ON TAILS OF FIXED POINTS OF THE SMOOTHING TRANSFORM IN THE
BOUNDARY CASE
DARIUSZ BURACZEWSKI
Abstract. Let {Ai } be a sequence of random positive numbers, such that only N first of them are
strictly positive, where N is a finite a.s. random number. In this paper we investigate nonnegative
P
solutions of the distributional equation Z =d N
and
i=1 Ai Zi , where Z, Z1 , Z2 , . . . are independent
£ PN
¤
identically distributed random variables, independent of N, A1 , A2 , . . .. We assume E
Ai =
i=1
£ PN
¤
1 and E
i=1 Ai log Ai = 0 (the boundary case), then it is known that all nonzero solutions
have infinite mean. We obtain new result concerning behavior of their tails.
1. Introduction
Let {An }n∈N be a sequence of random positive numbers. We assume that only first N of them
are nonzero, where N is some random number, finite almost surely. For any random variable Z, let
{Zn }n∈N be a sequence of i.i.d. copies of Z independent both on N and {An }n∈N . Then we define
PN
new random variable Z ∗ = i=1 Ai Zi and the map Z → Z ∗ is called the smoothing transform.
A random variable Z is said to be fixed point of the smoothing transform if Z ∗ has the same
distribution as Z, i.e.
(1.1)
Z =d
N
X
Ai Zi .
i=1
There exists an extensive literature, where the problems of existence, uniqueness and asymptotic behavior of solutions of (1.1) were studied. It turns out that the answer depends heavily on properties
of the function
¸
·X
N
Aθi .
v(θ) = log E
i=1
£ PN
¤
£ PN
¤
One assumes usually v(1) = 1 and v (1) < 0, i.e. E
i=1 Ai = 1 and E
i=1 Ai log Ai < 0. Then
if N is nonrandom Durrett and Liggett [6] proved existence of solutions of (1.1). Their results were
later extended by Liu [10] to the case where N is random and v(0) = log E[N ] > 0 (this value could
be infinite). Let us mention that in this case all nonzero solutions of (1.1) have finite mean. Fixed
points of the smoothing transform were characterized by Biggins and Kyprianou [3]. Also their
asymptotic properties are well described. Durrett and Liggett [6] studied behavior of the Laplace
transform of Z close to 0. The tail of Z was described by Guivarc’h [8] (for nonrandom N) and Liu
[11, 12] (for random N ). They proved that if v(χ) = 1 for some χ > 1 and some further hypotheses
are satisfied then the limit limx→∞ xχ P(Z > x) exists, is strictly positive and finite.
0
In this paper we study ’the boundary case’, when v(1) = 0 and v 0 (1) = 0. Existence of fixed points
of (1.1) was proved by Durrett and Liggett [6] and Liu [10]. Uniqueness was studied by Biggins and
Kyprianou [2]. It is known that all the solutions have infinite mean. To our knowledge, up to now,
This research project has been partially supported Marie Curie Transfer of Knowledge Fellowship Harmonic
Analysis, Nonlinear Analysis and Probability (contract number MTKD-CT-2004-013389) and by KBN grant N201
012 31/1020.
1
2
D. BURACZEWSKI
no estimates of tails of fixed points in the boundary case have been obtained. All known results
were
£
¤
stated in terms of the Laplace transform. Let Z be a solution of (1.1) and let φ(λ) = E e−λZ be
its Laplace transform. Then, under some further hypothesis it was proved by Durrett and Liggett
[6], Liu [12], Biggins and Kyprianou [2] that
(1.2)
lim
λ→0+
1 − φ(λ)
=C
λ| log λ|
for some positive constant C. Finiteness of C was discussed in [2].
Our main result is the following
Theorem 1.3. Assume
·X
¸
N
(1.4)
E
Ai = 1,
i=1
(1.5)
·X
¸
N
E
Ai log Ai = 0,
(1.6)
¸
·X
N
1−δ1
E
Ai
< ∞, for some δ1 > 0,
(1.7)
·µ X
¶1+δ2 ¸
N
E
Ai
< ∞, for some δ2 > 0,
(1.8)
E[N ] > 1 (it could be infinite).
i=1
i=1
i=1
Let Z be nonnegative and nonzero solution of (1.1) then if Ai are aperiodic
£
¤
lim xP Z > x = C0 ,
x→∞
for some finite and strictly positive constant C0 .
If Ai are periodic, then there exist two positive constants C1 and C2 such that
£
¤
£
¤
C1 = lim inf xP Z > x ≤ lim sup xP Z > x = C2 .
x→∞
x→∞
To prove the theorem, following ideas of Guivarc’h [8] and Liu [12], we reduce the problem to
study tails of solutions of the random difference equation R =d AR + B, where (A, B) and R are
independent. In Section 2 we describe all information on the random difference equation, that are
needed for our purpose and in Section 3 we present complete proof of Theorem 1.3.
2. Random difference equation in the critical case
Given a probability measure µ on R+ × R we define the Markov chain on R
X0
=
0,
Xn
=
An Xn−1 + Bn ,
where the random pairs {(An , Bn )} are independent and identically
distributed
according to the
£
¤
measure µ. This process is usually considered under assumption E log A1 < 0. Then, if additionally
£
¤
E log+ |B| < ∞, there exists a unique stationary measure ν of {Xn }. The tail of ν is well
understood, namely it was proved by Kesten [9] (see also Goldie [7] for much simpler argument)
that limt→∞ ν(|x| > t) = C+ for some positive constant C+ , where α is the unique positive number
such that EAα
1 = 1. Exactly this result was used by Guivarc’h [8] and Liu [12] to study solutions of
(1.1) with finite mean.
THE SMOOTHING TRANSFORM IN THE BOUNDARY CASE
3
£
¤
However, in this paper we will refer to the ’critical case’, when E log A1 = 0. Then there is no
finite stationary measure, but Babbilot, Bougerol, Elie [1] proved that if
• P[A
+¤B1 = x] < 1 for all x ∈ Rd ,
£ 1 = 1] < 1 and+ P[A1 x2+ε
• E (| log A1 | + log |B1 |)
< ∞, for some ε > 0
then there exists a unique (up to a constant factor) invariant Radon measure ν of {Xn }, i.e. the
measure ν on Rd satisfying
(2.1)
µ ∗ ν(f ) = ν(f ),
for any positive measurable function f , where
Z
Z
µ ∗ ν(f ) =
R+ ×R
f (ax + b)ν(dx)µ(da db).
Rd
Recently precisely behavior of the measure ν at infinity has been described:
Lemma 2.2 ([4, 5]). Assume that hypotheses above are satisfied and moreover
¤
£
E A−δ + Aδ + |B|δ < ∞
for some δ > 0. If A1 is aperiodic, then there exists a strictly positive and finite constant C+ such
that
β
lim ν(x : αt < |x| ≤ βt) = C+ log ,
t→∞
α
for any pair 0 < α < β.
Furthermore, if A1 is periodic and the group generated by the support of A1 is {enp : n ∈ Z} for
some p > 0, then
lim ν(x : t < |x| ≤ enp t) = nC+ ,
t→∞
for every n ≥ 1 and some positive constant C+ .
3. Proof of Theorem 1.3
Let (Ω, F, P) be a probability space on which random variables {Ai }i∈N , N , {Zi }i∈N are supported. We denote by E the expected value with respect to
£ P. Let¤ η be the law of Z. We define the
measure ν on R+ putting ν(dx) = xη(dx), then ν(f ) = E f (Z)Z for any bounded and compactly
supported function f . Measure ν is unbounded on R+ , however it is a Radon measure. Using
ideas of Guivarc’h [8] and Liu [12] we will prove that ν satisfies (2.1) for some appropriately chosen
probability measure µ on R+ × R. We cannot use directly their proofs. Guivarc’h assumed Ai to be
independent of each other, N to be a constant and obtained the random recurrence equation just
by a simple algebraic transformation of measures. Whereas Liu introduced the Peyriere’s measure,
which cannot be defined here. However we follow the approach of Liu [12], p. 276.
e = Ω × N, and let Fe be the σ-field on Ω
e being the direct product of F and B, where B
Define Ω
e Let δi be
is the Borel σ-field on N. We denote by ω an element of Ω and by (ω, i) an element of Ω.
e
the Dirac measure on N, i.e. δi (k) = 0 if k 6= i and δi (i) = 1. For every U ∈ F we define
·X
¸
Z
N
e
P(U ) = E
Ai (ω) 1U (ω, j)δi (dj) ,
i=1
N
e is a probability measure on Ω.
e We write Ee for its expected value. Thus,
then, in view of (1.4), P
P
e
e
e
we have defined a new probability space (Ω, F, P).
4
D. BURACZEWSKI
e we define
Given (ω, i) ∈ Ω
e
Z(ω,
i) =
e
A(ω, i) =
e
B(ω,
i) =
Zi (ω),
Ai (ω),
X
Aj (ω)Zj (ω).
j6=i
e independent. Moreover for every nonnegative
e B)
e are P
Lemma 3.1. Random variables Ze and (A,
functions h and g on R+ × R and R respectively:
(3.2)
£
¤
e
EeP g(Z)
=
(3.3)
£
¤
e B)
e
EeP h(A,
=
£
¤
E g(Z) ,
·X
N
³
´¸
X
E
Ai h Ai ,
Aj Zj .
i=1
j6=i
In particular Ze and Z have the same law η.
Proof. We write
·X
¸
Z ³
N
£
¤
¡
¡
¢´
e
e
e
e
e
e
EeP h(A, B)g(Z) = E
Ai (ω)
h A(ω, j), B(ω, j))g Z(ω, j) δi (dj)
i=1
N
·X
N
´ ¡ ¢¸
³
X
= E
Aj Zj g Zi
Ai h Ai ,
i=1
j6=i
·X
N
³
´¸ £
X
¤
Ai h Ai ,
Aj Zj E g(Zi ) .
= E
i=1
j6=i
Putting g = 1 we obtain (3.3) and next taking h = 1 we have (3.2). Therefore
£
¤
£
¤ £
¤
e B)g(
e Z)
e = Ee h(A,
e B)
e Ee g(Z)
e ,
EeP h(A,
P
P
e B).
e
that proves independence of Ze and (A,
¤
Next we define a probability measure µ on R+ × R:
h
i
e B)
e ,
µ(U ) = EeP 1U (A,
for every Borel set U ⊂ R+ × R.
Lemma 3.4. Measures ν and µ satisfy (2.1).
¡
¢
Proof. Take arbitrary compactly supported function f on R and let h (a, b), x = f (ax + b)x be a
e B)
e and Ze are independent and moreover
function on R+ × R × R. In Lemma 3.1 we proved that (A,
THE SMOOTHING TRANSFORM IN THE BOUNDARY CASE
5
that Ze and Z have the same distribution, applying these observations we have
Z
Z
µ ∗ ν(f ) =
f (ax + b)ν(dx)µ(da db)
R+ ×R R+
Z
Z
¡
¢
=
h (a, b), x η(dx)µ(da db)
R+ ×R R+
h ¡
h ¡
i
¢i
e B),
e Z
e = Ee f A
eZe + B)
e Ze
= Ee h (A,
P
P
=
·X
¸
N
³
X
E
Ai f Ai Zi +
Aj Zj )Zi
=
· ³X
¸
N
N
´X
E f
Ai Zi
Ai Z i
=
£
¤
E f (Z)Z = ν(f )
i=1
i=1
j6=i
i=1
¤
The next step is to prove that the measure µ satisfies hypotheses of Lemma 2.2, but first we will
prove that the random variable Z has moments smaller than 1. The proof is classical, however we
give here all the details for the reader’s convenience.
Lemma 3.5. For every α < 1:
£ ¤
E Z α < ∞.
Proof. First we will show that there exist constants C and M such that
£
¤ C log t
P Z>t ≤
,
t
(3.6)
for t > M .
For this purpose we will use the asymptotic of the Laplace transform of Z given in (1.2). Fix ε and
δ such that
¯
¯
¯ 1 − e−x
¯
¯
¯ < δ,
−
1
for |x| ≤ ε
¯ x
¯
and
1 − φ(λ) ≤ (C + δ)λ| log λ|
for λ < ε.
Then for λ < ε and t > 1 we have
·
¸
·
¸
1 − φ(λ)
1 − e−λZ
1 − e−λt
(C + δ)| log λ| ≥
=E
≥ tE
1[t,∞) (Z) .
λ
λ
λt
Putting λ =
ε
t
in the inequality above we obtain
·
¸
£
¤
1 − e−ε
(C + δ)(log t − log ε) ≥ tE
1[t,∞) (Z) ≥ t(1 − δ)P Z > t .
ε
Hence
£
¤ (C + δ)(log t − log ε)
.
P Z>t ≤
(1 − δ)t
But for t > 1ε , log t − log ε < 2 log t, thus we obtain (3.6), with M = 1ε .
6
D. BURACZEWSKI
Finally for α < 1 we write
£ ¤
E Zα =
·
¸
∞
£
¤ X
E Z α · 1(0,1) +
E Z α · 1[2n ,2n+1 ) (Z)
n=0
≤
1+
∞
X
£
¤
2α(n+1) P Z ≥ 2n
n=0
dlog2 M e
≤
1+
X
∞
X
2α(n+1) + C
n=0
n=dlog2 M e+1
n
2n(1−α)
and the expression above is finite.
¤
Lemma 3.7. The measure µ satisfies hypotheses of Lemma 2.2.
Proof. First notice that by (1.5) we have
·X
¸
N
¤
£
e =E
EeP log A
Ai log Ai = 0.
i=1
£
¤
e would be equal to 1 almost surely, then we had E PN Ai = E[N ], but the left hand
Next if A
i=1
side of this equation by (1.4) is equal to 1, whereas the right one, by (1.8) is strictly larger than 1.
e is positive
e +B
e = x a.s., then x = Be , but by definition B
Assume now that for some x: Ax
e
1−A
e changes the sign, since we already know: E[log A]
e = 0 and A
e 6≡ 1. Therefore
a.s., whereas 1 − A
such a point x cannot exist.
Finally we have to check moments conditions. Take δ2 as in (1.7), then
¶1+δ2 ¸
¸
·µ X
·X
N
N
i
δ2
δ2
e
Ai
< ∞.
EeP A
Ai · Ai ≤ E
=E
h
i=1
i=1
Next applying (1.6) we obtain
·X
¸
N
i
1−δ1
−δ1
e
Ai
< ∞.
EeP A
=E
h
i=1
e we consider the σ-field generated by N and {Ai }: F0 = σ(N, A1 , A2 , . . .).
To estimate moments of B
α
Take α = 1 − δ1 and ε such that α−ε
= 1 + δ2 . We may assume αε < 1. We are going to estimate for
¯P
¯ε
every i the conditional expectation of ¯ j6=i Aj Zj ¯ with respect to F0 . For this purpose, we use first
ε
concavity of the function x 7→ x α and the Jensen inequality, then the inequality |a+b|α ≤ |a|α +|b|α ,
which is valid for α < 1, independence Zj of F0 and finally Lemma 3.5. Thus, we obtain
¯ ¸¶ αε
·¯ X
µ ·X
µ ·¯ X
¯ε ¯¯ ¸
¯α ¯¯ ¸¶ αε
¯
¯
¯
¯
¯
α¯
E ¯
≤ E
Aj Zj ¯ ¯¯F0
≤
E ¯
Aj Zj ¯ ¯¯F0
Aα
Z
j j ¯F 0
j6=i
j6=i
≤
µX
j6=i
j6=i
¶ε
µX
¶ αε
N
£ α¤ α
α
Aα
·
E
Z
≤
C
.
A
j
j
j
j=1
THE SMOOTHING TRANSFORM IN THE BOUNDARY CASE
7
α
Next we use the Hölder inequality with parameters p = α−ε
and q = αε and we estimate
" · N
#
·X
N
¯ε ¸
¯ε ¯¯ ¸
¯X
X ¯¯ X
£ ε¤
¯
¯
¯
e
Aj Zj ¯ ¯¯F0
EeP |B|
= E
Ai ¯
Aj Zj ¯ = E E
Ai ¯
"
i=1
N
X
i=1
j6=i
j6=i
#
·¯ X
¶ αε ¸
·X
µX
N
N
¯ε ¯¯ ¸
¯
¯ ¯
Ai E ¯
Aj Zj ¯ ¯F0 ≤ CE
Ai ·
Aα
j
=
E
≤
·µ X
·X
¸ q1
¶1+δ2 ¸ p1
N
N
1−δ1
CE
Ai
·E
Aj
i=1
i=1
j6=i
i=1
j=1
j=1
and in view of (1.6) and (1.7) the value above is finite.
¤
Now we are ready to conclude.
Proof of Theorem 1.3. Assume Ai are aperiodic. Fix β > 1. In view of Lemma 3.4 hypotheses of
Lemma 2.2 are fulfilled, therefore for every ε there exists M such that
¯
¯
¯
¯
¯ν(t, βt) − C0 log β ¯ < ε
for every t > M . Next we estimate the tail of Z. We have
£
¤
tP Z > t =
≤
≤
∞
∞ Z
X
X
£ n
¤
n+1
t·
P tβ < Z ≤ tβ
=t·
n=0
∞
X
1
n
β
n=0
Z
n=0
tβ n+1
xη(dx) =
tβ n
tβ n+1
η(dx)
tβ n
∞
X
1 ¡ n n+1 ¤
ν tβ , tβ
βn
n=0
∞
X
β(C0 log β + ε)
C0 log β + ε
=
.
n
β
β−1
n=0
Hence passing with ε to 0 and next with β to 1 we obtain
£
¤
lim sup tP Z > t ≤ C0 .
t→∞
Analogously we justify
£
¤
lim inf tP Z > t ≥ C0 ,
t→∞
that proves the Theorem in the aperiodic case. If Ai are periodic, then the same arguments gives
the result
¤
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8
D. BURACZEWSKI
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D. Buraczewski, Institute of Mathematics, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland,
E-mail address: [email protected]