z∞| < ∞ for
0 ≤ ρ < sup{θ : Eeθ|Q||M | < 1} (this fact follows from Theorem 2.1 [18]; this work
implicitly contains some other results related to moments).
Note that so far the existence of the f -moments of Z∞ has not been investigated (the
only exception being the case f(x) = xa, a ∈ (0, 1]).
In the sequel, we assume that
P{M = 0} = 0, P{Q = 0} < 1,
and the distribution of Z∞ is nondegenerate.
The second main result is as follows.
Theorem 2. Let f satisfy Assumption A or B.
a) If
P{|M | ≤ 1} = 1 and P{|M | = 1} < 1, (6)
assume Assumption A holds. If the integral in (5) converges, then
Eg(|Q|) <∞ ⇒ Ef(|Z∞|) <∞;
if EX ∈ (0,∞), then
Eg(|Q|) <∞ ⇐ Ef(|Z∞|) <∞.
In particular, if EX ∈ (0,∞) and E log+ |Q| < ∞, then both implications hold and we
have the equivalence. If f � g then we have the equivalence under the weaker assumption
that the integral in (5) converges.
b) If P{|M | > 1} > 0, assume that Assumption B holds. If EX ∈ (0,+∞] then
b1) if r > 0 then
Ef(|Q|) <∞, Ef(|M | ∨ 1) <∞, E|M |r < 1 ⇒ Ef(|Z∞|) <∞;
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
456 A. M. IKSANOV, U. RÖSLER
b2) if r = 0 then
Ef(|Q|) <∞, Ef(|M | ∨ 1) <∞ ⇒ Ef(|Z∞|) <∞.
If EX ∈ (0,+∞) in both cases r > 0 and r = 0 the converse implications hold, and we
in fact have the equivalence.
The rest of the paper is organized as follows. Section 2 contains some relevant prop-
erties of functions f and g. In Section 3, after giving a preliminary result, we study
the f -moments of perpetuities and prove Theorem 2. In Section 4, we provide a careful
description of a change of measure construction and prove Theorem 1.
2. Properties of functions f and g, and examples. To give a better feeling of the
results obtained, we first point out some pairs (f, g) which satisfy Assumption A. These
examples are taken from Section 3 [11].
1. For p ∈ (0, 1),
f(x) = xp, g(x) = xp/p.
2. For p ∈ (0, 1],
f(x) = e−1x1{x∈[0,e)} + logp x1{x≥e}, g(x) = (p+ 1)−1(logp+1 x− 1)1{x≥e};
for p > 1,
f(x) = p(p− 1)e1−px1{x∈[0,ep−1)} + (logp x+ (p− 1)2)1{x≥ep−1},
g(x) = ((p+ 1)−1(logp+1 x− (p− 1)p) + (p− 1)2(log x− p+ 1))1{x≥ep−1},
therefore, for p > 0,
f(x) � logp x, g(x) � logp+1 x.
3. For β > 0 and c := (β/e)β − β,
f(x) = ββe−β−exp(β)x1{x∈[0,exp(eβ))} + (logβ log x+ c)1{x≥exp(eβ)},
g(x) =
log x logβ log x− βeβ − β
log x∫
eβ
logβ−1 udu+ c(log x− eβ)
1{x≥eβ},
therefore,
f(x) � logβ log x, g(x) � log x logβ log x.
As it follows from Theorems 1 (a) and 2 (a), it is important to know when f � g, if
(f, g) satisfy Assumption A. A simple sufficient condition for this to hold was given in
Corollary 1.2 [11]: if there exists an α ∈ (0, 1) such that x−αf(x) does not decrease for
large x, then f � g.
Now we would like to explain the point of using Assumption B. To prove Theo-
rem 2 (b), it would be highly desirable if functions f possessed two properties: (4) and
f(x) �
∫ x
0
(f(u)/u)du. Assumption B appears to be the weakest possible one to en-
sure that these properties do hold. The next lemma collects some properties of functions
satisfying Assumption B.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
SOME MOMENT RESULTS ABOUT THE LIMIT OF A MARTINGALE RELATED ... 457
Lemma 1. Let f satisfy Assumption B. Then
a) f and g(x) :=
∫ x
0
(f(u)/u)du satisfy Assumption A with B = d = 0; moreover,
f � g;
b) lim
x→∞
f(x)
logε x
= ∞ for every ε > 0.
Proof. The first part of (a) is obvious. Let us verify that f � g. Since f(x)/x is
nonincreasing, we have
g(x) =
x∫
0
(f(u)/u)du ≥ (f(x)/x)
x∫
0
du = f(x).
Using now (3), we obtain
g(x) =
1∫
0
(f(tx)/t)dt ≤ pf(x)
1∫
0
(f(t)/t)dt = constf(x).
From these two inequalities, we obtain the needed. The following result can be de-
rived from the proof of Proposition 2 [13]: if H : R
+ → R
+ is a convex function
with concave derivative, H(0) = 0, and there exists a positive constant c such that
H(xy) ≤ cH(x)H(y) for all x, y ∈ R
+, then
lim
x→∞
H(x)
x logε x
= ∞ for every ε > 0. (7)
Set H(x) :=
∫ x
0
f(u)du. Since
2−1xf(x) ≤ H(x) ≤ xf(x), (8)
we have
H(xy) ≤ xyf(xy)
(3)
≤ pxyf(x)f(y) ≤ 4pH(x)H(y).
Therefore, the so defined H possesses all the properties described above. This gives (7)
and, in view of (8), the statement follows.
As was indicated in Introduction, some results related to our Theorem 1 were given
in [14]. Kuhlbusch studied the φ-moments of W when φ is a regularly varying function
with index α subject to additional restrictions. If α ∈ (1, 2), his Theorems 4.5.1 and 4.4.1
are contained in our Theorem 1 (a) and Theorem 1 (b) correspondingly. Indeed, it is well
known that given a regularly varying function t with index α ∈ (1, 2), there exists a
concave function z such that t(x) � xz(x). On the other hand, a concave function need
not be regularly or slowly varying. Although it is a quite obvious fact, we propose a
simple example (due to Professor Oleg Zakusylo) of positive, increasing, and concave
function which is not regularly varying. Define
q(x) := 2−kx+ 2k+1 − 3, if x ∈ [4k, 4k+1), k = 0, 1, . . . .
The first three stated properties are easily seen. To check the absence of regular variation,
set xn := 4n, yn := 3 · 4n, n = 1, 2, . . . . Then lim
n→∞
q(2xn)
q(xn)
=
4
3
and lim
n→∞
q(2yn)
q(yn)
=
8
5
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
458 A. M. IKSANOV, U. RÖSLER
Another example could be given which, however, requires more computations (omitted
here). Take
q(x) := xβ(1 + aβ sin(log x) + bβ cos(log x)), β ∈ (0, 1),
with appropriate parameters aβ , bβ .
3. Moments of perpetuities. Let {(ξk, ηk), k = 1, 2, . . .} be independent copies of
a random vector (ξ, η). Set Rn := ξ1 + . . . + ξn, n = 1, 2, . . . , R0 := 0. Lemma 2
given next is needed for the proof of Proposition 3. Note that this lemma generalizes
Proposition 7 [19].
Lemma 2. Assume that lim
n→∞
Rn = ∞ and ζ := sup
k≥0
(−Rk + ηk+1) <∞ a.s. Then
P{ζ > x} ≥ P{η > x} +
x∫
−∞
P{sup
k≥0
(−Rk) > x− y}dP{η ≤ y}, x ∈ R.
Proof. For every n = 1, 2, . . . , put Mn := sup
{
k ≥ 0 : −Rk = max
0≤l≤n
(−Rl)
}
.
Then
ζ ≥ max
0≤k≤n
(−Rk + ηk+1) ≥ −RMn
+ ηMn+1.
Therefore,
P{ζ > x} ≥ P{−RMn + ηMn+1 > x} =
=
n∑
m=0
P{−RMn + ηMn+1 > x,Mn = m} =
=
n∑
m=0
P{−Rm + ηm+1 > x, Mn = m} =
=
n∑
m=0
∫
R
P{−Rm > x− y,Mn = m}dP{ηm+1 ≤ y} =
=
∫
R
n∑
m=0
P{−Rm > x− y,Mn = m}dP{η ≤ y} =
=
∫
R
P
{
max
0≤k≤n
(−Rk) > x− y
}
dP{η ≤ y}.
Since Rk drifts to ∞, sup
k≥0
(−Rk) < ∞ a.s. Letting n → ∞ and using Fatou’s lemma
allow us to conclude that
P{ζ > x} ≥
∫
R
P
{
sup
k≥0
(−Rk) > x− y
}
dP{η ≤ y} =
= P{η > x} +
x∫
−∞
P
{
sup
k≥0
(−Rk) > x− y
}
dP{η ≤ y}.
The proof is complete.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
SOME MOMENT RESULTS ABOUT THE LIMIT OF A MARTINGALE RELATED ... 459
Consider independent and identically distributed random vectors
(M̀k, Q̀k) := (M2k−1M2k,M2k−1Q2k +Q2k−1), k = 1, 2, . . . ,
and set
Π̀0 := 1, Π̀n := M̀1M̀2 . . . M̀n, n = 1, 2, . . . .
Proposition 3. Assume that h does not decrease, has one-sided derivatives which
coincide a.e., and for large x and some c > 0,
h(2x) ≤ ch(x). (9)
Then Eh(|Z∞|) <∞ implies:
Eh(|Q|) <∞ and either Eh
(
sup
n≥0
|Πn|
)
<∞ or Eh
(
sup
n≥0
|Π̀n|
)
<∞;
either Eh
(
sup
n≥1
|Πn−1||Qs
n|
)
<∞ or Eh
(
sup
n≥1
|Π̀n−1||Q̀s
n|
)
<∞, (10)
whereQs
n := Qn−Q′
n; (Mn, Qn) d= (Mn, Q
′
n); givenMn,Qn andQ′
n are conditionally
independent; Q̀s
n and Q̀′
n have the same meaning but in terms of M̀n and Q̀n.
Proof. As in [8, p. 1212, 1213], by using a regular conditional distribution forQ given
M , we can construct the sequence {Q′
j , j = 1, 2, . . .} such that (Mj , Q
′
j), j = 1, 2, . . . ,
are independent copies of (M,Q); given Mj , Q′
j and Qj are conditionally independent.
Consider two cases:
1. Q is not a Borel function of M . Let us define conditionally symmetrized random
variables
Qs
j := Qj −Q′
j , Zs
n :=
n∑
k=1
Πk−1Q
s
k, n = 1, 2, . . . .
Note that L(Qs
j) �= δ0. Let Bn = σ(M1, . . . ,Mn), n = 1, 2, . . . . By symmetrization
inequalities [20] for n = 1, 2, . . .
P
{
max
1≤k≤n
|Πk−1Q
s
k| > x|Bn
}
≤
≤ 2P {|Zs
n| > x|Bn} ≤ 4P
{∣∣∣∣∣
n∑
k=1
Πk−1Qk
∣∣∣∣∣ > x/2|Bn
}
.
Taking expectations and then letting n go to ∞ gives
P
{
sup
k≥1
|Πk−1Q
s
k| > x
}
≤ 2P {|Zs
∞| > x} ≤ 4P {|Z∞| > x/2} , x ∈ R. (11)
These inequalities hold for all x, as the distribution of Z∞ is continuous and the sequence{
max
1≤k≤n
|Πk−1Q
s
k|, n = 1, 2, . . .
}
is monotone.
Assume that Eh(|Z∞|) <∞. Condition (11) implies
Eh
(
(1/2)sup
k≥1
|Πk−1Q
s
k|
)
<∞.
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460 A. M. IKSANOV, U. RÖSLER
Taking into account (9), we have
∞ > Eh
(
sup
k≥1
|Πk−1Q
s
k|
)
= Eq
(
sup
k≥0
(−Sk + log |Qs
k+1|)
)
, (12)
where q(x) := h(ex), Sk := − log |Πk|, k = 0, 1, . . . , is a random walk with a step
distributed like X. Since |Z∞| < ∞ a.s., Proposition 2 ensures that lim
k→∞
Sk = ∞ a.s.
According to Lemma 2, we have
P
{
sup
k≥0
(−Sk + log |Qs
k+1|) > x
}
≥
≥ P{log |Qs| > x} +
x∫
−∞
P
{
sup
k≥0
(−Sk) > x− y
}
dP{log |Qs| ≤ y}, x ∈ R. (13)
Let q′ be any of one-sided derivatives of q. In view of (12),
∞ >
∫
R
q′(x)P
{
sup
k≥0
(−Sk + log |Qs
k+1|) > x
}
dx
(13)
≥
(13)
≥
∫
R
q′(x)
x∫
−∞
P
{
sup
k≥0
(−Sk) > x− y
}
dP{log |Qs| ≤ y}dx =
=
∫
R
dP{log |Qs| ≤ y}
∞∫
0
q′(x+ y)P
{
sup
k≥0
(−Sk) > x
}
dx =
= Eq
(
logU + sup
k≥0
(−Sk)
)
= Eh
(
Usup
k≥0
|Πk|
)
,
where U is a random variable which is independent of sup
k≥0
|Πk| and distributed like |Qs|.
Therefore, Eh
(
sup
k≥0
|Πk|
)
<∞ and Eh(|Qs
1|) <∞.
The latter inequality ensures E
(
h(|Qs
1|)|B1
)
<∞ a.s. Hence,
Eh(|Q−Q∗|) <∞, (14)
where Q∗ is independent copy of Q. In the same way as formula (5.7) in [20] has been
proved, we can verify that there exists a ∈ R such that
2P
{
|Q−Q∗| > x− a
}
≥ P
{
|Q| > x
}
. (15)
Monotonicity of h and condition (9) imply that, for fixed b ∈ R,
h(x) � h(x− b). (16)
From this, (14) and (15), we have Eh(|Q|) <∞.
2. Q = r(M) for some Borel function r. We have
2n∑
k=1
Πk−1Qk =
n∑
k=1
Π̀k−1Q̀k,
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where the random variables on the right-hand side were defined before the proposition. If
Q̀k = s(M̀k) for some Borel function s, then
M1Q2 +Q1 = s(M1M2)
and, by Proposition 1 [21], either Q + cM = c a.s. for some c or (M,Q) = (1, c1) for
some c1. The first of these is excluded by our assumption before Theorem 2. The second is
incompatible with |Z∞| <∞ a.s. by Proposition 1. Therefore, Q̀k is not a Borel function
of M̀k. Now, the first part of the proof can be applied (on (M̀k, Q̀k) instead of (Mk, Qk))
which yields Eh(|Q̀|) < ∞, Eh
(
sup
n≥0
Π̀n
)
< ∞ and Eh
(
sup
n≥1
|Π̀n−1||Q̀s
n|
)
< ∞.
Taking into account (9), (16) and the equality
Eh(|Q̀|) =
∫
R
∫
R
Eh(|mQ2 + q|)P
{
M1 ∈ dm, Q1 ∈ dq
}
,
we conclude that Eh(|Q|) <∞. Proposition 3 has been proved.
Now we are ready to give proof of Theorem 2(a).
Proof of Theorem 2 (a). Necessity. Assume that
Eg(|Q|) <∞. (17)
In view of (6), lim
n→∞
Πn = 0 a.s. By Proposition 2, this together with the assumption that
the integral in (5) converges, ensures that |Z∞| <∞ a.s.
Condition (17) implies J :=
∫ ∞
d
P{|Q| > v}g′(v)dv < ∞. By assumption, m :=
:= E|M | ∈ (0, 1). Let us check that for arbitrary n ∈ N and fixed c > f(d),
In := (1 −m)E
(
f
∣∣∣∣∣
n∑
k=1
|Πk−1Qk|
∣∣∣∣∣ ∨ c
)
≤ J. (18)
We have
J =
∞∫
d
P{|Q| > v}(f(v)/v)dv ≥ (19)
≥ (1 −m)
∞∫
d
P{|Q| > v}E
(
n∑
k=1
|Πk−1|f ′(|Πk−1|v)
)
dv ≥
≥ (1 −m)E
n∑
k=1
∞∫
d/|Πk−1|
P{|Q| > v}|Πk−1|f ′(|Πk−1|v)dv ≥ (20)
≥ (1 −m)
∞∫
c
n∑
k=1
EP{|Qk| > f−1(x)/|Πk−1|}dx =
= (1 −m)
n∑
k=1
∞∫
c
P{f(|Πk−1||Qk|) > x}dx =
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462 A. M. IKSANOV, U. RÖSLER
= (1 −m)E
n∑
k=1
(f(|Πk−1||Qk|) ∨ c) ≥ (21)
≥ (1 −m)E
(
f
(
n∑
k=1
|Πk−1||Qk|
)
∨ c
)
≥
≥ (1 −m)E
(
f
∣∣∣∣∣
n∑
k=1
Πk−1Qk
∣∣∣∣∣ ∨ c
)
which proves (18). Inequality (19) above has been obtained as follows: f ′ does not
increase, the sequence |Πk(ω)|, k = 0, 1, . . . , does not increase and 0 < Πk(ω) ≤ 1 a.s.
Hence,
f(v) ≥ E
v∫
|Πn|v
f ′(y)dy = E
n∑
k=1
|Πk−1|v∫
|Πk|v
f ′(y)dy ≥
≥ (1 −m)vE
n∑
k=1
|Πk−1|f ′(|Πk−1|v).
Inequality (20) follows by change of variable x = f(v|Πk−1|). Inequality (21) follows
from the fact that the function x → |x| is subadditive, and the functions x → x ∨ c and
f(x) are subadditive and nondecreasing. (Take for simplicity of explanation n = 2 and
set x := |Q1|, y := |M1||Q2|. Inequality (21) is implied by the inequalities (f(|x|)∨c)+
+ (f(|y|) ∨ c) ≥ (f(|x|) + f(|y|) ∨ c) ≥ (f(|x+ y|) ∨ c) ≥ (|f(x+ y)| ∨ c).)
Thus, In is bounded from the above by the constant J that does not depend on n.
By the assumptions of the theorem and Proposition 2, the series
∑n
k=1
|Πk−1||Qk|
is a.s. convergent. Since f is continuous, we have that as n → ∞, the sequence
f
(∣∣∣∑n
k=1
Πk−1Qk
∣∣∣) converges a.s. to f
(∣∣∣∑∞
k=1
Πk−1Qk
∣∣∣). An appeal to Fatou’s
lemma gives
Ef(|Z∞|) = Ef
(∣∣∣∣∣
∞∑
k=1
Πk−1Qk
∣∣∣∣∣
)
≤ J <∞.
Sufficiency. Assume that Ef(|Z∞|) < ∞. By Proposition 3, Ef(|Q|) < ∞. Thus,
if f � g, combining this observation with the previous part of the theorem, we deduce
that Ef(|Z∞|) < ∞ ⇔ Ef(|Q|) < ∞ under the sole condition that the integral in (5)
converges.
Consider now the general case. If the support of a distribution of Q is bounded, then
|Z∞| has finite moments of all positive integer orders and the result of the theorem is
trivial. Hence, in what follows, we assume that the support of a distribution of |Q| is
unbounded from the above. In that case, there exists s > d such that P{|Q| > t} > 0 for
all t ≥ s.
Assume that the distribution of M is nondegenerate. By Proposition 3, either
f
(
sup
n≥1
|Πn−1||Qs
n|
)
<∞ or Ef
(
sup
n≥1
|Π̀n−1||Q̀s
n|
)
<∞.
We only invesigate the second (harder) possibility. The strong law of large numbers
implies that there almost surely exists L > 0 such that |Π̀k| ≥ e−4µk for k ≥ L,
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SOME MOMENT RESULTS ABOUT THE LIMIT OF A MARTINGALE RELATED ... 463
where µ := E(− log |M |) ∈ (0,∞). Therefore, we can choose k0 < ∞ such that
|Π̀k| ≥ e−4µ(k∨k0) for every k = 0, 1, . . . . Since
sup
k≥1
|Π̀k−1||Q̀s
k| ≥ sup
k≥k0+2
|Π̀k−1||Q̀s
k| ≥ sup
k≥k0+2
e−4µ(k−1)|Q̀s
k|,
sup
k≥k0+2
e−4µ(k−1)|Q̀s
k|
d= e−4µk0sup
k≥1
e−4µk|Q̀s
k|,
and f does not decrease, Ef
(
sup
n≥1
|Π̀n−1||Q̀s
n|
)
<∞ implies
Ef
(
sup
k≥1
e−4µk|Q̀s
k|
)
<∞. (22)
Since µ ∈ (0,∞) and |Z∞| < ∞ almost surely, Proposition 2 allows us to conclude
that E log+ |Q| <∞. The function x→ log(1 + |x|) is subadditive and log(1 + |xy|) ≤
≤ log(1+|x|)+log(1+|y|). Therefore, E log+ |Q̀| = E log+ |Q1+M1Q2| <∞. Taking
conditional expectations and using the symmetrization inequality allow us to check that
E log+ |Q̀s| <∞. The latter implies
C :=
∞∏
k=1
P{e−4µk|Q̀s
k| ≤ s} > 0.
Furthermore, we have for t ≥ s
P
{
sup
k≥1
e−4µk|Q̀s
k| > t
}
=
∞∑
k=1
P
{
|Q̀s| > e4µkt
} k−1∏
j=1
P
{
|Q̀s| ≤ e4µjt
}
≥
≥
∞∑
k=1
P
{
|Q̀s| > e4µkt
} k−1∏
j=1
P
{
|Q̀s| ≤ e4µjs
}
≥ C
∞∑
k=1
P
{
|Q̀s| > e4µkt
}
. (23)
Assume now that the distribution of M is degenerate. By assumption, P{|M | = 1} <
< 1. Consequently, P{|M | = γ} = 1 for some γ ∈ (0, 1). An easy calculation reveals
that, in this case, the analogue of (23) holds with e−4µ = γ.
Recall that g′(u) = u−1f(u) for u > s. Let us show that
∞∫
s
g′(u)P{e−4µ|Q̀s| > u}du <∞. (24)
Inequality (24) follows from the inequalities
∞
(22)
>
∞∫
s
f ′(t)P{sup
k≥1
e−4µk|Q̀s
k| > t}dt
(23)
≥
(23)
≥ C
∞∫
s
f ′(t)
∞∑
k=1
P{e−4µk|Q̀s
k| > t}dt ≥
≥ C
e4µ − 1
∞∫
s
f ′(t)
∞∫
e4µt
P{|Q̀s| > z}
z
dzdt ≥
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464 A. M. IKSANOV, U. RÖSLER
≥ const +
C
e4µ − 1
∞∫
s
f(t)
t
P{e−4µ|Q̀s| > t}dt.
Now (24) implies Eg(|Q̀s|) <∞. By taking conditional expectations and using (15), we
can prove that
Eg(|Q̀|) = Eg(|Q1 +M1Q2|) <∞.
The latter implies Eg(|Q|) < ∞ (a similar situation has been treated at the end of the
proof of Proposition 3). The proof of Theorem 2 (a) is complete.
For later use, it is worth recording the following corollary which can be read from the
previous proof.
Corollary 1. Assume that Assumption A and (6) hold, and EX ∈ (0,∞) and
E log+ |Q| <∞. Then (10) implies Eg(|Q|) <∞.
Proposition 4. Assume that Assumption B holds and lim
n→∞
Πn = 0 a.s. Then
Ef(|Z∞|) <∞ ⇐ Ef(|Q|) <∞, Ef
(
sup
n≥0
|Πn|
)
<∞.
Proof. Define the random times N0 := 0,
Ni+1 := inf{n > Ni : |Πn| < |ΠNi
|}, i = 0, 1, . . . .
Clearly, ENi <∞, i = 1, 2, . . . . For k = 1, 2, . . . set
M ′
k := |MNk−1+1| . . . |MNk
|, Π′
0 := 1, Π′
k := M ′
1 . . .M
′
k,
Q′
k := |QNk−1+1| + |MNk−1+1||QNk−1+2| + . . .+ |MNk−1+1| . . . |MNk−1||QNk
|.
Then we have that (M ′
k, Q
′
k) are independent copies of
(
|ΠN1 |,
N1∑
k=1
|Πk−1||Qk|
)
and,
moreover,
∞∑
k=1
|Πk−1||Qk| =
∞∑
k=1
Π′
k−1Q
′
k. (25)
Let us show that we can use the implication ⇒ of Theorem 2(a) on the vector
(
|ΠN1 |,
N1∑
k=1
|Πk−1||Qk|
)
. Since |ΠN1 | ∈ (0, 1) a.s. and P{|ΠN1 | = 1} = 0, it remains to
verify that the integral in (5)
(
with (|M |, |Q|) replaced by
(
|ΠN1 |,
∑N1
k=1
|Πk−1||Qk|
))
converges. By Lemma 1, f � g, where g(x) =
∫ x
0
(f(u)/u)du, and f grows faster than
any power of logarithm. Thus, if we can show that
Ef
(
N1∑
k=1
|Πk−1||Qk|
)
<∞, (26)
then
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1) (26) implies that E log+
(∑N1
k=1
|Πk−1||Qk|
)
< ∞; this, in turn, allows us to
conclude that the integral in (5) converges and, therefore, the needed part of Theorem 2 (a)
applies;
2) since (26) is equivalent to Eg
(∑N1
k=1
|Πk−1||Qk|
)
<∞, by Theorem 2 (a), (26)
implies Ef
(∑∞
k=1
Π′
k−1Q
′
k
)
<∞ and, hence, Ef
(∑∞
k=1
|Πk−1||Qk|
)
<∞ in view
of (25).
Let U be a random variable distributed like |Q| and independent of sup
k≥0
|Πk|. We now
prove (26):
Ef
(
N1∑
k=1
|Πk−1||Qk|
)
≤ E
N1∑
k=1
f(|Πk−1||Qk|) =
(subadditivity of f and N1 <∞ a.s.)
=
∞∑
n=1
E
(
n∑
k=1
f(|Πk−1||Qk|)
)
1{N1=n} =
∞∑
n=1
E (f(|Πn−1||Qn|)) 1{N1>n−1} =
(the change of order of summation is justified by the fact that all summands are nonnega-
tive)
=
∞∑
n=1
∞∫
0
E(f(|Πn−1|q)1{N1>n−1})dP{|Qn| ≤ q} =
(Qn is independent of both Πn−1 and 1{N1>n−1})
=
∞∫
0
∞∑
n=0
Ef(|Πn|q)1{N1>n}dP{U ≤ q} = EN1
∞∫
0
Ef(qsup
k≥0
|Πk|)dP{U ≤ q} =
(
this, is in fact, Lemma 2 of [22]:
Ef
(
qsup
k≥0
|Πk|
)
= Ef
(
q exp(− inf
k≥0
Sk)
)
=
= (EN1)−1
∞∑
n=0
Ef(q exp(−Sn))1{S0≤0,...,Sn≤0} =
= (EN1)−1
∞∑
n=0
Ef(q|Πn|)1{N1>n};
Keener assumed that ES1 exists, but this condition is not needed
)
= EN1Ef(Usup
k≥0
|Πk|) ≤ pEN1Ef(|Q|)Ef(sup
k≥0
|Πk|) <∞
(we have used (3) and the assumptions of the proposition). The proof of Proposition 4 is
finished.
We are now ready to give proof of Theorem 2 (b).
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466 A. M. IKSANOV, U. RÖSLER
Proof of Theorem 2 (b). Necessity. Let Ef(|Q|) < ∞, Ef(|M | ∨ 1) < ∞, and
E|M |r < 1 if r > 0. Using Lemma 1 (a), we conclude that Eg(|M | ∨ 1) < ∞, where
g(x) =
∫ x
0
(f(u)/u)du. Let us check that
ã :=
∞∫
0
h(x)P{log+ |M | > x}dx <∞,
where h(x) = f(ex). Note first that
g(ex) =
x∫
−∞
f(eu)du = const +
x∫
0
f(eu)du.
Further, we have ã = E
∫ log+ |M |
0
f(ex)dx = −const + Eg(elog
+ |M |) = −const +
+ Eg(|M | ∨ 1) <∞. Inequality (4) now implies that
∞∫
0
ψ(x)P{log+ |M | > x}dx <∞.
According to Theorem 2 of [23], the latter condition together with the conditions E|M |r <
< 1, if r > 0, and E log |M | ∈ [−∞, 0) allows us to conclude that Eψ
(
sup
n≥0
(−Sn)
)
<
< ∞, where Sn = − log |Πn|. From (4) it follows that
∞ > Ef
(
exp(sup
n≥0
(−Sn)
)
= Ef
(
sup
n≥0
|Πn|
)
.
It remains to apply Proposition 4 to conclude that Ef(|Z∞|) <∞.
Sufficiency. Let now Ef(|Z∞|) <∞. By Proposition 3, ∞ > Ef(|Q|) and either
∞ > Ef
(
sup
n≥0
|Πn|
)
= Eh
(
sup
n≥0
(−Sn)
)
(27)
or
∞ > Ef
(
sup
n≥0
|Π̀n|
)
= Eh
(
sup
n≥0
(−S̀n)
)
, (28)
where S̀n = − log |Π̀n| is the random walk with a step distributed like − log |M1M2|,
and h(x) = f(ex). Let us check that (28) ensures Ef(|M | ∨ 1) < ∞ and, if r > 0,
E|M |r < 1. The proof for (27) is simpler.
In view of (4), we have
Eψ
(
sup
n≥0
(−S̀n)
)
<∞. (29)
By the assumption of the theorem, E(−S̀1) = E log |M1M2| ∈ (−∞, 0). An appeal to
Theorem 1 [23] allows us to conclude that
∞∫
0
ψ(x)P{log+ |M1M2| > x}dx <∞.
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According to (4), b̂ :=
∫ ∞
0
h(x)P{log+ |M1M2| > x}dx <∞. But
b̂ = E
log+ |M1M2|∫
0
f(ex)dx = −const + Eg(elog
+ |M1M2|) =
= −const + Eg(|M1M2| ∨ 1)
and f � g. Therefore, Ef(|M1M2| ∨ 1) <∞ and
Ef(|M | ∨ 1) ≤ (P{|M | > 1})−1
Ef(|M1M2| ∨ 1) <∞.
If r > 0, then (29) implies 1 > E|M1M2|r = (E|M |r)2 (see Remark 1 [23]). The proof
of Theorem 2 (b) is finished.
4. Proofs related to the BRW. Let tr be a rooted family tree associated with a point
process M. We say that (tr, X) is a labelled tree if each individual (vertex) θ ∈ tr\{0} is
assigned its displacement X(θ) from its parent. The BRW defines a probability measure
µ on the set of labelled trees. Lyons in [7] constructed a new probability measure µ̂∗
on the set of infinite labelled trees with distinguished rays (a ray is an infinite line of
descent starting from the root). It is under this measure µ̂∗ we can successfully bound the
martingale limit W by a perpetuity from the above, and sup
k≥0
Wk by a largest summand of
a perpetuity from below.
Under µ̂∗, the usual family tree is replaced by a size-biased tree that has a ray with a
special status. This single ray is often called a trunk or spine.
For k = 1, 2, . . . , let vk be the individual belonging to the trunk and sitting at the k-th
generation, v0 be an initial ancestor; and let Avk,i, i = 1, 2, . . . , be the displacements of
children of vk from vk. For k = 1, 2, . . . , let M̂k be the random variable which gives the
displacement of vk from her mother divided by m(γ) and Q̂k = m−1(γ)
∑
i
eγAvk−1,i .
Note that, by construction, the random vectors (M̂k, Q̂k), k = 1, 2, . . . , are independent
and identically distributed1 .
Let G be the σ-field generated by the reproduction of vk, k = 1, 2, . . ., i.e., by the
sequence of independent point processes M̂k with pointsAvk−1,i, i = 1, 2, . . . . Set Π̂0 =
= 1 and Π̂k := M̂1 . . . M̂k, k = 1, 2, . . . .Now we can write the two essential inequalities
which, in fact, were found by Lyons:
Eµ̂∗(Wn|G) ≤
n∑
k=1
Π̂k−1Q̂k, (30)
sup
k≥0
Wk ≥ sup
k≥0
Π̂kQ̂k+1 under µ̂∗. (31)
1 Note that the proof of Theorem 2 [3] gives erroneous impression that M̂k and Q̂k are independent as
well. Fortunately, this gap does not affect the proof of that result.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
468 A. M. IKSANOV, U. RÖSLER
Proof of Theorem 1 (b). Necessity. Under µ̂∗, M̂1
d= Z and Q̂1 has the size-biased
distribution corresponding to the distribution ofW1. Thus, the assumptions of the theorem
can be rewritten in terms of M̂ and Q̂ as follows:
µ̂∗{M̂ > 1} > 0, Eµ̂∗ log M̂ ∈ [−∞, 0), Eµ̂∗f(Q̂) <∞
and if r > 0
Eµ̂∗M̂r < 1.
Since W1 =
∑K
i=1
Yi, we have Yi ≤W1, i = 1, 2, . . . . Hence,
Eµ̂∗f(M̂ ∨ 1) = Ef(Z ∨ 1) = E
∞∑
i=1
Yif(Yi ∨ 1) ≤ EW1f(W1 ∨ 1) <∞.
We conclude that all the assumptions of necessity of Theorem 2 (b) hold. Therefore, the
right-hand side of (30) converges µ̂∗ a.s. to a random variable Ẑ∞, say and Eµ̂∗f(Ẑ∞) <
<∞. From the results of Lyons it follows thatWn converges toW µ̂∗ a.s. (apply Fatou’s
lemma to (30) to conclude Eµ̂∗
(
lim inf
n→∞
Wn|G
)
≤ Ẑ∞; hence, lim inf
n→∞
Wn <∞ under µ̂∗;
from the change of measure construction it follows that 1/Wn is a positive µ̂∗ martingale
with respect to an appropriate filtration; hence the statement). By Fatou’s lemma, we have
Eµ̂∗(W |G) ≤ Ẑ∞.
Since f is nondecreasing and concave, then using Jensen’s inequality and applying the
expectation operator one more results in
Eµ̂∗(f(W )|G) ≤ f(Eµ̂∗(W |G)) ≤ f(Ẑ∞),
Eµ̂∗f(W ) ≤ Eµ̂∗f(Ẑ∞). (32)
We have already proved that the right-hand side of (32) is finite. Hence, we have
Eµ̂∗f(W ) <∞.
Using the Laplace – Stieltjes transforms, it can be easily checked that EWnf(Wn) =
= Eµ̂∗f(Wn), n = 1, 2, . . . , implies
EWf(W ) = Eµ̂∗f(W ). (33)
This completes the proof of this part of Theorem 1 (b).
To prove Theorem 1 (b) in the reverse direction, we need a lemma. It proposes a
µ̂∗-counterpart of the inequality obtained in Lemma 2 [12].
Lemma 3. For each a > 0 small enough, there exists B > 1 such that whenever
t > 1, the following inequalities hold:
µ̂∗{W > t} ≤ µ̂∗
{
sup
n≥0
Wn > t
}
≤ Bµ̂∗{W > at}.
In particular, for any nonnegative, nondecreasing and anti-starshaped (in particular, con-
cave) function h,
Eµ̂∗h(W ) <∞ ⇔ Eµ̂∗h
(
sup
n≥0
Wn
)
<∞.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
SOME MOMENT RESULTS ABOUT THE LIMIT OF A MARTINGALE RELATED ... 469
Proof. The left-hand side inequality is obvious. Let us prove the rest. It can be
checked that under µ̂∗, the following equality of distributions holds:
W
d=
1
mn(γ)
eγvnW +
1
mn(γ)
′∑
|u|=n
eγAuVu, (34)
where
∑′
|u|=n
denotes the summation over all individuals of the n-th generation (of the
size-biased tree) but vn; given the information about first n generations in the size-biased
tree Vu are independent copies of a random variable V with distribution P{W ∈ dx}
which is also independent of W .
In what follows, we write E and P instead of Eµ̂∗ and µ̂∗. We can choose c, d > 0
such that r := E(V ∧ c) = E(W ∧ d) ∈ (0, 1). Fix any a ∈ (0, r). Consider the
events En :=
{
max
0≤i≤n−1
Wi ≤ t, Wn > t
}
, n = 1, 2, . . .. Without loss of generality,
we can assume that the set {u : |u| = n} is enumerated in some way such that vn is
the first individual. Keeping this in mind, denote by {αk, k = 1, 2, . . .} realizations of{
1
mn(γ)Wn
eγAu , |u| = n
}
. Note that
∑
k
αk = 1. Define the event
D :=
1
mn(γ)Wn
eγvn(W ∧ d) +
1
mn(γ)Wn
′∑
|u|=n
eγAu(Vu ∧ c) > a
.
Then almost surely
P{D|Fn} = P
{
η := α1(W ∧ d) +
∞∑
k=2
αk(Vk ∧ c) − a > 0
}
≥ 1
B
,
where
1
B
:=
(r − a)2
(E(V ∧ c− a)2) ∨ (E(W ∧ d− a)2)
∈ (0, 1). To get the latter inequality,
we have used
P{η > 0} ≥ (Eη)2
Eη2
,
which is applicable as Eη = r − a > 0. Eη2 is estimated as follows:
Eη2 = α2
1E(W ∧ d− a)2 + E(V ∧ c− a)2
∞∑
k=2
α2
k + 2
(
E(V ∧ c− a)
)2 ∑
1≤i<j
αiαj ≤
≤
(
(E(W ∧ d− a)2
)
∨
(
E(V ∧ c− a)2)
) ∞∑
k=1
α2
k + 2
(
E(V ∧ c− a)
)2 ∑
1≤i<j
αiαj ≤
≤
(
(E(W ∧ d− a)2
)
∨
(
E(V ∧ c− a)2)
) ∞∑
k=1
α2
k + 2
∑
1≤i<j
αiαj
=
=
(
(E(W ∧ d− a)2
)
∨
(
E(V ∧ c− a)2)
)
.
Since En ∈ Fn, we have P{D
⋂
En} = EP{D|Fn}1En ≥ (1/B)P{En}. If P{En} �=
�= 0, the latter implies
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
470 A. M. IKSANOV, U. RÖSLER
P{D|En} ≥ 1
B
. (35)
For t > 1, we have
P{W > at|En}
(34)
≥ P
1
mn(γ)Wn
eγvnW +
1
mn(γ)Wn
′∑
|u|=n
eγAuVu >
at
Wn
∣∣∣∣En
≥
≥ P{D|En}
(35)
≥ 1
B
.
The inequality
P{W > at} ≥
∞∑
n=1
P{W > at|En}P{En} ≥
(
1
B
)
P
{
sup
n≥0
Wn > t
}
completes the proof of the first part of Lemma 3. To prove the second part, we should only
note that the implication ⇒ follows from the inequality Eh(sup
n≥0
Wn) ≤ BEh(W/a) ≤
≤ (B/a)Eh(W ).
Proof of Theorem 1 (b). Sufficiency. Assume now that E logZ ∈ (−∞, 0) and
EWf(W ) < ∞. Then Eµ̂∗ log M̂ ∈ (−∞, 0) and in view of (33), Eµ̂∗f(W ) < ∞.
Therefore, by Lemma 3, Eµ̂∗f
(
sup
k≥0
Wk
)
<∞. In view of (31),
∞ > Eµ̂∗f
(
sup
k≥0
Wk
)
≥ Eµ̂∗f
(
sup
k≥0
Π̂kQ̂k+1
)
.
Let us now apply Proposition 3 on the pair (M̂, Q̂) to get Eµ̂∗f(Q̂) < ∞ and either
Eµ̂∗f
(
sup
n≥0
Π̂n
)
< ∞ or Eµ̂∗f
(
sup
n≥0
`̂Πn
)
< ∞ . Exactly the same analysis as in the
proof of Theorem 1 (b) (implication ⇐) shows that Eµ̂∗f(M̂ ∨ 1) < ∞ and, if r > 0,
Eµ̂∗M̂r < 1. It remains to recall that
Eµ̂∗f(Q̂) = EW1f(W1), Eµ̂∗f(M̂ ∨ 1) = Ef(Z ∨ 1), Eµ̂∗M̂r = EZr.
However, the condition Ef(Z∨1) <∞ can be omitted as it is implied by EW1f(W1) <
< ∞. The proof is complete.
Proof of Theorem 1 (a) goes the similar but simpler way as that of Theorem 1 (b).
The only difference is that while Theorem 1 (b) uses Theorem 2 (b) and Proposition 3,
Theorem 1 (a) appeals to Theorem 2 (a) and Corollary 1. We can use these statements
as the condition M(−∞,−γ−1 logm(γ)) = 0 a.s. implies that the distribution of the
random variable Z (and hence of M̂ ) is concentrated on [0, 1]. Recall that, throughout the
paper, we assumed that P{Z = 1} < 1.
Acknowledgement. A part of this work was done while A. M. Iksanov was visiting
Kiel in December 2004. He gratefully acknowledges the hospitality and financial support
received from Christian-Albrechts-Universität zu Kiel.
1. Rösler U. A fixed point theorem for distributions // Stochast. Process. and Appl. – 1992. – 42. – P. 195 –
214.
2. Liu Q. Fixed points of a generalized smoothing transformation and applications to the branching random
walk // Adv. Appl. Probab. – 1998. – 30. – P. 85 – 112.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
SOME MOMENT RESULTS ABOUT THE LIMIT OF A MARTINGALE RELATED ... 471
3. Iksanov A. M. Elementary fixed points of the BRW smoothing transforms with infinite number of sum-
mands // Stochast. Process. and Appl. – 2004. – 114. – P. 27 – 50.
4. Kingman J. F. C. The first birth problem for an age-dependent branching process // Ann. Probab. – 1975.
– 3. – P. 790 – 801.
5. Biggins J. D. Martingale convergence in the branching random walk // J. Appl. Probab. – 1977. – 14. –
P. 25 – 37.
6. Liu Q. Sur une équation fonctionnelle et ses applications: une extension du théorème de Kesten – Stigum
concernant des processus de branchement // Adv. Appl. Probab. – 1997. – 29. – P. 353 – 373.
7. Lyons R. A simple path to Biggins’ martingale convergence for branching random walk // Classical and
Modern Branching Processes / Eds K. B. Athreya, P. Jagers: IMA Vol. in Math. and Appl. – Berlin:
Springer, 1997. – 84. – P. 217 – 221.
8. Goldie C. M., Maller R. A. Stability of perpetuities // Ann. Probab. – 2000. – 28. – P. 1195 – 1218.
9. Sgibnev M. S. On the existence of submultiplicative moments for the stationary distributions of some
markovian random walks // J. Appl. Probab. – 1999. – 36. – P. 78 – 85.
10. Alsmeyer G., Rösler U. On the existence of φ-moments of the limit of a normalized supercritical Galton –
Watson process // J. Theor. Probab. – 2004. – 17. – P. 905 – 928.
11. Iksanov A. M. On some moments of the limit random variable for a normalized supercritical Galton –
Watson process // Focus on Probab. Theory. – New York: Nova Sci. Publ., 2004. – P. 151 – 158.
12. Biggins J. D. Growth rates in the branching random walk // Z. Wahrscheinlichkeitstheor. und verw. Geb.
– 1979. – 48. – S. 17 – 34.
13. Rösler U., Topchii V. A., Vatutin V. A. Convergence conditions for the weighted branching process //
Discrete Math. and Appl. – 2000. – 10. – P. 5 – 21.
14. Kuhlbusch D. Moment conditions for weighted branching processes: PhD thesis. – Univ. Münster, 2004.
15. Hardy R., Harris S. C. Spine proofs for Lp-convergence of branching-diffusion martingales. – 2004. –
Math. Preprint. – Univ. Bath. Available online at http://www.bath.ac.uk/∼massch/Research/Papers/spine-
Lp-cgce.pdf.
16. Kellerer H. G. Ergodic behaviour of affine recursions III: positive recurrence and null recurrence // Techn.
Repts. – Math. Inst. Univ. München, 1992.
17. Vervaat W. On a stochastic difference equation and a representation of non-negative infinitely divisible
random variables // Adv. Appl. Probab. – 1979. – 11. – P. 750 – 783.
18. Goldie C. M., Grübel R. Perpetuities with thin tails // Ibid. – 1996. – 28. – P. 463 – 480.
19. Araman V. F., Glynn P. W. Tail asymptotics for the maximum of perturbed random walk. – 2004. – Preprint.
Available online at http://pages.stern.nyu.edu/∼varaman.
20. Feller W. An introduction to probability theory and its applications. – New York: John Wiley & Sons,
1966. – Vol. 2.
21. Grincevičius A. K. A random difference equation // Lith. Math. J. – 1981. – 21. – P. 302 – 306.
22. Keener R. A note on the variance of a stopping time // Ann. Statist. – 1987. – 15. – P. 1709 – 1712.
23. Sgibnev M. S. Submultiplicative moments of the supremum of a random walk with negative drift // Statist.
Probab. Lett. – 1997. – 32. – P. 377 – 383.
Received 10.10.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4
|
| id | umjimathkievua-article-3467 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:43:08Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/52/7f5c79abaacacc9a7b1b53f6b6c88052.pdf |
| spelling | umjimathkievua-article-34672020-03-18T19:55:25Z Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities Деякі результати про моменти границі мартингала, пов'язаного з надкритичним гіллястим випадковим блуканням, та розв'язків деяких стохастичних різницевих рівнянь Iksanov, O. M. Іксанов, О. М. Let $\mathcal{M}_{(n)},\quad n = 1, 2,...,$ be the supercritical branching random walk in which the family sizes may be infinite with positive probability. Assume that a natural martingale related to $\mathcal{M}_{(n)},$ converges almost surely and in the mean to a random variable $W$. For a large subclass of nonnegative and concave functions $f$ , we provide a criterion for the finiteness of $\mathbb{E}W f(W)$. The main assertions of the present paper generalize some results obtained recently in Kuhlbusch’s Ph.D. thesis as well as previously known results for the Galton-Watson processes. In the process of the proof, we study the existence of the $f$-moments of perpetuities. Нехай $\mathcal{M}_{(n)},\quad n = 1, 2,...,$ — надкритичне випадкове блукання, у якому розмір родини може бути нескінченним з додатною ймовірністю. Припустимо, що стандартний мартингал, пов'язаний з $\mathcal{M}_{(n)},$ збігається майже напевно і в середньому до випадкової величини $W$. Для великого підкласу невід'ємних та вгнутих функцій $f$ наведено критерій скінченності $\mathbb{E}W f(W)$. Основні твердження роботи узагальнюють деякі результати, отримані в дисертації Кульбуша, а також результати, відомі для процесів Гальтона-Ватсона. У процесі доведення досліджується існування $f$ - моментів розв'язків деяких стохастичних різницевих рівнянь. Institute of Mathematics, NAS of Ukraine 2006-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3467 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 4 (2006); 451–471 Український математичний журнал; Том 58 № 4 (2006); 451–471 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3467/3671 https://umj.imath.kiev.ua/index.php/umj/article/view/3467/3672 Copyright (c) 2006 Iksanov O. M. |
| spellingShingle | Iksanov, O. M. Іксанов, О. М. Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities |
| title | Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities |
| title_alt | Деякі результати про моменти границі мартингала, пов'язаного з надкритичним гіллястим випадковим блуканням, та розв'язків деяких стохастичних різницевих рівнянь |
| title_full | Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities |
| title_fullStr | Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities |
| title_full_unstemmed | Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities |
| title_short | Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities |
| title_sort | some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3467 |
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