Regular variation in the branching random walk
initial ancestor located at the origin of the real line. For n = 0, 1, . . . , let Wn be the moment generating function of Mn normalized by its mean. Denote by AWn any of the following random variables: maximal function, square function, L1 and a.s. limit W, supn≥0 |W − Wn|, supn≥0 |Wn+1 − Wn|. Un...
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Iksanov, A. Polotskiy, S. 2009-11-10T14:49:23Z 2009-11-10T14:49:23Z 2006 Regular variation in the branching random walk / A. Iksanov, S. Polotskiy // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 38–54. — Бібліогр.: 25 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4440 519.21 initial ancestor located at the origin of the real line. For n = 0, 1, . . . , let Wn be the moment generating function of Mn normalized by its mean. Denote by AWn any of the following random variables: maximal function, square function, L1 and a.s. limit W, supn≥0 |W − Wn|, supn≥0 |Wn+1 − Wn|. Under mild moment restrictions and the assumption that {W1 > x} regularly varies at ∞, it is proved that P{AWn > x} regularly varies at ∞ with the same exponent. All the proofs given are non-analytic in the sense that these do not use Laplace–Stieltjes transforms. The result on the tail behaviour of W is established in two distinct ways. en Інститут математики НАН України Regular variation in the branching random walk Article published earlier |
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Regular variation in the branching random walk Iksanov, A. Polotskiy, S. |
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Regular variation in the branching random walk |
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regular variation in the branching random walk |
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Iksanov, A. Polotskiy, S. |
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Iksanov, A. Polotskiy, S. |
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initial ancestor located at the origin of the real line. For n = 0, 1, . . . , let Wn be the moment generating function of Mn normalized by its mean. Denote by AWn any of
the following random variables: maximal function, square function, L1 and a.s. limit
W, supn≥0 |W − Wn|, supn≥0 |Wn+1 − Wn|. Under mild moment restrictions and
the assumption that {W1 > x} regularly varies at ∞, it is proved that P{AWn > x}
regularly varies at ∞ with the same exponent. All the proofs given are non-analytic in the sense that these do not use Laplace–Stieltjes transforms. The result on the tail behaviour of W is established in two distinct ways.
|
| issn |
0321-3900 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/4440 |
| citation_txt |
Regular variation in the branching random walk / A. Iksanov, S. Polotskiy // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 38–54. — Бібліогр.: 25 назв.— англ. |
| work_keys_str_mv |
AT iksanova regularvariationinthebranchingrandomwalk AT polotskiys regularvariationinthebranchingrandomwalk |
| first_indexed |
2025-11-25T15:10:05Z |
| last_indexed |
2025-11-25T15:10:05Z |
| _version_ |
1850516044491259904 |
| fulltext |
Theory of Stochastic Processes
Vol. 12 (28), no. 1–2, 2006, pp. 38–54
UDC 519.21
ALEKSANDER IKSANOV AND SERGEY POLOTSKIY
REGULAR VARIATION IN THE BRANCHING RANDOM WALK
Let {Mn, n = 0, 1, . . . } be the supercritical branching random walk starting with one
initial ancestor located at the origin of the real line. For n = 0, 1, . . . , let Wn be the
moment generating function of Mn normalized by its mean. Denote by AWn any of
the following random variables: maximal function, square function, L1 and a.s. limit
W , supn≥0 |W − Wn|, supn≥0 |Wn+1 − Wn|. Under mild moment restrictions and
the assumption that {W1 > x} regularly varies at ∞, it is proved that {AWn > x}
regularly varies at ∞ with the same exponent. All the proofs given are non-analytic
in the sense that these do not use Laplace–Stieltjes transforms. The result on the
tail behaviour of W is established in two distinct ways.
An introduction, notation, and results
Let M be a point process on R, i.e. a random, locally finite counting measure.
Explicitly,
M(A)(ω) :=
J(ω)∑
i=1
δXi(ω)(A),
where J := M(R), {Xi : i = 1, J} are the points of M, A is any Borel subset of R and
δx is the Dirac measure concentrated at x. We assume that M has no atom at +∞, and
the J may be deterministic or random, finite or infinite with positive probability.
Let {Mn, n = 0, 1, . . . } be a branching random walk (BRW), i.e. the sequence of
point processes which, for any Borel set B ⊆ R, are defined as follows: M0(B) = δ0(B),
Mn+1(B) :=
∑
r
Mn,r(B − An,r), n = 0, 1, . . . ,
where {An,r} are the points of Mn, and {Mn,r} are independent copies of M. The more
detailed definition of the BRW can be found in, for example, [3,17,22].
In the case where P{J < ∞} = 1 we assume that EJ > 1. In the contrary case,
the condition holds automatically. Thus, we only consider the supercritical BRW. As a
consequence, P{Mn(R) > 0 for all n} > 0.
In what follows, we use the notation that is generally accepted in the literature on the
BRW: Au denotes the position of a generic point u = i1 . . . in on R; the record |u| = n
means that u is a point of Mn; the symbol
∑
|u|=n denotes the summation over all points
of Mn; Fn = σ(M1, . . . ,Mn) denotes the σ-field generated by {Mk, k = 1, . . . , n}; F0
is the trivial σ-field.
Define the function
m(y) := E
∫
R
eyxM(dx) = E
∑
|u|=1
eyAu ∈ (0,∞], y ∈ R,
2000 AMS Mathematics Subject Classification. Primary 60G42, 60J80; Secondary 60E99.
Key words and phrases. Branching random walk, supercritical case, perpetuity.
38
REGULAR VARIATION IN THE BRANCHING RANDOM WALK 39
and assume that there exists a γ > 0 such that m(γ) < ∞. Set Yu := eγAu/m|u|(γ) and
Wn := m(γ)−n
∫
R
eγxMn(dx) =
∑
|u|=n
Yu.
As is well known (see, for example, [12]), the sequence {(Wn,Fn), n = 0, 1, . . .} is a
non-negative martingale. Notice that W0 = EWn = 1.
Let {dn, n = 1, 2, . . .} be the martingale difference sequence, i.e.
Wn = 1 +
n∑
k=1
dk, n = 1, 2, . . .
The square function S and maximal function W ∗ are defined by
S :=
(
1 +
∞∑
k=1
d2
k
)1/2
and W ∗ := sup
n≥0
Wn.
Set also
Sn :=
(
1 +
n∑
k=1
d2
k
)1/2
, n = 1, 2, . . . and Δ := sup
n≥1
|dn|.
Recall that since Wn is a non-negative martingale all the defined variables are a.s. finite
(for the finiteness of S for general L1-bounded martingales, we refer to [1] or to Theorem 2
on p.390[11]).
When the martingale Wn is uniformly integrable, we denote, by W∞ = W , its L1 and
a.s. limit and then define
M := sup
n≥0
|W − Wn| = sup
n≥0
|
∞∑
k=n+1
dk|.
Lemma 1 [21] (see also [2] for a slightly different proof in the case J < ∞ a.s.) states
that there exist r ∈ (0, 1) and θ = θ(r) > 1 such that, whenever t > 1,
(1) P{W > t} ≤ P{W ∗ > t} ≤ θP{W > rt}.
This suggests that the tail behaviours of W and W ∗ are quite similar.
Let now {fn :=
∑n
k=0 gk, n = 0, 1, . . . } be any martingale. It is well known that
the distributions of maximal f∗ := sup
n≥0
|fn| and square S(f) := (
∑∞
k=0 g2
k)1/2 functions
are close in many respects. The evidence in favor of such a statement is provided by,
for example, the (moment) Burkholder—Gundy—Davis inequality (Theorem 1.1 [10])
or the distribution function inequalities like (34) of this paper. From [9] and [10] and
many other subsequent works, it follows that there exist a subset H of the set of all
martingales and a class A of operators on martingales such that the distributions of A1h
and A2h are close in an appropriate sense whenever Ai ∈ A and h ∈ H. Often, it can
be possible to express this closeness via moment or distribution function inequalities like
those mentioned above. Keeping this in mind, it would not be an unrealistic conjecture
that the regular variation of P{A1h > x} is equivalent to that of P{A2h > x}, where Ai
and h belong to some subsets of operators and martingales, respectively, that may be
different from A and H. On the other hand, let us notice that, as far as we know, the
conjecture does not follow from previously known results on martingales.
The aim of this paper is to prove a variant of the conjecture for the martingales Wn and
operators Ai, i = 1, 5 given as follows: A1W = W ∗, A2W = Δ, A3W = S, A4W = W∞,
A5W = M .
40 ALEKSANDER IKSANOV AND SERGEY POLOTSKIY
In addition to the notation introduced above, other frequently used notations and
conventions include: L(t) denotes a function that slowly varies at infinity; 1A denotes
the indicator function of the set A; f(t) ∼ g(t) is the abbreviation of the limit relation
lim
t→∞
f(t)
g(t) = 1; x+ := max(x, 0); x ∧ y = min(x, y); x ∨ y = max(x, y); we write Pn{·}
instead of P{·|Fn} and En{·} instead of E{·|Fn}; the record ”const” denotes a constant,
whose value is of no importance and may be different on different appearances.
Now we are ready to state our result.
Proposition 1.1. Assume that there exist β > 1 and ε > 0 such that
(2) kβ := E
∑
|u|=1
Y β
u < 1, E
∑
|u|=1
Y β+ε
u < ∞ and
(3) P{W1 > x} ∼ x−βL(x).
Then
(I) P{W ∗ > x} ∼ P{Δ > x} ∼ P{S > x} ∼ (1 − kβ)−1P{W1 > x};
(II) Wn converges almost surely and in mean to a random variable W and
(4) P{W > x} ∼ (1 − kβ)−1P{W1 > x};
P{M > x} ∼ (1 − kβ)−1P{W1 > x}.
Remark 1.1 We are not aware of any papers on branching processes which investigate
the tail behaviour of random variables like Δ, M , or S. [21] is the only paper we know
of that deals with the tail behaviour of random variables like W ∗.
Remark 1.2 When γ = 0 and J < ∞ a.s., Wn reduces to the (supercritical) normalized
Galton—Watson process. In this case, (4) was proved in [5] for non-integer β and in
[13] for integer β. When γ > 0, J < ∞ a.s. and M(−∞,−γ−1 log m(γ)) = 0 a.s., W
can be viewed as a limit random variable in the Crump-Mode branching process. In this
case (4) was established in [6] for non-integer β. The technique used in the last three
cited works is purely analytic (based on using the Laplace—Stieltjes transforms and the
Abel—Tauberian theorems) and completely different from ours. On the other hand, let
us notice that the above-mentioned analytic approach was successfully employed and
further developed by the second-named author. In 2003, in an unpublished diploma
paper, he proved (4) for non-integer β for the general case treated here.
Our desire to find a non-analytic proof of (4) was a starting point for the development
of this paper. In the course of writing, two different (non-analytic) proofs were found.
One of these proofs given in Section 2 falls within the general scope of the paper. The
second given in Section 3 continues a line of research initiated in [17] , [22], [18]. Here,
an underlying idea is that the martingale Wn and the so-called perpetuities have many
features in common. In particular, several non-trivial results on perpetuities (however, it
seems, only those related to perpetuities with not all moments finite) can be effectively
exploited to obtain similar results on the limiting behaviour of Wn. Maybe, we should
recall that, in modern probability, a perpetuity means a random variable
B1 +
∞∑
k=2
A1A2 · · ·Ak−1Bk,
provided the latter series absolutely converges, and where {(Ak, Bk) : k = 1, 2, . . .} are
independent identically distributed random vectors.
The paper is structured as follows. In Section 2, we prove Proposition 1.1. Here, an
essential observation is that, given Fn, Wn+1 looks like a weighted sum of independent
identically distributed random variables. This allows us to exploit the well-known result
REGULAR VARIATION IN THE BRANCHING RANDOM WALK 41
[25] on the tail behaviour of such sums under the regular variation assumption. The
second key ingredient of the proof is the use of the distribution function inequalities
for martingales. In Section 3, we give another proof of (4) which rests on a relation
between the BRW and perpetuities. Here, the availability of Grincevičius—Grey [16]
result on the tail behaviour of perpetuities is crucial. Finally, in Section 4, we discuss the
applicability of Proposition 1.1 to several classes of point processes. The section closes
with two remarks which show that (2) and (3) are not the necessary conditions for a
regular variation of the tails of W ∗, W and a related random variable.
Proof of Proposition 1.1
(I) We will prove the result for W ∗ and Δ simultaneously. To this end, let Q and Q̃
be independent identically distributed random variables, whose distribution is supported
by (a,∞), a > −∞. Assume that P{Q > x} ∼ x−βL(x) for β > 1. In particular, this
assumption ensures that E|Q| < ∞ and P{|Q| > x} ∼ P{Q > x}. With a slight abuse of
the notation, set Qs := |Q| − |Q̃|. Then
(5) 1 − F (x) := P{|Qs| > x} ∼ 2x−βL(x).
Indeed, 1−F (x) = 2
∫ ∞
0 (1−G(x+ y))dG(y), where G(x) = P{|Q| ≤ x}, x ≥ 0. Now (5)
follows from the monotonicity of 1−G, the relation 1−G(x + y) ∼ 1−G(x), y ∈ R and
the Fatou lemma.
The equality
Et(Z) = E
∑
|u|=1
Yut(Yu),
which is assumed to hold for all bounded Borel functions t, defines the distribution of a
random variable Z. More generally,
(6) Et(Z1 · · ·Zn) = E
∑
|u|=n
Yut(Yu),
where Z1, Z2, . . . are independent copies of the Z. Notice that we can permit for (6)
to hold for any Borel function t. In that case, we assume that if the right-hand side is
infinite or does not exist, the same is true for the left-hand side.
Under the assumptions of the theorem, the function kx := E
∑
|u|=1 Y x
u is log-convex
for x ∈ (1, β), k1 = 1 and kβ < 1. Therefore,
(7) kβ−ε < 1 for all ε ∈ (0, β − 1).
Also we can pick a δ ∈ (0, β − 1) such that kβ+δ < 1. By using these facts and equality
(6), we conclude that, with this δ,
(8) E
∑
|u|=n
Y β−δ
u = kn
β−δ < 1 and E
∑
|u|=n
Y β+δ
u = kn
β+δ < 1.
Let us notice, for later needs, that we can choose δ as small as needed. Among other
things, (8) implies that, for x ∈ [1, β + δ],
(9)
∑
|u|=n
Y x
u < ∞ a.s.
Until a further notice, we fix an arbitrary n ∈ N. Put
Tn := |
∑
|u|=n
YuQu| and Xn :=
∑
|u|=n
Yu|Qu|.
42 ALEKSANDER IKSANOV AND SERGEY POLOTSKIY
Given Fn, let {Qu : |u| = n} and {Qs
u : |u| = n} be conditionally independent copies
of the random variables Q and Qs, respectively. In view of (9), an appeal to Lemma
A3.7[25] allows us to conclude that
(10) Pn{Tn > x} ∼
∑
|u|=n
Y β
u P{|Q| > x} a.s.
The cited lemma assumes that each term of the series on the left-hand side has zero
mean, but this condition is not needed in the proof of the result used here.
Denote, by μFn
n , the conditional median of Xn w.r.t. Fn, i.e. μFn
n is a random variable
that satisfies
Pn{Xn − μFn
n ≥ 0} ≥ 1/2 ≤ Pn{Xn − μFn
n ≤ 0} a.s.
Let also μn denote the usual median of Xn. Since μFn
n ≥ 0 a.s., relation (10) yields
lim sup
x→∞
Pn{Tn > x + μFn
n }
P{|Q| > x} ≤
∑
|u|=n
Y β
u a.s.
If we could prove that, for large x,
(11)
Pn{Tn > x + μFn
n }
P{|Q| > x} ≤ Un a.s. and EUn < ∞,
where Un is a random variable, then using the Fatou lemma yields
(12) lim sup
x→∞
E
Pn{Tn > x + μFn
n }
P{|Q| > x} = lim sup
x→∞
P{Tn > x + μn}
P{|Q| > x} ≤ E
∑
|u|=n
Y β
u
(6)
= kn
β .
Since P{|Q| > x + μn} ∼ P{|Q| > x}, (12) implied that
lim sup
x→∞
P{Tn > x}
P{|Q| > x} ≤ kn
β .
On the other hand, by using (10) and the Fatou lemma, the reverse inequality for the
lower limit follows easily. Therefore, as soon as (11) is established, we get
(13) P{Tn > x} ∼ kn
βP{|Q| > x}.
We now intend to show that (11) holds with
(14) Un = const
⎛
⎝ ∑
|u|=n
Y β−δ
u +
∑
|u|=n
Y β+δ
u
⎞
⎠
for an appropriate small δ that satisfies (8). Notice that
(15) EUn = const(kn
β−δ + kn
β+δ) < ∞.
By the triangle inequality and the conditional symmetrization inequality,
(16) (1/2)Pn{Tn > x + μFn
n } ≤ (1/2)Pn{Xn > x + μFn
n } ≤ Pn{|
∑
|u|=n
YuQs
u| > x}.
Let us show that, for x > 0,
Pn{|
∑
|u|=n
YuQs
u| > x} ≤
REGULAR VARIATION IN THE BRANCHING RANDOM WALK 43
(17)
≤ Pn{ sup
|u|=n
Yu|Qs
u| > x} + x−2En
⎛
⎝ ∑
|u|=n
Y 2
u (Qs
u)21{Yu|Qs
u|≤x}
⎞
⎠ := I1(n, x) + I2(n, x).
Let {Y (k)Qs(k) : k = 1, 2, . . . } be any enumeration of the set {YuQs
u : |u| = n}. The
inequality E|Q| < ∞ implies that the series
∑
|u|=n YuQu is absolutely convergent. There-
fore,
∑
|u|=n YuQu =
∑∞
k=1 Y (k)Qs(k). Define
τx :=
{
inf{k ≥ 1 : Y (k)|Qs(k)| > x}, if sup
k≥1
Y (k)|Qs(k)| > x;
+∞, otherwise .
For any fixed m ∈ N and x > 0,
Pn{|
m∑
k=1
Y (k)Qs(k)| > x} ≤
≤ Pn{τx ≤ m − 1} + Pn{|
m∑
k=1
Y (k)Qs(k)| > x, τx ≥ m} ≤
≤ Pn{ sup
1≤k≤m−1
Y (k)|Qs(k)| > x} + Pn{|
τx∧m∑
k=1
Y (k)Qs(k)| > x} ≤
(by the Markov inequality)
≤ Pn{ sup
1≤k≤m−1
Y (k)|Qs(k)| > x} + x−2En
(
m∑
k=1
Y (k)Qs(k)1{τx≥k}
)2
≤
(EnQs(k) = 0 and, given Fn, Qs(k) and 1{τx≥k} are independent)
≤ Pn{ sup
1≤k≤m−1
Y (k)|Qs(k)| > x} + x−2En
m∑
k=1
Y 2(k)(Qs(k))21{Y (k)|Qs(k)|≤x}.
If the distribution of Qs is continuous, sending m → ∞ then completes the proof of (17).
Assume now that the distribution of Qs has atoms. Let R be a random variable with a
uniform distribution on [−1, 1] which is independent of Qs. Given Fn, let {Ru : |u| = n}
be conditionally independent copies of R which are also independent of {Qs
u : |u| = n}.
Since, for all t > 0,
P{|Qs| > t} ≤ 2P{|Qs||R| > t/2},
we have by Theorem 3.2.1[23]
(18) Pn{|
∑
|u|=n
YuQs
u| > t} ≤ 4Pn{|
∑
|u|=n
YuQs
uRu| > t/4},
and the distribution of QsR is (absolutely) continuous. Now we can apply the already
established part of (17) to the right-hand side of (18). Strictly speaking, when the
distribution of Qs has atoms, (17) should be written in a modified form: additional
constants should be added, and Qs
u should be replaced with Qs
uRu. On the other hand,
a perusal of the subsequent proof reveals that only the regular variation of P{|Qs| > x}
plays a crucial role. Therefore, to simplify the notation, we prefer to keep (17) in its
present form. This does not cause any mistakes as P{|QsR| > x} ∼ E|R|βP{|Qs| > x}.
44 ALEKSANDER IKSANOV AND SERGEY POLOTSKIY
Assume temporarily that 1 − F (x) regularly varies with index −β, β ∈ (1, 2). Set
T (x) :=
∫ x
0
y2dF (y). By Theorem 1.6.4[7],
T (x) ∼ β
2 − β
x2(1 − F (x)) ∼ β
2 − β
x2−βL1(x).
Also by Theorem 1.5.3[7], there exists a non-decreasing S(x) such that
(19) T (x) ∼ S(x).
For any Ai > 0 and δ defined in (8), there exists an xi > 0 such that, whenever x ≥
xi, i = 1, 2, 3,
(20) xβ+δ(1 − F (x)) ≥ 1/A1;
(21) xβ−2+δS(x) ≥ 1/A2;
(22) T (x) ≤ (A3 +
β
2 − β
)x2(1 − F (x)) := Bx2(1 − F (x)).
Also for any Ai > 1 and the same δ as above, there exists an xi > 0 such that, whenever
x ≥ xi and ux ≥ xi, i = 4, 5, 6,
(23)
1 − F (ux)
1 − F (x)
≤ A4(u−β+δ ∨ u−β−δ);
(24)
T (ux)
T (x)
≤ A5(u2−β+δ ∨ u2−β−δ);
(25)
T (ux)
T (x)
≤ A6
S(ux)
S(x)
.
Inequalities (23) and (24) follow from Potter’s bound Theorem 1.5.6 (iii)[7]; (25) is im-
plied by (19). Set x0 := max
1≤i≤6
xi and assume that x0 > 1.
To check (11) and (14), we consider three cases: (a) β ∈ (1, 2); (b) β > 2, β �= 2n, n ∈
N; (c) β = 2n, n ∈ N.
(a) For any fixed x ≥ x0 and y > 0,
I1(n, x/y)
1 − F (x)
≤
∑
|u|=n
Pn{Yu|Qs
u| > x/y}
1 − F (x)
=
∑
|u|=n
1 − F (x(yYu)−1)
1 − F (x)
=
=
∑
|u|=n
· · · 1{yYu>x/x0} +
∑
|u|=n
· · · 1{yYu≤x/x0} =: I11(n, x, y) + I12(n, x, y).
Since
(yYu)β+δ ≥ (yYu)β+δ1{yYu>x/x0} ≥ (x/x0)β+δ1{yYu>x/x0},
we get
I11(n, x, y) ≤ (x0y)β+δ
∑
|u|=n Y β+δ
u
xβ+δ(1 − F (x))
(20)
≤ A1x
β+δ
0 yβ+δ
∑
|u|=n
Y β+δ
u .
Further,
I12(n, x, y)
(23)
≤ A4
∑
|u|=n
(yYu)β−δ ∨ (yYu)β+δ ≤
REGULAR VARIATION IN THE BRANCHING RANDOM WALK 45
≤ A4
⎛
⎝yβ−δ
∑
|u|=n
Y β−δ
u + yβ+δ
∑
|u|=n
Y β+δ
u
⎞
⎠ ;
I2(n, x/y)
1 − F (x)
= y2
∑
|u|=n
Y 2
u
∫ x(yYu)−1
0
z2dF (z)
x2(1 − F (x))
(22)
≤ By2
∑
|u|=n
Y 2
u T (x(yYu)−1)
T (x)
=
= By2
⎛
⎝ ∑
|u|=n
· · · 1{yYu>x/x0} +
∑
|u|=n
· · · 1{yYu≤x/x0}
⎞
⎠ =:
=: By2(I21(n, x, y) + I22(n, x, y)).
I21(n, x, y)
(25)
≤ A6
∑
|u|=n
Y 2
u S(x(yYu)−1)
S(x)
1{yYu>x/x0} ≤
≤ A6(x0y)β−2+δS(x0)
∑
|u|=n
Y β+δ
u
xβ−2+δS(x)
(21)
≤ A2A6(x0y)β−2+δS(x0)
∑
|u|=n
Y β+δ
u ;
I22(n, x, y)
(24)
≤ A5
∑
|u|=n
Y 2
u ((yYu)β−2−δ ∨ (yYu)β−2+δ) ≤
≤ A5
⎛
⎝yβ−2−δ
∑
|u|=n
Y β−δ
u + yβ−2+δ
∑
|u|=n
Y β+δ
u
⎞
⎠ .
Thus, according to (17), we have proved that, for x ≥ x0 and y > 0,
(26)
Pn{|
∑
|u|=n YuQs
u| > x/y}
P{|Qs| > x} ≤ const
⎛
⎝yβ−δ
∑
|u|=n
Y β−δ
u + yβ+δ
∑
|u|=n
Y β+δ
u
⎞
⎠ .
In particular, since P{|Qs| > x} ∼ 2P{|Q| > x}, setting y = 1 in (26) and using (16) lead
to (11), with Un being a multiple of the right-hand side of (26).
In the remaining cases, we only investigate the situation where β ∈ (2, 4) and β = 2.
For other values of β, inequality (11) with Un satisfying (14) follows by induction.
(b) β ∈ (2, 4). Given Fn, let {Ñ, Nu : |u| = n} be conditionally independent copies
of a random variable N with normal (0, 1) distribution. Using the approach similar to
that exploited to obtain (18)(this fruitful argument has come to our attention from [25])
allows us to conclude that, for x > 0 and appropriate positive constants c1, c2,
Pn{|
∑
|u|=n
YuQs
u| > x} ≤
(27) ≤ c1Pn{|
∑
|u|=n
YuNuQs
u| > c2x} = c1Pn
⎧⎪⎨
⎪⎩|Ñ |
⎛
⎝ ∑
|u|=n
Y 2
u (Qs
u)2
⎞
⎠
1/2
> c2x
⎫⎪⎬
⎪⎭ .
Notice that P{(Qs)2 > x} regularly varies with index −β/2 ∈ (−2,−1). Also it is obvious
that, if needed, we can reduce δ in (8) to ensure that kβ−2δ < 1 and kβ+2δ < 1. Therefore,
we can use (26) with Yu replaced with Y 2
u and Qs
u replaced with (Qs
u)2, which gives after
a little manipulation that, for x ≥ x
1/2
0 ,
Pn{|
∑
|u|=n YuQs
u| > x}
P{|Qs| > x} ≤
46 ALEKSANDER IKSANOV AND SERGEY POLOTSKIY
≤ const
⎛
⎝ ∑
|u|=n
Y β+2δ
u E|N |β+2δ +
∑
|u|=n
Y β−2δ
u E|N |β−2δ
⎞
⎠ .
By using the same argument as in case (a), we can check that (11) and (14) have been
proved.
(c) β = 2. In the same manner as we have established (26), it can be proved that, for
y > 0 and large x,
Pn{
∑
|u|=n Y 2
u (Qs
u)2 > x/y}
P{(Qs)2 > x} ≤ const
⎛
⎝y2−2δ
∑
|u|=n
Y 2−2δ
u + y2+2δ
∑
|u|=n
Y 2+2δ
u
⎞
⎠ .
Hence, an appeal to (27) assures that (11) and (14) hold in this case too.
We have a representation
(28) Wn+1 =
∑
|u|=n
YuW
(u)
1 ,
where, given Fn, W
(u)
1 are (conditionally) independent copies of W1. Each element of
the set {W (u)
1 : |u| = n} is constructed in the same way as W1, the only exception being
that while W1 is defined on the whole family tree, W
(u)
1 is defined on the subtree with
root u.
We only give a complete proof for Δ. The analysis of W ∗ is similar but simpler, and
hence omitted. From (28), we conclude that |dn+1| is the same as Tn with Qu = W
(u)
1 −1.
Hence, by (10),
1{ max
1≤k≤n
|dk|≤x}Pn{|dn+1| > x} ∼
∑
|u|=n
Y β
u P{|W1 − 1| > x} ∼
(29) ∼
∑
|u|=n
Y β
u P{W1 > x} a.s.,
and, by (13),
(30) P{|dn+1| > x} ∼ kn
βP{|W1 − 1| > x} ∼ kn
βP{W1 > x}.
Recall that
P{Δ > x} = P{|d1| > x} +
∞∑
n=1
P{ max
1≤k≤n
|dk| ≤ x, |dn+1| > x} =
= P{|d1| > x} + E
∞∑
n=1
1{ max
1≤k≤n
|dk|≤x}Pn{|dn+1| > x}.
Using this relation and (29) and applying the Fatou lemma twice allows us to conclude
that
lim inf
x→∞
P{Δ > x}
P{W1 > x} ≥ 1 +
∞∑
n=1
Elim inf
x→∞
1{ max
1≤k≤n
|dk|≤x}Pn{|dn+1| > x}
P{W1 > x} ≥
≥ 1 +
∞∑
n=1
E
⎛
⎝ ∑
|u|=n
Y β
u
⎞
⎠ = (1 − kβ)−1.
REGULAR VARIATION IN THE BRANCHING RANDOM WALK 47
To complete the proof for Δ, we must calculate the corresponding upper limit. For
this, it suffices to check that, for large x and large n ∈ N,
(31)
P{|dn+1| > x}
P{W1 > x} ≤ Cn,
where Cn is a summable sequence, and use the dominated convergence theorem. Taking
the expectation in (11) allows us to conclude that, for n = 1, 2, . . . and large x,
P{|dn+1| > x + μn}
P{W1 > x} ≤ constEUn,
where μn is the median of Vn :=
∑
|u|=n Yu|W (u)
1 − 1|, and EUn is given by (15). The
family of distributions of Vn is tight. In view of (30),
P{|dn+1| > x + y} ∼ P{|dn+1| > x} locally uniformly in y.
Therefore, (31) holds with Cn = const EUn, and the result for Δ has been proved.
For later needs, let us notice here that, in the same way as above, we can prove that,
for fixed n ∈ N,
(32) P{ sup
m≥n
Wm > x} ∼ kn
β (1 − kβ)−1P{W1 > x}.
Consider now the square function S. Since S ≥ Δ a.s., and we have already proved
that P{Δ > x} ∼ (1 − kβ)−1P{W1 > x},
lim inf
x→∞
P{S > x}
P{W1 > x} ≥ 1
1 − kβ
.
Therefore, we must only calculate the upper limit. We begin with showing that, for any
n ∈ N,
(33) lim sup
x→∞
P{Sn > x}
P{W1 > x} ≤
n−1∑
m=0
km
β .
We use induction on n.
(1) If n = 1, then S1 ≤ W1, and (33) is obvious.
(2) Assume that (33) holds for n = j and show that it holds for n = j + 1. For every
x > 0 and ε ∈ (0, 1),
P{Sj+1 > x} ≤ P{S2
j > (1 − ε)x2} + P{d2
j+1 > (1 − ε)x2}+
+P{S2
j > εx2, d2
j+1 > εx2} =
(write the latter P as EPj and use the Fj-measurability of Sj)
= P{Sj > (1 − ε)1/2x} + P{|dj+1| > (1 − ε)1/2x} + E1{Sj>ε1/2x}Pj{|dj+1| > ε1/2x}.
According to (10) with Qu replaced by W
(u)
1 − 1,
lim
x→∞1{Sj>ε1/2x}
Pj{|dj+1| > ε1/2x}
P{W1 > x} = 0 a.s.,
and there exists a δ1 > 0 such that, for large x,
1{Sj>ε1/2x}
Pj{|dj+1| > ε1/2x}
P{W1 > x} ≤ ε−β/2
∑
|u|=n
Y β
u + δ1 a.s.
48 ALEKSANDER IKSANOV AND SERGEY POLOTSKIY
Therefore, by the dominated convergence,
lim
x→∞E1{Sj>ε1/2x}
Pj{|dj+1| > ε1/2x}
P{W1 > x} = 0.
By the inductive assumption and (30),
lim sup
x→∞
P{Sj+1 > x}
P{W1 > x} ≤ (1 − ε)−β/2
j−1∑
m=0
km
β + (1 − ε)−β/2kj
β = (1 − ε)−β/2
j∑
m=0
km
β .
Sending ε → 0 proves (33).
For m = 0, 1, . . . and fixed n ∈ N, set W̃m := Wm∨n and F̃m; = Fm∨n. Choose
ρ ∈ (0,
√
3) so small that ν := 2ρ2
3−ρ2 2β+1 ∈ (0, 1). Applying Theorem 18.2[8] (in the
notation of that paper, we take β = 2 and δ = ρ) to the non-negative martingale
(W̃m, F̃m) gives
P{(
∞∑
m=n+1
d2
m)1/2 > 2x} ≤
≤ P{ sup
m≥n
W̃m > ρx} + P{(
∞∑
m=n+1
d2
m)1/2 > 2x, sup
m≥n
W̃m ≤ ρx} ≤
(34) ≤ P{ sup
m≥n
Wm > ρx} +
2ρ2
3 − ρ2
P{(
∞∑
m=n+1
d2
m)1/2 > x}.
By Potter’s bound, we can take y > 0 such that P{W1>x}
P{W1>2x} ≤ 2β+1 for x ≥ y. Set
A(y) := sup
x≥y
P{ sup
m≥n
Wm>ρx}
P{W1>2x} . In view of (32), A(y) < ∞ and lim
x→∞A(x) = kn
β
1−kβ
(
2
ρ
)β
. Now
we have, for x ≥ y,
P{(∑∞
m=n+1 d2
m)1/2 > 2x}
P{W1 > 2x} ≤ A(y) + ν
P{(∑∞
m=n+1 d2
m)1/2 > x}
P{W1 > x} .
Iterating the latter inequality gives that, for k = 0, 1, . . . ,
sup
x∈[2ky,2k+1y]
P{(∑∞
m=n+1 d2
m)1/2 > x}
P{W1 > x} ≤
≤ A(y)(1 + ν + · · · + νk−1) + νk sup
x∈[y,2y]
P{(∑∞
m=n+1 d2
m)1/2 > x}
P{W1 > x} .
Let k → ∞ to obtain
lim sup
x→∞
P{(∑∞
m=n+1 d2
m)1/2 > x}
P{W1 > x} ≤ A(y)(1 − ν)−1.
Now sending y → ∞ gives
(35) lim sup
x→∞
P{(∑∞
m=n+1 d2
m)1/2 > x}
P{W1 > x} ≤ kn
β
(1 − kβ)(1 − ν)
(
2
ρ
)β
= const kn
β .
For any λ ∈ (0, 1) and any n ∈ N,
P{S > x} ≤ P{Sn > (1 − λ)1/2x} + P{(
∞∑
k=n+1
d2
k)1/2 > λ1/2x}.
REGULAR VARIATION IN THE BRANCHING RANDOM WALK 49
Therefore,
lim sup
x→∞
P{S > x}
P{W1 > x}
(33),(35)
≤ (1 − λ)−β/2
n−1∑
m=0
km
β + const λ−β/2kn
β .
Let n → ∞ and then λ → 0 to get the desired bound for the upper limit:
lim sup
x→∞
P{S > x}
P{W1 > x} ≤ 1
1 − kβ
.
This completes the proof for S.
(II) From the already proved relation
(36) P{W ∗ > x} ∼ (1 − kβ)−1P{W1 > x} ∼ (1 − kβ)−1x−βL(x),
it follows that EW ∗ < ∞, which ensures, in turn, the uniform integrability of Wn.
Let us now prove (4). Since W ∗ ≥ W a.s., E(W ∗ − x)+ ≥ E(W − x)+ for any x ≥ 0.
Relation (36) and Proposition 1.5.10[7] yield
E(W ∗ − x)+ ∼ (β − 1)−1(1 − kβ)−1xP{W1 > x}.
Therefore,
(37) lim sup
x→∞
E(W − x)+
xP{W1 > x} ≤ 1
(β − 1)(1 − kβ)
.
For each x ≥ 1, we define the stopping time νx by
νx :=
{
inf{n ≥ 1 : Wn > x}, if W ∗ > x;
+∞, otherwise.
The random variable W closes the martingale Wn. Hence, for each x ≥ 1,
E(W − x)1{νx<∞} = E(Wνx − x)1{νx<∞},
and, hence,
E(W − x)+ ≥ E(Wνx − x)+1{νx<∞}.
We now transform the right-hand side into a more tractable form
E(Wνx − x)+1{νx<∞} =
∞∑
k=1
E(Wk − x)+1{νx=k} = E
∞∑
k=1
Ek−1((Wk − x)+1{νx≥k}) =
= E
∞∑
k=1
1{νx≥k}Ek−1(Wk − x)+ = E
νx∑
k=1
Ek−1(Wk − x)+.
From (28) and (10) with Q replaced by W1, it follows that, for k = 2, 3, . . . ,
Pk−1{Wk > y} ∼
∑
|u|=k−1
Y β
u P{W1 > y} a.s.
An appeal to Proposition 1.5.10[7] gives that, for k = 2, 3, . . . ,
Ek−1(Wk − y)+ ∼ (β − 1)−1
∑
|u|=k−1
Y β
u yP{W1 > y} a.s.
Since lim
x→∞νx = +∞ a.s., using the Fatou lemma allows us to conclude that
lim inf
x→∞
E(W − x)+
xP{W1 > x} ≥ lim inf
x→∞
E
∑νx
k=1 Ek−1(Wk − x)+
xP{W1 > x} =
50 ALEKSANDER IKSANOV AND SERGEY POLOTSKIY
=
1
β − 1
⎛
⎝1 +
∞∑
k=2
E
⎛
⎝ ∑
|u|=k−1
Y β
u
⎞
⎠
⎞
⎠ =
1
(β − 1)(1 − kβ)
.
Combining the latter inequality with (37) yields
E(W − x)+ ∼ (β − 1)−1(1 − kβ)−1xP{W1 > x},
which by the monotone density theorem (see Theorem 1.7.2[7]) implies (4).
The result for M immediately follows from
P{W > x} ∼ P{W ∗ > x} ∼ (1 − kβ)−1P{W1 > x},
as |W − 1| ≤ M ≤ W ∗ a.s. The proof of the theorem is finished.
The second proof of (4) in the case where β > 2
Assume that the assumptions of Proposition 1.1 hold with β > 2. By (36) and
Theorem 1.6.5[7],
EW ∗(W ∗ − x)+ ∼ β(β − 1)−1(β − 2)−1(1 − kβ)−1x2−βL(x).
Since, for each x > 0, EW ∗(W ∗ − x)+ ≥ EW (W − x)+
(38) lim sup
x→∞
EW (W − x)+
x2−βL(x)
≤ β
(β − 1)(β − 2)(1 − kβ)
.
Lyons [24] constructed a probability space with probability measure Q and proved the
equality
EQ(Wn|G) = 1 +
n∑
k=1
Πk−1(Sk − 1) Q a.s.,
where EQ is the expectation with respect to Q; Π0 := 1, Πk := M1M2 · · ·Mk, k = 1, 2, . . . ;
{(Mk, Sk) : k = 1, 2, . . . } are Q-independent copies of a random vector (M, S), whose
distribution is defined by the equality
(39) E
∑
|u|=1
Yuh(Yu,
∑
|v|=1
Yv) = Eh(M, S)
which is assumed to hold for any nonnegative Borel bounded function h(x, y); G is the
σ-field that can be explicitly described (we only note that σ((Mk, Sk), k = 1, 2, . . . ) ⊂ G).
Moreover, for any Borel function r with the obvious convention when the right-hand side
is infinite or does not exist,
(40) EQr(Wn) := EWnr(Wn) and EQr(W ) := EWr(W ).
Lyons explains his clever argument in a quite condensed form. More details clarifying
his way of reasoning can be found in [4] and [22].
Since P{W1 > x} regularly varies with exponent −β, β > 2, EW 2
1 < ∞. Also by (7),
E
∑
|u|=1 Y 2
u < 1. By Proposition 4[17], the last two inequalities together ensure that
EW 2 < ∞. In view of (40), EQW = EW 2, and hence EQW < ∞. In Lemma 4.1[22],
it was proved that (1) holds with P replaced by Q. This implies that EQW ∗ < ∞ iff
EQW < ∞. Therefore, we have checked that EQW ∗ < ∞ which implies by the dominated
convergence theorem that
EQ(W |G) = 1 +
∞∑
k=1
Πk−1(Sk − 1) =: R Q a.s.
REGULAR VARIATION IN THE BRANCHING RANDOM WALK 51
By Jensen’s inequality, for any convex function g,
EQ(g(W )|G) ≥ g(EQ(W |G)) = g(R) Q a.s.
Setting g(u) := (u − x)+, x > 0 and taking expectations yields
(41) EW (W − x)+
(40)
= EQ(W − x)+ ≥ EQ(R − x)+.
It follows from (39) that EQMβ−1 = kβ < 1, EQMβ−1+ε = kβ+ε < ∞ and
Q{S − 1 > t} =
∫ ∞
t+1
ydP{W1 ≤ y}.
Using the latter equality and Theorem 1.6.5[7] leads to
Q{S − 1 > t} ∼ β(β − 1)−1t1−βL(t).
Therefore, Theorem 1[16] can be applied to the perpetuity R, which gives
Q{R > t} ∼ β(β − 1)−1(1 − EQMβ−1)−1t1−βL(t).
By Proposition 1.5.10[7],
EQ(R − x)+ =
∫ ∞
x
Q{R > t}dt ∼ β
(β − 1)(β − 2)(1 − kβ)
x2−βL(x).
An appeal to (41) now results in
lim inf
x→∞
EW (W − x)+
x2−βL(x)
≥ β
(β − 1)(β − 2)(1 − kβ)
.
Combining this with (38) yields
EQ(W − x)+
(40)
= EW (W − x)+ ∼ β
(β − 1)(β − 2)(1 − kβ)
x2−βL(x).
By the monotone density theorem,
Q{W > x} ∼ β
(β − 1)(1 − kβ)
x1−βL(x).
Since Q{W > x} =
∫ ∞
x
ydP{W ≤ y}, integrating by parts gives
xP{W > x}
Q{W > x} = 1 − x
Q{W > x}
∫ ∞
x
y−2Q{W > y}dy.
By Proposition 1.5.10[7], the right-hand side tends to (β−1)β−1 when x → ∞. Therefore,
P{W > x} ∼ (β − 1)(βx)−1Q{W > x} ∼ (1 − kβ)−1x−βL(x) as desired.
Remark 3.1. This argument seems not to work as just described when β ∈ (1, 2]. If
β ∈ (1, 2), we can get a bound for the upper limit
lim sup
x→∞
EW (W ∧ x)
x2−βL(x)
≤ β
(β − 1)(2 − β)(1 − kβ)
.
However, we do not know how the corresponding lower limit could be obtained. In fact,
we have not been able to find any random variable ξ with the appropriate tail behaviour
and such that W ≥ ξ in some strong or weak sense.
52 ALEKSANDER IKSANOV AND SERGEY POLOTSKIY
Miscellaneous comments
We begin this section with discussing the following problem: which classes of point
processes satisfy both (2) and (3) and which do not.
Let h : [0,∞) → [0,∞) be a nondecreasing and right-continuous function with
h(+0) > 0.
Example 4.1 Let {τ0 := 0, τi, i ≥ 1} be the renewal times of an ordinary renewal
process. In addition to the conditions imposed above on h, assume that h(0) is finite.
Consider the point process M with points {Ai = γ−1 log h(τi), i = 1, 2, . . .}, where γ > 0
is chosen so that E
∑∞
i=1 h(τi) = 1. According to Theorem 1[14], W1 =
∑∞
i=1 h(τi) has
exponentially decreasing tail. Thus, while we can find h and {τi} such that (2) holds,
(3) always fails.
The situation where P{W1 > x} ∼ x−βL(x) and EW β
1 < ∞ is not terribly interesting.
However, if this is the case, Proposition 1.1 yields the one-way implication of a well-known
moment result (see [17] and [21])
E
∑
|u|=1
Y β
u < 1, EW β
1 < ∞ ⇔ EW β < ∞, E(W ∗)β < ∞.
In the subsequent examples in addition to (2) and (3), we require that EW β
1 = ∞.
Examples 4.2 and 4.3 essentially show that when the number of points in a point process
is infinite, and the points are independent or constitute an (inhomogeneous) Poisson flow,
E
∑
|u|=1 Y β
u < ∞ implies EW β
1 < ∞, β > 1.
Example 4.2 Assume that M is a point process with independent points {Ci}, and
E
∑∞
i=1 Yi = 1 and, for some β > 1, E
∑∞
i=1 Y β
i < ∞, where Yi = eγCi and γ > 0. Then
EW β
1 < ∞.
In this case, W1 =
∑∞
i=1 Yi. Hence, we must check that E(
∑∞
i=1 Yi)β < ∞. By using
the cβ-inequality, let us write the (formal) inequality
E(
∞∑
i=1
Yi)β = E
∞∑
i=1
Yi(Yi +
∑
k
=i
Yk)β−1 ≤
(42) ≤ (2β−2 ∨ 1)(E
∞∑
i=1
Y β
i + (E
∞∑
i=1
Yi)E(
∞∑
i=1
Yi)β−1).
For any β > 1, there exists n ∈ N such that β ∈ (n, n + 1]. We will use induction
on n. If β ∈ (1, 2], then E
∑∞
i=1 Yi < ∞ implies E(
∑∞
i=1 Yi)β−1 < ∞. Hence by (42),
E(
∑∞
i=1 Yi)β < ∞. Assume that the conclusion is true for β ∈ (n, n + 1] and prove it for
β ∈ (n+1, n+2]. Since E
∑∞
i=1 Yi < ∞ and E
∑∞
i=1 Y β
i < ∞, we have E
∑∞
i=1 Y β−1
i < ∞,
which, by the assumption of induction, implies E(
∑∞
i=1 Yi)β−1 < ∞. It remains to apply
(42).
Example 4.3 Let {τi, i ≥ 1} be the arrival times of a Poisson process with intensity
λ > 0. Consider a Poisson point process M with points {Bi} and assume that, for any
a ∈ R, M(a,∞) ≥ 1 a.s. Then there exists a function h, as described at the beginning of
this section, that additionally satisfies h(0) = ∞, and γ > 0 such that h(τi) = eγBi and
EW1 = E
∑∞
i=1 h(τi) = λ
∫ ∞
0
h(u)du = 1. The distribution of W1 is infinitely divisible
with zero shift and the Lévy measure ν given as follows: ν(dx) = λh←(dx), where h←
is a generalized inverse function. Since λE
∑∞
i=1 hβ(τi) =
∫ ∞
0 xβν(dx) and, as is well
known from the general theory of infinitely divisible distributions,
∫ ∞
1 xβν(dx) < ∞
implies EW β
1 < ∞, we conclude that E
∑∞
i=1 eγβBi < ∞ implies EW β
1 < ∞. In the next
example, we point out a class of point processes which satisfy (2) and (3), and EW β
1 = ∞.
REGULAR VARIATION IN THE BRANCHING RANDOM WALK 53
Example 4.4 Let K be a nonnegative integer-valued random variable with P{K >
x} ∼ x−βL(x), β > 1, and let {Di, i ≥ 1} be independent identically distributed random
variables which are independent of K. If M is a point process with points {Di, i = 1, K}
and there exists a γ > 0 such that E
∑K
i=1 eγDi = 1 and E
∑K
i=1 eγβDi < 1, then,
according to Proposition 4.3[15], we have P{W1 > x} ∼ EeγβD1x−βL(x). We conclude
with two remarks that fit the context of the present paper.
Remark 4.1 The tail behaviour of P{W > x} and P{W ∗ > x} has been investigated
in [17] and [21]. In particular, it follows from those works that, when the condition
E
∑
|u|=1 Y β
u < 1 fails, the condition P{W1 > x} ∼ x−βL(x) is not a necessary one for
either P{W > x} ∼ x−βL(x) or P{W ∗ > x} ∼ x−βL(x) to hold.
Remark 4.2 Let {Xi} be the points of a point process, and let V be a random variable
satisfying the distributional equality
V
d=
∞∑
i=1
XiVi,
where V1, V2, . . . are conditionally independent copies of V on {Xi}. The distribution
of V is called a fixed point of the smoothing transform (see [17] for more details and
[19] and [20] for an interesting particular case). It is known and can be easily checked
that the distribution of W is a fixed point of the smoothing transform with {Xi : i =
1, 2, . . . } = {Yu : |u| = 1}. Thus, (4) can be reformulated as a result on the tail behaviour
of the fixed points with finite mean. The tail behaviour of fixed points with infinite mean
deserves a special mention. Typically, their tails regularly vary with index α ∈ (0, 1),
or
∫ x
0 P{V > y}dy slowly varies. This follows from Proposition 1(b)[17] and Proposition
8.1.7[7].
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E-mail : iksan@unicyb.kiev.ua
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