A note on random walk in random scenery

0.18 MB·

by  Amine Asselah

A Note on Random Walk in Random Scenery.
Amine Asselah & Fabienne Castell
5 C.M.I., Universit´e de Provence,
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0 39 Rue Joliot-Curie,
2
F-13453 Marseille cedex 13, France

n
a [email protected] & [email protected]
J

5

] Abstract
R
P
We consider a random walk in random scenery X = η(S )+ +η(S ),n N ,
. n 0 n
h where a centered walk S ,n N is independe{nt of the scene·r·y· η(x),x ∈Zd},
t { n ∈ } { ∈ }
a consisting of symmetric i.i.d. with tail distribution P(η(x) > t) exp( c tα), with
m ∼ − α
1 α < d/2. We study the probability, when averaged over both randomness, that
[ ≤
{Xn > ny} for y > 0, and n large. In this note, we show that the large deviation
1 estimate is of order exp( c(ny)a), with a = α/(α+1).
v −
8
6 Keywords and phrases: random walk, random scenery, large deviations, local times.
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1 AMS 2000 subject classification numbers: 60K37,60F10,60J55.
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Running head: Random Walk in Random Scenery.
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0
/
h
t 1 Introduction
a
m
: We consider a centered random walk S ,k N on Zd. When S = x, we denote the law
v k 0
{ ∈ }
i of the walk by P and the expectation with respect to this law by E . Each site x Zd
X x x
∈
is associated with a random variable η(x), and we assume that the scenery η(x),x Zd
r
a { ∈ }
consists of symmetric i.i.d. unbounded random variables, independent of the random walk.
We denote the law of the scenery by P , and by E the expectation with respect to this law.
η η
The random walk in random scenery (RWRS) is the process X ,n N defined by
n
{ ∈ }
n n
X := η(S ) = l (x)η(x), where l (x) = 1I . (1.1)
n k n n Sk=x
Xk=0 xX∈Zd Xk=0
RWRS has been introduced by Kesten, Spitzer [9], and Borodin [4, 5] as a case-study for
sums of dependent random variables, in order to exhibit new scaling and new self-similar
limiting laws. Indeed, the convergence in law of X , studied for d = 2 in [9, 4, 5], and for
n
6
d = 2 by Bolthausen [3], needs a super-diffusive scaling in dimensions 1 and 2. In terms of
1
the mean square, the dominant orders are the following.
n3/2 for d = 1,
E E X2 nlog(n) for d = 2, (1.2)
0 ⊗ η n ≃ 
n for d 3.

(cid:2) (cid:3) ≥
Recently, the moderate and large deviations for X have been studied in [6, 1, 2, 7] for
n
the Brownian motion in various sceneries, and in [10, 11] in the original random walk setting.
We distinguish three regimes depending on the tail parameter α of the scenery variable
P (η(x) > t) exp( c tα), for t large. (1.3)
η α
∼ −
When α < 1, η(x) has no exponential moments, and is called a heavy-tail variable. In
•
a recent paper [11], van der Hofstad, Gantert and K¨onig deal with X = l (x)η(x),
n n
by conditioning on the local times l (x),x Zd , thus obtaining a weighted sum of
n
{ ∈ } P
i.i.d. heavy-tail variables. Using classical heavy-tail estimates (as those of [13]), they
show that X > ny is realized as only one term of the series reaches level ny. Thus,
n
{ }
in terms of logarithmic equivalence ( ),
≈
P P [X ny] P P [l (0)η(0) ny].
0 η n 0 η n
⊗ ≥ ≈ ⊗ ≥
Now, recall that for a time k n, the local time at site 0, l (0), satisfies the following
n
≪
property
1
P (l (0) = k) exp( κ k), with κ := log , (1.4)
0 n ≈ − 0 0 P (H < )
(cid:18) 0 0 ∞ (cid:19)
where H = inf n 1,X = 0 . Thus, for y > 0, [11] shows that for an explicit J > 0
0 n
{ ≥ }
ny
P P (l (0)η(0) > ny) sup P (l (0) = k)P η(0) >
0 η n 0 n η
⊗ ≈ k
k=1,2,...
(cid:16) (cid:17)
ny
exp inf κk +c ( )α
α
≈ −k≥1 k
(cid:18) (cid:19)
(cid:16) α (cid:17)
exp J(ny)α+1 . (1.5)
≈ −
(cid:16) (cid:17)
In the optimal strategy l (0) is of order (ny)α/(α+1).
n
When α > max(d/2,1), a different behavior holds: for all y > 0
•
P0 Pη[Xn ny] exp nd+d2J(y) , ( with a J(y) > 0 known explicitly). (1.6)
⊗ ≥ ≈ −
(cid:16) (cid:17)
This result is proved in [2] for Brownian motion in a bounded scenery (i.e. α = ),
∞
in [7] for a Gaussian scenery (α = 2 and d 3), and in [10] for a random walk in a
≤
general scenery. The best strategy to realize X > ny is the following.
n
{ }
– Force the random walk to spend all its time in a ball of radius r with 1 r
n n
≪ ≪
√n, in such a way that for x in this ball, l (x) is of order n/rd. This has a cost
n n
of order exp( n/r2)
− n
2
– Require the scenery to satisfy η(x) rdy. This has a cost of order
kxk≤rn ≥ n
exp( rd).
− n P
The exponent d/(d+2) appears as one sets equal n/r2 and rd. Thus, in the optimal
n n
strategy, the walk spends a time n2/(d+2) on each site of a ball of about nd/(d+2) sites.
Observe that when η(x) satisfies (1.3), then l (x)η(x) has a heavy tail (see (1.5)). Also
n
X = η(x)l (x) is a sum of about n-terms (in dimensions d 3). However, the
n n
≥
variables l (x)η(x),x Zd are not independent, and the extreme value of the sum
n
P{ ∈ }}
does not dominate.
The regime 1 α < d/2 is the purpose of this note.
• ≤
Our main result is the following.
Proposition 1.1 Let S ,n N be a walk with centered independentincrements with finite
n
{ ∈ }
exponential moments. Assume that η(x),x Zd are symmetric i.i.d. variables with tail
{ ∈ }
parameter α with 1 α < d/2, and whose law has a density decreasing on R+. There are
≤
c ,c > 0, such that when n is large enough
1 2
exp( c (ny)α/(α+1)) logP P [X ny] exp( c (ny)α/(α+1)). (1.7)
1 0 η n 2
− ≤ ⊗ ≥ ≤ −
In the course of deriving the upper bound, we rely on a localization lemma of independent
interest.
Lemma 1.2 Assume d 3. There is a constant κ > 0 such that for any Λ Zd, and any
d
≥ ⊂
t > 0
t
P (l (Λ) > t) exp κ , (1.8)
0 ∞ ≤ − d Λ 2/d
(cid:18) | | (cid:19)
where l (Λ) is the total sojourn time of the walk in the region Λ.
∞
Note that the recent paper [8] gives a representation of P (l (Λ) > t), in terms of the
0 ∞
eigenvalues and eigenvectors of the matrix whose entries are the Green function restricted
to Λ. It is not clear to us how to deduce Lemma 1.2 from the type of representation of [8].
This note is organized as follows. We specify the model in Section 2. In Section 3, we
deal with the lower bound. In Section 4, we deal with the upper bound. Finally, we have
gathered in the Appendix the proof of Lemma 1.2, and the proof of some technical facts.
2 Model
Assumptions on the random walk. We assume that the increments of the walk are
centered, with finite exponential moments, i.e.
k
S = ξ , ξ i.i.d , E[ξ ] = 0, E[exp(λξ )] < for all λ Rd. (2.1)
k j j 1 1
∞ ∈
j=1
X
3
It is then easy to see that there exist constants C,c > 0, such that for all n,
P max S n Cexp( cn). (2.2)
0 k
k≤n k k ≥ ≤ −
(cid:20) (cid:21)
Assumptions on the scenery. Besides our basic tail assumption (1.3), we make assump-
tions on the law of the scenery whose goal is to simplify the technical parts. Thus, we say
that a random variable with value in R is bell-shaped, if its law has a density with respect
to Lebesgue which is even, and decreasing on R+. Throughout the paper, we will assume
that η(x);x Zd are i.i.d and bell-shaped, with following handy consequence, proved in
{ ∈ }
the Appendix.
Lemma 2.1 When η(x),x Zd have independent bell-shaped densities, then for any Λ
{ ∈ }
finite subset of Zd, and any y > 0
P α η(x) > y P β η(x) > y , if 0 α β for all x Λ. (2.3)
x x x x
≤ ≤ ≤ ∈
! !
x∈Λ x∈Λ
X X
A typical use of Lemma 2.1 is the following bound
y y
P η(x) > P α η(x) > y P η(x) > . (2.4)
x
minα ≤ ≤ maxα
x! ! x!
Λ Λ Λ
X X X
Some notations. Throughout the paper, we set a := α/(α+ 1) and b := 1/(α+1), and
for x Zd, x := max x . Finally, when considering the variables η(x),x Λ for
i=1,···,d i
∈ k k | | { ∈ }
a finite region Λ of cardinality L, we will sometimes use the notation η ,1 j L .
j
{ ≤ ≤ }
3 Lower Bound
We show in this section the following simple estimate.
Lemma 3.1 There is a constant c > 0 such that, for any y > 0, and n large
1
P l (x)η(x) > ny exp( c (ny)a). (3.1)
n 1
≥ −
!
x∈Zd
X
Proof. The bound (3.1) is obtained by using Lemma 2.1. Thus,
ny
P l (x)η(x) > ny P (l (0)η(0) > ny) = P (l (0) = k)P η(0) >
n n 0 n η
≥ k
!
Xx Xk>0ny (cid:16) (cid:17)
P (l (0) = k)P η(0) > for any k. (3.2)
0 n η
≥ k
(cid:16) (cid:17)
We choose an n-depending k, for instance k = [(ny)a]. Since k /(ny)a 1 as n tends to
n n
→
infinity, we have for any ǫ > 0 and n large that
α
ny ny
P η(0) > exp c (1+ǫ)
η α
k ≥ − k
(cid:18) n(cid:19) (cid:18) (cid:18) n(cid:19) (cid:19)
4
exp( c (1+2ǫ)(ny)a).
α
≥ −
Now, if we set κ = log(1/P (H < )), then
0 0 0
∞
n
P (l (0) = k ) P (H )kn
0 n n 0 0
≥ ≤ k
n
n kn
e−κ0 P ( < H < ) . (3.3)
0 0
≥ − k ∞
(cid:18) n (cid:19)
Thus, for any κ > κ , we have for n large enough
0
P l (x)η(x) > ny exp( (κ+(1+2ǫ)c )(ny)a) (3.4)
n α
≥ −
!
x
X
This concludes the proof.
4 Upper Bound
The case α = 1 is special and much simpler than α > 1. Thus, we will treat the former
specifically in Remark 4.4. Henceforth, we assume that α > 1 and we recall that a :=
α/(α+1),andb = 1 a < a. Weconsiderasubdivisionof[b,a],b = b < b < < b = a,
1 2 N+1
− ···
and a decomposition of y > 0 into positive constants y ,y ,...,y ,y summing up to y.
↓ 0 N ↑
{ }
We will specify N and b ,y ,i = 1,...,N after we partition the range of the walk , into
i i n
{ } R
N +3 sets. For 1 i N, we set
≤ ≤
:= x : yanbi l (x) < yanbi+1 , (4.1)
i n n
D { ∈ R ≤ }
and for a small constant z to be chosen later
:= x : znb l (x) < yanb , (4.2)
0 n n
D { ∈ R ≤ }
and lastly, for the two sets at the extremities
:= x : l (x) < znb , and := x : l (x) (yn)a . (4.3)
↓ n n ↑ n n
D { ∈ R } D { ∈ R ≥ }
Thus,
N
l (x)η(x) > ny l (x)η(x) > ny l (x)η(x) > ny = .
n n i n ↓ ↑
{ } ⊂ { } { } {D 6 ∅}
xX∈Zd i[=0 xX∈Di [ xX∈D↓ [
(4.4)
Thus, if we define := max S < n , and recall that P ( c) is negligible compared
k≤n k 0
A { || || } A
to exp( na) by (2.2), then
−
N
P l (x)η(x) > ny P ( c)+ P , l (x)η(x) > ny
n 0 n i
≤ A A
! !
xX∈Zd Xi=0 xX∈Di
5
+P , l (x)η(x) > ny +P ( = ). (4.5)
n ↓ 0 ↑
A  D 6 ∅
xX∈D↓
 
We will now estimate each terms separately in the next Section. However, in the course of
obtaining an upper bound, we will fall on the following requirement: we will need a positive
β, independent of n, such that for i N
≤
1/α 2/d
βy y n(a−bi+1)−(1−δ0)(a−bi) with the positive constant δ := − < 1. (4.6)
i 0
≤ 1 2/d
−
Thus, a simple choice of b ,y which fulfills (4.6) is y := βyn(1−δ0)(a−bN), and
i i N
{ }
δ /2
i < N, y = βyn−ǫ0(a−bi+1), and (a b ) = (1+ǫ )(a b ), with ǫ = 0 . (4.7)
i i 0 i+1 0
∀ − − 1 δ /2
0
−
To explicit further the choices in (4.7), we introduce more notations:
z = a b , z = b b ,...,z = b b . (4.8)
1 N 2 N N−1 N 2 1
− − −
Thus, (4.7) is fulfilled when z = ǫ z and for i > 2
2 0 1
z = ǫ (z + +z ) = (1+ǫ )z = (1+ǫ )i−2z = (1+ǫ )i−2ǫ z . (4.9)
i 0 1 i−1 0 i−1 0 2 0 0 1
···
Note that for i < N
z
a b = z + +z = N−i+1 = (1+ǫ )N−i−1z .
i+1 1 N−i 0 1
− ··· ǫ
0
The condition on y in (4.7) will be fulfilled if we choose z = χ/log(n), for a constant χ to
i 1
be tuned later. Indeed, we obtain y = βyexp(χ(1 δ )), and
N 0
−
i < N, y = βyexp log(n)z ǫ (1+ǫ )N−i−1 = βyexp χǫ (1+ǫ )N−i−1 . (4.10)
i 1 0 0 0 0
∀ − −
(cid:0) (cid:1) (cid:0) (cid:1)
Thus, since
∞ ∞
(1+ǫ )i = , and exp χǫ (1+ǫ )i < , (4.11)
0 0 0
∞ − ∞
i=1 i=1
X X (cid:0) (cid:1)
one can find N finite, of order log(log(n)), χ > 0 and β > 0 independent of n (or rather χ
n
and β can be chosen to converge to positive constants, and we omit the subscript n) such
n
that
N N
χ
a b = z = (1+ǫ )N−1, and y +y +y = y. (4.12)
i 0 i ↓ ↑
− log(n)
i=1 i=0
X X
Actually, the choice of y ,y is arbitrary since we are not after the exact constant in front
↓ ↑
of the speed na. For instance, we choose y = y = y/3.
↓ ↑
6
4.1 Contribution of
D↓
Lemma 4.1 We set for any z > 0, (z) = x : l (x) znb . Then, for any y > 0, we
↓ n ↓
D { ≤ }
have
1
lim lim logP l (x)η(x) > ny = . (4.13)
z→0n→∞ na  n ↓ −∞
x∈XD↓(z)
 
Proof. We fix z > 0 and l (x) : x (z) and integrate over the η, to obtain for λ 0
n ↓
{ ∈∈ D } ≥
λ l (x)
n
P l (x)η(x) > ny exp( ny ) E [exp λη(x) ]. (4.14)
η n ↓ ≤ − ↓znb η znb
x∈XD↓(z) x∈YD↓(z) (cid:18) (cid:19)
 
Note that by hypothesis (1.3), there is δ > 0 such that
ν(δ) = E [η2(x)exp(δ η(x) ) < . (4.15)
η
| | ∞
Also, it is an obvious fact that for 0 θ δ
≤ ≤
θ2η(x)2
exp(θη(x)) 1+θη(x)+ eδ|η(x)|. (4.16)
≤ 2
Thus, after taking expectation in (4.16)
θ2
E [exp(θη(x))] 1+ ν(δ) eθ2ν(δ)/2. (4.17)
η
≤ 2 ≤
Back to estimating (4.14), we choose λ δ and use (4.17) to obtain
≤
E [exp λη(x)ln(x) ] exp λ2ν(δ) x∈D↓(z)ln(x)2 . (4.18)
η
znb ≤ 2 (znb)2
x∈YD↓(z) (cid:18) (cid:19) P !
Now, l (x)2 zn1+b. Thus,
x∈D↓(z) n ≤
P
n1−b ν(δ)λ2
P l (x)η(x) > ny exp sup y λ (4.19)
n ↓ ↓
  ≤ − z { − 2 }
x∈XD↓(z) (cid:18) 0≤λ≤δ (cid:19)
 
The results follows since for any y > 0, the supremum is positive, and z can be sent to zero.
↓
4.2 Contributions of
D↑
Lemma 4.2 For given in 4.3, there is C > 0 such that for n large
↑ d
D
P ( = ) C ne−κ0(yn)a. (4.20)
0 ↑ d
D 6 ∅ ≤
7
Proof. First, note that
P ( = ) = P l (x) > (yn)a for some x P (x )P (l (0) (yn)a).
0 ↑ 0 n n 0 n 0 ∞
D 6 ∅ ∈ R ≤ ∈ R ≥
x∈Zd
(cid:0) (cid:1) X
(4.21)
Now, for κ := 1/log(P (H < )), it is clear that
0 0 0
∞
P (l (0) (yn)a) e−κ0(yn)a. (4.22)
0 ∞
≥ ≤
Thus, we conclude by recalling standard estimates `a la Dvoretzky-Erd¨os (see for instance
Theorem 6.2 of [12]) which establish that E [ ] is of order n in dimension larger than 2.
0 n
R
4.3 Contributions of for i = 0,...,N.
i
D
Lemma 4.3 Fix i = 0,...,N. We have a constant c > 0 such that for y > 0, and n large
2
P , l (x)η(x) > ny exp( c (ny)a).
n i 2
A ≤ −
(cid:0) xX∈Di (cid:1)
Proof. We first treat the case i > 0. Note that y−an1−bi, and on there are at most
i
|D | ≤ A
nd possible choices for since the walk does not exit a region of radius n. Thus, using
|Di| Di
Lemma 2.1 (and (2.4))
(cid:0) (cid:1)
y−an1−bi L
ny
P , l (x)η(x) > ny P ( , = L) P η > i
n i 0 i η j
A ≤ A |D | max l (x) : x
n i !
(cid:0) xX∈Di (cid:1) XL=1 Xj=1 { ∈ D }
y−an1−bi nd L ny
sup P ( = Λ) P η > i
0 i η j
≤ L=1 (cid:18)L(cid:19)Λ:|Λ|=L D j=1 nbi+1ya!
X X
y−an1−bi L
ny
(nd)L sup P (l (Λ) > Lyanbi) P η > i . (4.23)
≤ Λ:|Λ|=L 0 ∞ η j nbi+1ya!
L=1 j=1
X X
By using Lemma 1.2, we have
(nd)L sup P0 l∞(Λ) > Lyanbi exp κdyanbiL1−d2 +Llog(nd) , (4.24)
≤ −
Λ:|Λ|=L
(cid:16) (cid:17)
(cid:0) (cid:1)
and the combinatorial factor ndL is negligible when
nbiL1−d2 Llog(n). (4.25)
≫
Since L y−an1−bi and b b = 1/(α+1), (4.25) requires n large and
i
≤ ≥
2 d
(1 b ) < b = α < . (4.26)
i i
d − ⇐ 2
Thus, the combinatorial factor is always innocuous when α < d/2.
8
Let B > 0 be a fixed large constant. We say that L is large when nbiL1−2d > Bna, and
this case poses obviously no problem since the term P (l (Λ) > Lnbiya) suffices to obtain
0 ∞
the right speed. Thus, we assume that L is small, that is:
nbiL1−d2 Bna. (4.27)
≤
Thus, we consider for a fixed i = 1,...,N
L Anγi, with γ := a−bi, and A = B1−12/d (4.28)
≤ i 1 2
− d
We want to evaluate P ( η(x) > n1−bi+1y y−a, L) when L is as in (4.28). First,
η Di i |Di| ≤
note that nγi n1−bi+1y , when n is large enough. Indeed, first rewrite
i
≪ P
γ
i
1 b γ = 1 a +(a b +(1 δ )(a b )).
i+1 i i+1 0 i
− − − − α − − −
(cid:16) (cid:17)
Then, by noting that 1 a γ /α 1 a γ /α bδ , and using (4.6),
i 1 0
− − ≥ − − ≥
n1−bi+1−γiyi = n1−a−γαina−bi+1+(1−δ0)(a−bi)yi n1−a−γα1βy = nbδ0βy. (4.29)
≥
Hence for L satisfying (4.28), n1−bi+1y L, and using standard Large Deviations estimates
i
≫
(see lemma 5.4 in the appendix), for all ǫ > 0 and n sufficiently large,
L y n1−bi+1y α
P η > n1−bi+1 i exp c (1 ǫ)L i
η j ya ≤ − α − Lya
!
j=1 (cid:18) (cid:18) (cid:19) (cid:19)
X n1−bi+1−γi(1−1/α)y α
i
exp c (1 ǫ)A
≤ − α − Aya
(cid:18) (cid:18) (cid:19) (cid:19)
βα
exp c (1 ǫ) (ny)a , using (4.6). (4.30)
≤ − α − Aα−1
(cid:18) (cid:19)
By the same arguments, we can treat the case . Indeed, note that γ = (a b)(1
0 0
D − −
2/d)−1 < a, and for L small, we have
L y naα(y /ya)α
P η > n1−b 0 exp c 0 , (4.31)
η j ya ≤ − α Lα−1
!
j=1 (cid:18) (cid:19)
X
which is negligible since aα (α 1)γ > a.
0
− −
Remark 4.4 When α = 1, then b = a = 1/2. Thus, the range of the walk is divided
into three sets: , and as in (4.3) and (4.2) respectively. Also, we can choose
↓ ↑ 0
D D D
y = y = y = y/3. Now, our treatment for , only assumed small exponential
↓ ↑ 0 ↓ ↑
D D
moments for the walk, which hold in this case. To treat note that only the case L small
0
D
may pose problem. However, since γ = 0, L small means L A for a large constant. It is
0
≤
easy to see that those terms are of the correct order since, there is a constant c¯such that
L y (ny)1/2
P η > n1−a 0 LP (η > ) Lexp( c¯(ny)1/2). (4.32)
η j ya ≤ η 1 3L ≤ −
!
j=1
X
9
5 Appendix
5.1 On bell-shaped densities
We recall that a density f is bell-shaped if it is even and decreasing on R+. Our first
observation is the following.
Lemma 5.1 If f,g are two bell-shaped densities, so is their convolution f g.
∗
Proof. First, it is obvious that f g is even. Indeed, by the eveness of both f and g
∗
f g( t) = f( t s)g(s)ds = f(t+s)g( s)ds = f(t s)g(s)ds = f g(t).
∗ − − − − − ∗
R R R
Z Z Z
Now, assume that f is differentiable. Then,
∞
(f g)′(t) = f′(t s)g(s)ds = f′(s)g(t s) = f′(s)(g( t s ) g(t+s))ds, (5.1)
∗ − − | − | −
ZR ZR Z0
where we used the oddness of f′ and the eveness of g. Now, for t,s 0, we have g( t s )
≥ | − | −
g(t+s) 0, and f′(s) 0 implying that (f g)′(t) 0.
≥ ≤ ∗ ≤
Now let ϕ ,ǫ > 0 be a differentiable bell-shaped approximate identity. By what we just
ǫ
{ }
saw, ϕ f is a bell-shaped differentiable density. So is in turn (ϕ f) g. Thus, for any
ǫ ǫ
∗ ∗ ∗
0 t T, we have (ϕ f) g(t) (ϕ f) g(T). By pointwise convergence, as ǫ tends to
ǫ ǫ
≤ ≤ ∗ ∗ ≥ ∗ ∗
0, we obtain that f g(t) f g(T).
∗ ≥ ∗
By induction, using Lemma 2.1, we obtain the following corollary.
Corollary 5.2 If η ,i = 1,...,n are independent bell-shaped variables and S = α η +
i 1 1
{ }
+α η , with positive α , then S has a bell-shaped density.
n n i
··· { }
Finally, the useful result is the following.
Lemma 5.3 Let η ,i = 1,...,n be independent bell-shaped variables and 0 α β for
i i i
{ } ≤ ≤
i = 1,...,n. then for any y > 0, we have (2.3).
Proof. We prove the Lemma by induction on the number of β larger than α . Thus, it is
i i
enough to show that x P(S +xη ) is increasing on R+ when S is a bell-shaped variable
n
7→
independent of η .
n
First note that for symmetric independent ξ,η, we have for y > 0
∞
P(ξ +η > y) = P(ξ > y)+ P(η > z)(f ( y z ) f (y +z))dz. (5.2)
ξ ξ
| − | −
Z0
The proof is concluded as we apply (5.2) to ξ = S and η = xη , and as we note that
n
f ( y z ) f (y +z) 0 and x P(η > z/x) is increasing.
S S n
| − | − ≥ 7→
10