Selection from a stable box
Alexander Aue
1
István Berkes
2
Lajos Horváth
3
Abstract: Let {Xi } be independent, identically distributed random variables. It is well–
known that the functional CUSUM statistics and its randomly permuted version both converge
weakly to a Brownian bridge if second moments exist.
Surprisingly, an infinite variance counterpart does not hold true. In the present paper, we
let {Xi } be in the domain of attraction of a strictly α–stable law, α ∈ (0, 2). While the
functional CUSUM statistics itself converges to an α–stable bridge, and so does the permuted
version as long as both the {Xi } and the permutation are random, the situation turns out
to be more delicate if a realization of the {Xi} is fixed and randomness is restricted to the
permutation. Here, the conditional characteristic function of the permuted CUSUM statistics
converges in probability to a random and nondegenerate limit.
AMS 2000 Subject Classification: Primary 60F17, 60G52; Secondary 60E07.
Keywords and Phrases: Stable distributions; CUSUM; Order statistics, Functional limit
theorems, Permutation principle
1
Department of Mathematics, University of Utah, 155 South 1440 East, Salt Lake City, UT 84112–0090,
USA, email: aue@math.utah.edu
2
Department of Statistics, Graz University of Technology, Steyrergasse 17/IV, A–8010 Graz, Austria, email:
berkes@tugraz.at
3
Department of Mathematics, University of Utah, 155 South 1440 East, Salt Lake City, UT 84112–0090,
USA, email: horvath@math.utah.edu
Corresponding author: A. Aue, phone: +1–801–581–5231, fax: +1–801–581–4148.
Research partially supported by NATO grant PST.EAP.CLG 980599, NSF–OTKA grant INT–0223262 and
Hungarian National Foundation for Scientific Research Grants T 043037 and T 037886
1
Introduction
Let X, X1 , . . . , Xn be independent, identically distributed random variables. CUSUM–based
procedures, frequently used to test for time homogeneity of an underlying random phenomenon, are functionals of the process
⌊nt⌋
1 X
Zn (t) =
(Xi − X̄n ),
sn i=1
t ∈ [0, 1],
(1.1)
where ⌊·⌋ denotes integer part and
n
1X
Xi
X̄n =
n i=1
n
and
s2n
1 X
=
(Xi − X̄n )2
n − 1 i=1
are the sample mean and the sample variance, respectively. The asymptotic properties of
{Zn (t)} are immediate consequences of Donsker’s invariance principle [see Billingsley (1968)].
D[0,1]
Denote by −→ convergence in the Skorohod space D[0, 1]. We state the following fundamental
theorem.
Theorem 1.1 If EX 2 < ∞, then
D[0,1]
Zn (t) −→ B(t)
(n → ∞),
where {B(t) : t ∈ [0, 1]} is a Brownian bridge.
One of the main problems in testing for a homogeneous data generating process is to obtain
approximations for the critical values. It is well–known, however, that the convergence of functionals of Zn (t) to their limit distributions is often quite slow [sometimes even the convergence
speed is unknown], raising the question if Theorem 1.1 is reasonably applicable in case of small
sample sizes. To avoid these complications, along with bootstrap methods, the permutation
principle has been suggested to simulate critical values. For a comprehensive recent survey on
this topic we refer to Hušková (2004). Let π = (π(1), . . . , π(n)) be a random permutation of
(1, . . . , n) which is independent of X = (X1 , X2 , . . .). Then, the permuted version of Zn (t) is
defined by
Zn,π (t) =
1 X
(Xπ(i) − X̄n ),
sn i≤nt
t ∈ [0, 1].
(1.2)
One can think of Zn,π (t) as a CUSUM process whose summands are chosen from X1 , . . . , Xn
by drawing from a box without replacement. The following result establishes the limiting
behavior of Zn,π (t) and, hence, shows that the permutation method works to simulate critical
values. Note also that, by empirical evidence, the convergence to the limit is faster than for
Zn (t).
2
Theorem 1.2 If EX 2 < ∞, then, for almost all realizations of X,
D[0,1]
Zn,π (t) −→ B(t)
(n → ∞),
where {B(t) : t ∈ [0, 1]} is a Brownian bridge.
The proof of Theorem 1.2 can be found in Billingsley (1968, p. 209–214).
In this paper we are interested in the asymptotics of the analogues of Zn (t) and Zn,π (t) in
the infinite variance case EX 2 = ∞. To this end, let α ∈ (0, 2] and recall that a random
variable ξα is strictly α–stable if its characteristic function has the form

exp(−t2 /2)
if α = 2,


exp(−|t|α [1 − iβ sgn(t) tan( 21 α)])
if α ∈ (0, 1) ∪ (1, 2), β ∈ [−1, 1],
φα (t) =


exp(−|t|)
if α = 1.
The definition includes the normal law [α = 2] and the Cauchy law [α = 1], where the latter is
moreover assumed to be symmetric. Note that E[ξα ] exists only for α ∈ (1, 2] and E[ξα2 ] only
for α = 2. The class of stable distributions has become more and more important in theory as
well as in applications. Stable laws appear in a natural way in areas such as radio engineering,
electronics, biology and economics [see Zolotarev (1986, Chapter 1)]. For other expositions on
α–stable random variables and processes we refer to Samorodnitsky and Taqqu (1994), and
Bingham et al. (1987).
In what follows, let α ∈ (0, 2) and consider independent, identically distributed random
variables {Xi } which are in the domain of attraction of a strictly α–stable law. This means
that there exists a function L(x), slowly varying at infinity, such that
n
X
1
D
(Xi − µα ) −→ ξα
1/α
n L(n) i=1
(n → ∞),
(1.3)
where ξα is a strictly α–stable random variable and µα = 0 if α ∈ (0, 1]. If (1.3) holds, then
there is a p ∈ [0, 1] such that the tail probabilities of X and |X| satisfy the relations
P {X > y}
=p
y→∞ P {|X| > y}
lim
and
P {X < −y}
= q,
y→∞ P {|X| > y}
lim
(1.4)
where q = 1 − p. See Feller (1966) and Petrov (1995) for details.
To define the counterparts of Zn (t) and Zn,π (t) in the α–stable case, introduce the process
An (t) =
1 X
(Xi − X̄n ),
Tn i≤nt
t ∈ [0, 1],
(1.5)
and its permuted version
An,π (t) =
1 X
(Xπ(i) − X̄n ),
Tn i≤nt
t ∈ [0, 1],
3
(1.6)
where Tn = max1≤i≤n |Xi |, and π = (π(1), . . . , π(n)) is a random permutation of (1, . . . , n),
independent of X. InP
contrast to the finite variance case, Tn is a natural norming factor in
(1.5) and (1.6), since 1≤i≤n (Xi − µα )/Tn has a nondegenerate limit distribution.
One might now expect that {An (t)} and {An,π (t)} follow an asymptotic pattern similar to
that of {Zn (t)} and {Zn,π (t)}. Consequently, given a realization of X, one might want to use
the permutation principle to simulate critical values that have a higher accuracy in case of small
sample sizes. Surprisingly, this turns out to be impossible. While a version of Theorem 1.1
holds true for {An (t)}, an adaptation of Theorem 1.2 cannot be given for almost all realizations
of X. On the contrary, we will show that, conditioned on X, the characteristic function of the
permuted process {An,π (t)} converges in distribution to a random and nondegenerate limit for
every fixed t. On the other hand, there is an averaging effect: if both X and π are assumed
to be random, An,π (t) and An (t) converge weakly to the same limit, namely to an α–stable
bridge.
The paper is organized as follows. In Section 2 we will formulate precisely our results. Their
proofs will be given in Section 3.
2
Results
First we study the limiting behavior for the sequence {An (t) : t ∈ [0, 1]}. Its weak convergence
can easily be derived from the joint convergence of


⌊nt⌋


X
1
Tn
.
(X
−
µ
)
:
t
∈
[0,
1]
and
i
α
1/α
1/α
 n L(n)

n
L(n)
i=1
To deal with these two sequences, we are going to apply the method developed in LePage
et al. (1981), who proved that any stable distribution can be represented as an infinite sum
of random variables constructed from partial sums of exponential random variables. More
specifically, let {Ei } be a sequence of independent, identically distributed exponential random
variables having expected value 1, and define the sequence {Zi } by
Zi = (E1 + . . . + Ei )−1/α ,
i ≥ 1.
(2.1)
Additionally, let {δi } be independent, identically distributed random variables, independent
of {Ei }, satisfying P {δi = 1} = p and P {δi = −1} = q with p and q specified in (1.4). Then,
define
 ∞
X



δk Zk
0 < α ≤ 1,


k=1
(2.2)
η=
∞
X



(δk Zk − (p − q)E[Zk I{0 < Zk ≤ 1}])
1 < α < 2,


k=1
and Z = Z1 . LePage et al. (1981, Theorem 1) showed that the sums in (2.2) converge with
D
probability one and η = ξα . Consider a random vector (Wα (t), Z) such that Wα (t) is a strictly
4
D
α–stable process, Z = Z1 and the joint distribution of (Wα (t), Z) is defined by the requirement
that for any 0 = t1 < . . . < tK = 1, we have
(Wα (t2 ) − Wα (t1 ), Wα (t3 ) − Wα (t2 ), . . . , Wα (tK ) − Wα (tK−1 ), Z)
D
1/α
1/α
1/α
1/α
=
(t2 − t1 ) η1 , (t3 − t2 ) η2 , . . . , (tK − tK−1 ) ηK−1 , max (ti+1 − ti ) Zi ,
1≤i≤K−1
where (η1 , Z1 ), . . . , (ηK−1, ZK−1) are independent, identically distributed random vectors, distributed as (η, Z). Using the above, we will prove the following theorem.
Theorem 2.1 If (1.3) holds, then


⌊nt⌋
X
2 [0,1]
Tn
1
 D−→

(Wα (t), Z)
(X
−
µ
),
i
α
n1/α L(n) i=1
n1/α L(n)
(n → ∞),
where the distribution of the vector (Wα (t), Z) is defined above.
The asymptotic behavior of {An (t)} is now an immediate consequence of Theorem 2.1.
Corollary 2.1 If (1.3) holds, then
D[0,1]
An (t) −→
1
Bα (t)
Z
(n → ∞),
where, for t ∈ [0, 1], Bα (t) = Wα (t) − tWα (1).
The process {Bα (t) : t ∈ [0, 1]} is sometimes called an α–stable bridge. Very little is known
about the distributions of functionals of Bα (t)/Z and therefore it is hard to determine critical
values needed to construct asymptotic test procedures. So it would be nice to apply the permutation method to obtain critical values for functionals of An (t). However, as the subsequent
series of theorems shows, Theorem 1.2 does not have an infinite variance counterpart.
Consider the process {An,π (t)} defined in (1.6) and let both X and π be random. Then, we
obtain the following result.
Theorem 2.2 If (1.3) holds, then
D[0,1]
An,π (t) −→
1
Bα (t)
Z
(n → ∞),
(2.3)
where, for t ∈ [0, 1], Bα (t) = Wα (t) − tWα (1).
Obviously, Theorem 2.2 is much weaker than Theorem 1.2, since it is valid only for the average
of the realizations. However, a stronger result, holding true for almost all realizations of X,
cannot be proved. To this end, let PX and EX denote conditional probability and expected
value given X.
5
Theorem 2.3 Let t ∈ (0, 1). If (1.3) holds, then, as n → ∞,
1
PX {An,π (t) ≤ x} −→ P
Bα (t) ≤ x
a.s.
Z
cannot hold true.
Our last result states that the conditional characteristic function of An,π (t) given X converges in distribution, for any fixed t ∈ (0, 1), to a nondegenerate random variable which can
be explicitly given via a power series whose coefficients involve the {Zi } defined in (2.1). For
any m ≥ 2, set
Rm =
∞
X
(δi Zi )m
and
i=1
Qm =
Rm
.
Zm
(2.4)
Let further
cm (t) =
m
X
k X
k
(−1)k−ℓ ℓm ,
k ℓ=0 ℓ
k
k+1 t
(−1)
k=1
t ∈ (0, 1), m ≥ 2.
Theorem 2.4 Let t ∈ (0, 1). If (1.3) holds, then
)
( ∞
m
X
Qm (is)
D
cm (t)
EX eisAn,π (t) −→ exp
m!
m=2
(n → ∞)
for all real |s| < 0.05.
The previous results concern the asymptotic properties of {An (t)} and {An,π (t)} but it is
possible to consider modifications of these processes replacing the normalization Tn by a more
general sequence
Tn(ν) =
n
X
i=1
|Xi − X̄n |ν
!1/ν
with some ν > α.
The corresponding CUSUM processes are then defined by
A(ν)
n (t) =
⌊nt⌋
1 X
(ν)
Tn
i=1
(Xi − X̄n ),
A(ν)
n,π (t) =
⌊nt⌋
1 X
(ν)
Tn
i=1
(Xπ(i) − X̄n ),
t ∈ [0, 1].
Analogues of Theorems 2.1 and
2.2 can be easily
by exploiting the joint convergence
P⌊nt⌋established
P⌊nt⌋
|Xi|ν . Theorems 2.3 and 2.4 also remain true
Xi and i=1
of the partial sum processes i=1
[with some modifications in the corresponding limit processes], so that, conditionally on X,
(ν)
the permuted sequence {An,π (t)} cannot converge weakly in D[0, 1].
6
3
Proofs
The proof of Theorem 2.1 is based on the following result due to LePage et al. (1981). Their
method relies heavily on a representation of the order statistics X1,n , . . . , Xn,n in terms of
partial sums of exponential random variables as given in the previous section, and makes use
of the fact that the upper extreme values of |X1 |, . . . , |Xn | are asymptotically independent of
the signs corresponding to these extremes.
Theorem 3.1 If (1.3) holds, then
1
1/α
N L(N)
N
X
i=1
(Xi − µα ), TN
!
D
−→ (η, Z)
as N → ∞.
Proof of Theorem 2.1. First, we establish the convergence of the finite dimensional distributions. Let 0 = t1 < t2 < . . . < tK = 1 and observe that
Tn =
max
max
1≤j≤K−1 ntj <i≤ntj+1
Then, clearly

⌊ntj+1 ⌋
X

(Xi − µα ),
i=⌊ntj ⌋+1
|Xi |.
max
ntj <i≤ntj+1

|Xi | ,
j = 1, . . . , K − 1,
(3.1)
are independent vectors. Since, by the specification of {Xi }, the distribution of the vector in
(3.1) depends only on the number ∆j,n = ⌊ntj+1 ⌋ − ⌊ntj ⌋ of terms Xi − µα but not on the
specific indices i, we get for any j = 1, . . . , K − 1, as n → ∞,


∆j,n
X
1
 (Xi − µα ), T∆j,n 
1/α
n L(n) i=1
=
1/α
∆j,n L(∆j,n )
n1/α L(n)


∆j,n
1
1/α
∆j,n L(∆j,n )

X
i=1
D
(Xi − µα ), T∆j,n  −→ (tj+1 − tj )1/α ηj , (tj+1 − tj )1/α Zj ,
using Theorem 3.1 and that L is slowly varying at infinity. Therein, {ηj } and {Zj } are independent copies of η and Z, respectively. We conclude that the finite dimensional distributions
converge.
P⌊nt⌋
(Xi − µα )/(n1/α L(n)) is proven in Gikhman and Skorohod (1969, p.
The tightness of i=1
480).
2
Proof of Theorem 2.2. Note that, by the assumptions on {Xi },
D
{An (t), t ∈ [0, 1]} = {An,π (t), t ∈ [0, 1]} ,
7
so the result follows from Theorem 2.1.
2
From now on, we fix a realization of X. Permuting the X’s is the same as selecting n elements
from X1 , X2 , . . . , Xn without replacement. The proof of Theorem 2.3 will be based on the order
statistics of X1 , . . . , Xn . Our first goal is to express An,π (t) in terms of X1,n , . . . , Xn,n . To do
so, let
(
1
if Xi,n is among the first ⌊nt⌋ elements chosen,
(n)
εi (t) =
0
otherwise,
(n)
that is, εi (t) identifies those of the order statistics which are selected by the permutation π
in the first ⌊nt⌋ positions. Then, it is easy to see that
X
(Xπ(i) − X̄n ) =
i≤nt
(n)
(n)
n
X
j=1
(n)
(Xj,n − X̄n )εj (t) =
(n)
n
X
j=1
(n)
(Xj,n − X̄n )ε̄j (t),
(n)
(n)
where ε̄j (t) = εj (t) − Eεj (t) = εj (t) − ⌊nt⌋/n is the centered version of εi (t).
Let VarX denote conditional variance with respect to X.
Proof of Theorem 2.3. In a first step, we compute the conditional expected value and
variance of An,π (t). Observe that
2
2 ⌊nt⌋
⌊nt⌋
E ε̄k (t) =
−
(3.2)
n
n
and
h
i ⌊nt⌋(⌊nt⌋ − 1) ⌊nt⌋ 2
⌊nt⌋(n − ⌊nt⌋)
(n)
(n)
=−
−
.
E ε̄j (t)ε̄k (t) =
n(n − 1)
n
n2 (n − 1)
Then, it is straightforward that
n
h
i
1 X
(n)
(Xj,n − X̄n )E ε̄j (t) = 0
EX [An,π (t)] =
Tn j=1
and, on applying (3.2) and (3.3),
VarX (An,π (t)) = VarX
=
n
1 X
(n)
(Xj,n − X̄n )ε̄j (t)
Tn j=1
!
n
h
i
1 X
(n)
(n)
(X
−
X̄
)(X
−
X̄
)E
ε̄
(t)ε̄
(t)
j,n
n
k,n
n
j
k
Tn2 j,k=1
n
1 X
(Xj,n − X̄n )2 + Dn ,
= (t(1 − t) + o(1)) 2
Tn j=1
8
(3.3)
where
X
1
|Dn | ≤
(Xj,n − X̄n )(Xk,n − X̄n )
2
(n − 1)Tn j6=k
!2
X
n
n
X
1
2
=
(X
−
X̄
)
(X
−
X̄
)
−
k,n
n
k,n
n
(n − 1)Tn2 k=1
k=1
=
n
X
1
(Xk,n − X̄n )2 ,
(n − 1)Tn2
k=1
since
that
Pn
k=1 (Xk,n
1
2/α
n L2 (n)
− X̄n ) = 0. It follows from LePage et al. (1981) [cf. also Theorem 3.1 above]
n
X
i=1
(Xi − µα )2 , Tn2
D
D
−2/α
where η ∗ = ξα/2 and Z ∗ = E1
D
VarX (An,π (t)) →
!
D
−→ (η ∗ , Z ∗ )
(n → ∞),
(3.4)
. Consequently, the preceding calculations imply that
η∗
Z∗
(3.5)
as n → ∞. It is clear that η ∗ /Z ∗ is a nondegenerate random variable. Similarly to (3.5) one
can show that
D
EX A4n,π (t) → ζ,
(3.6)
as n → ∞ with some random variable ζ. Fix now t ∈ (0, 1). Relation (3.6) implies that
the sequence {EX A4n,π (t), 1 ≤ n < ∞} is tight and thus for any given ǫ > 0 there exists a
K = K(ǫ) > 0 and for each n ≥ 1 an event Ωn with P {Ωn } ≥ 1 − ǫ such that
EX A4n,π (t) ≤ K
on Ωn .
(3.7)
The inequality in (3.7) implies
EX A2n,π (t)I{|An,π (t) ≥ y}
1 2
2
A (t)y I{|An,π (t) ≥ y}
= EX
y 2 n,π
1
≤ 2 EX A4n,π (t)
y
K
≤ 2
y
9
(3.8)
for any y > 0 on Ωn . Let us assume indirectly that, for all points x of continuity of some
distribution function F , the following convergence result holds true:
PX {An,π (t) ≤ x} −→ F (x)
a.s.
(n → ∞).
This implies that for any continuity point y of F we have
Z y
2
EX An,π (t)I{|An,π (t)| < y} →
x2 dF (x)
a.s.
(3.9)
(3.10)
−y
We choose y ≥ (K/ǫ)1/2 and conclude by Egorov’s theorem [cf. Hewitt and Stromberg (1969,
p. 158)] that there exists an event Ω∗ with P {Ω∗ } ≥ 1 − ǫ on which (3.10) is uniform and
consequently there exists a nonrandom n0 such that
Z y
2
EX A2 (t)I{|An,π (t)| < y} −
(3.11)
x dF (x) ≤ ǫ
n,π
−y
on Ω∗ if n ≥ n0 . By (3.8) and (3.11) we have
Z y
2
EX A2 (t) −
x dF (x) ≤ 2ǫ
on Ωn ∩ Ω∗ ,
n,π
(3.12)
−y
if n ≥ n0 . Since we can choose y as large as we want in (3.12), we conclude
Z ∞
P
2
EX An,π (t) →
x2 dF (x)
(3.13)
−∞
regardless if the integral is finite or infinite. Since η ∗ /Z ∗ is nondegenerate, (3.13) contradicts
(3.5).
2
The proof of Theorem 2.4 will be based on a series of lemmas. Since {Xi } is a sequence of
strictly α–stable random variables with α ∈ (0, 2), only the very small and very large order
statistics will contribute asymptotically to An,π (t), which is shown in Lemmas 3.1 and 3.3.
Lemma 3.1 If (1.3) holds, then
( (
)
)
n−K
X
1
(n)
lim lim sup P PX
(Xi,n − X̄n )ε̄i (t) ≥ ε ≥ δ = 0
1/α
K→∞ n→∞
n L(n) i=K
for all ε > 0 and δ > 0.
Proof. Without loss of generality, we can assume that µα = 0 for all α ∈ (0, 2). By (3.2) and
(3.3) we have, similarly as in the proof of Theorem 2.3,
"n−K
#
X
(n)
EX
(Xi,n − X̄n )ε̄i (t) = 0
i=K
10
and
VarX
n−K
X
i=K
(Xi,n −
(n)
X̄n )ε̄i (t)
!
≤C
n−K
X
2
Xi,n
+
nX̄n2
i=K
with some constant C > 0. Using (1.3) we get
!2
n
2
X
nX̄n
1
1
Xi
=
= oP (1)
2/α
2
1/α
n L (n)
n n L(n) i=1
!
(n → ∞).
Let U1,n ≤ U2,n ≤ . . . ≤ Un,n be the order statistics of n independent, uniformly distributed
random variables on [0, 1]. The [generalized] inverse of the distribution function of X is denoted
by Q. It is well–known that
n−K
X
2 D
Xi,n
=
i=K
n−K
X
Q2 (Ui,n ).
i=K
Using the method of Csörgő et al. (1986), one can show that, for any K ≥ 1,
n−K
X
1
D
Q2 (Ui,n ) −→ A(K)
2/α
2
n L (n) i=K
(n → ∞),
where A(K) is a random variable satisfying A(K) → 0 a.s. as K → ∞. The methodology
developed in Csörgő et al. (1986) is based on the fact that the upper and lower extreme values
of X1 , . . . , Xn are asymptotically independent.
The proof of Lemma 3.1 is complete after an application of Markov’s inequality.
2
Without loss of generality, we assume that all processes used in this paper are defined on
the same probability space. Next, we utilize a result of Berkes and Philipp (1979) to show that
(n)
the dependent random variables {εi } can be approximated with a sequence of independent
Bernoulli variables.
Lemma 3.2 If (1.3) holds, then, for each n and each t ∈ (0, 1), there are independent, identically distributed random variables δi,n (t), i = 1, . . . , n, independent of {Xi }, with
P {δi,n (t) = 1} =
⌊nt⌋
n
and
P {δi,n (t) = 0} =
n − ⌊nt⌋
n
(3.14)
such that
PX
(
PX
(
and
K
)
X
1
P
(n)
(Xi,n − X̄n )(εi (t) − δi,n (t)) > x −→ 0
n1/α L(n) i=1
n
)
X
1
P
(n)
(Xi,n − X̄n )(εi (t) − δi,n (t)) > x −→ 0
1/α
n L(n) i=n−K+1
for all x > 0 and K ≥ 1.
11
(n → ∞)
(n → ∞)
(3.15)
(3.16)
(n)
(n)
Proof. For i ≥ 1, let γ2i−1 = εi (t) and γ2i = εn−i+1 . It is easy to see that there is a constant
C > 0 such that, for ai = 0 or 1, i ≥ 1, we have
P γk = ak γk−1 = ak−1 , . . . , γ1 = a1 − P {γk = ak } ≤ C k
n
for all k = 2, . . . , n. Hence, applying Theorem 2 of Berkes and Philipp (1979), there exist
independent Bernoulli random variables δi,n (t) satisfying (3.14) such that
k
k
(n)
≤ 6C
P εk (t) − δk,n (t) ≥ 6C
n
n
and
k
k
(n)
P εn−k+1(t) − δn−k+1,n (t) ≥ 6C
≤ 6C
n
n
for k = 1, . . . , ⌊n/2⌋. Hence, we can estimate
)
(
K
X
1
(n)
PX
(Xi,n − X̄n )(εi (t) − δi,n (t)) ≥ x
n1/α L(n) i=1
K
≤C +I
n
)
K
X
K
1
|Xi,n − X̄n | ≥ x .
C
n n1/α L(n) i=1
(
We need to further estimate the right–hand side of the latter inequality. Note that, from (1.3),
we readily obtain
X̄n
1+1/α
n
L(n)
P
−→ 0
(n → ∞).
Along the lines of Corollary 3.1 in Csörgő et al. (1986), it can be verified that
!
K
X
1
D
|Xi,n | − bn,α −→ τ
(n → ∞),
1/α
n L(n) i=1
where τ is a nondegenerate random variable and {bn,α } is a sequence satisfying the condition
bn,α (n1+1/α L(n))−1 → 0 as n → ∞. Thus, we arrive at
" (
)#
K
X
K
1
E I C
|Xi,n − X̄n | ≥ x
−→ 0
(n → ∞),
n n1/α L(n) i=1
completing the proof of (3.15). The same argument proves (3.16).
2
(n)
The following lemma is the counterpart of Lemma 3.1, in which the ε̄i are replaced by
mean–corrected variables δ̄i,n (t) obtained from the independent sequence {δi,n (t)}.
12
Lemma 3.3 If (1.3) holds, then
( (
)
)
n−K
X
1
lim lim sup P PX
(Xi,n − X̄n )δ̄i,n (t) ≥ ε ≥ δ = 0
1/α
K→∞ n→∞
n L(n) i=K
for all ε > 0 and δ > 0, where δ̄i,n (t) = δi,n (t) − ⌊nt⌋/n.
Proof. The assertion follows along the lines of the proof of Lemma 3.1.
2
Lemma 3.4 Let t ∈ (0, 1). If (1.3) holds, then for all ε > 0,
P
PX {|An,π (t) − Bn (t)| > ε} −→ 0
(n → ∞),
(3.17)
where
n
⌊nt⌋
1 X
Xi,n − X̄n δi,n (t) −
Bn (t) =
Tn i=1
n
and δi,n (t), i = 1, . . . , n, t ∈ [0, 1], are defined in Lemma 3.2.
Proof. It follows readily from Lemmas 3.1–3.3.
2
To complete the proof of Theorem 2.4, it suffices to show, in view of (3.17), that
)
( ∞
X Qm (is)m
D
cm (t)
(n → ∞).
EX eisBn (t) −→ exp
m!
m=2
√
Let Si,n = Tn−1 (Xi,n − X̄n ). Clearly, |Si,n | ≤ 2. To lighten the notation we write z = −1z,
where z is real. Since {δi,n (t)} is a sequence of independent Bernoulli variables, we get
zBn (t)
EX e
n
Y
⌊nt⌋ zSi,n
=
exp(−zSi,n ⌊nt⌋/n) 1 +
e
−1
n
i=1
and therefore
ψn (t, z) = log EX ezBn (t)
n
n
X
⌊nt⌋ zSi,n
⌊nt⌋ X
e
−1
z
Si,n +
log 1 +
=−
n
n
i=1
i=1
=
n
X
i=1
⌊nt⌋ zSi,n
log 1 +
e
−1 .
n
The next lemma is an immediate consequence of the mean–value theorem applied to the
function log(1 + x).
13
Lemma 3.5 Let |z| <
ψn (t, z) −
n
X
i=1
1
2
and t ∈ (0, 1). If (1.3) holds, then
log 1 + t ezSi,n − 1
P
−→ 0
(n → ∞).
Proof. By the mean–value theorem for the real and imaginary parts,
⌊nt⌋
e |z|
zS
zS
i,n
i,n
log 1 + t e
− 1 − log 1 +
e
− 1 ≤
|Si,n |.
n
1−t n
Arguments already applied in the proof of Lemma 3.2 yield that
n
n
1X
1 X
P
|Si,n | =
|Xi,n − X̄n | −→ 0
n i=1
nTn i=1
(n → ∞),
completing the proof.
2
Lemma 3.6 For any t ∈ (0, 1) and |z| < 0.1 it holds
log (1 + t(e − 1)) =
z
∞
X
zj
j=1
j!
cj (t),
and the series is absolutely convergent, where
j
k k X
X
k
k+1 t
(−1)k−ℓ ℓj .
(−1)
cj (t) =
k ℓ=0 ℓ
k=1
Proof. We note that
∞
k
X
k+1 u
log(1 + u) =
(−1)
k
k=1
(3.18)
is absolutely convergent for any complex u with |u| < 1. Also |ex − 1| < 1 for all complex
|x| < 0.1. First using (3.18), then the binomial theorem and, finally, Taylor expansion for the
exponential function, we conclude
x
log (1 + t(e − 1)) =
=
∞
X
(−1)k+1
k=1
∞
X
(−1)
k=1
=
∞
X
k=1
(−1)
[t(ex − 1)]k
k
k
k+1 t
k
k
k+1 t
k
k X
k
ℓ=0
ℓ
k X
k
ℓ=0
ℓ
(−1)k−ℓ eℓx
k−ℓ
(−1)
∞
X
(xℓ)j
j=0
j!
∞
k ∞
X
xj X (−1)k+1 k X k
(−1)k−ℓ ℓj .
t
=
ℓ
j!
k
j=0
k=1
ℓ=0
14
On observing that
k X
k
(−1)k−ℓ ℓj = 0
l
ℓ=0
if k > j,
Lemma 3.6 is proved.
2
Lemma 3.7 Let t ∈ (0, 1) and |z| < 0.1. If (1.3) holds, then
!
n
∞
n
j
X
X
X
z
j
log 1 + t ezSi,n − 1 =
Si,n
cj (t).
j!
i=1
j=2
i=1
Proof. Since
n
X
Si,n = 0,
i=1
the assertion follows directly from Lemma 3.6.
Lemma 3.8 If (1.3) holds, then
!
n
n
X
X
D
3
2
Si,n
,
Si,n
, . . . −→ (Q2 , Q3 , . . .)
i=1
i=1
2
(n → ∞),
where Q2 , Q3 , . . . are defined in (2.4) and the convergence is meant in the sense of convergence
of the finite dimensional distributions.
Proof. If m = 2 then, by (1.3),
2 X
n
n
X
nX̄n2
1
1
2
2
=
(X
−
X̄
)
−
X
=
i
n
i
1/α
2/α
2
n L (n)
n L(n) i=1
n
i=1
If m > 2, the mean–value theorem implies
n
n
n
X
X
X
m
m
2m−1 |Xi |m−1 + |X̄n |m−1
Xi ≤ m|X̄n |
(Xi − X̄n ) −
i=1
i=1
i=1
!
n
X
|Xi |m−1 + n|X̄n |m .
= m2m−1 |X̄n |
i=1
On applying (1.3) once again, we obtain
n
|X̄n |
1/α
n L(n)
m
=
1
nm−1
n
X
1
Xi
n1/α L(n) i=1
15
!m
P
−→ 0
n
X
1
Xi
n1/α L(n) i=1
(n → ∞)
!2
P
−→ 0.
and
1
n1/α L(n)
m
|X̄n |
n
X
i=1
|Xi |m−1
as m → ∞, since (n1/α L(n))−(m−1)
to show that, as n → ∞,
1
n1/α L(n)
Tn ,
1
n1/α L(n)
m−1 X
n
n
X
1
1
1
P
Xi |Xi|m−1 −→ 0,
= 1/α
1/α
n n L(n)
n L(n)
i=1
Pn
i=1
2 X
n
|Xi |m−1 converges in distribution. Hence it is enough
Xi2 ,
i=1
i=1
1
n1/α L(n)
3 X
n
!
Xi3 , . . .
i=1
D
−→ (Z1 , R2 , R3 , . . .) ,
where R2 , R3 , . . . are defined in (2.4). But this follows from the proof of Theorem 1 in LePage
et al. (1981) and the proof is complete.
2
Proof of Theorem 2.4. Applying Lemmas 3.5 and 3.7, it is enough to prove that
!
∞
n
∞
X
X
zj X j
zj
D
Qj cj (t)
(n → ∞).
Si,n cj (t) −→
j!
j!
j=2
i=1
j=2
For any M ≥ 2, Lemma 3.8 implies
!
M
n
M
X
X
zj X j
zj
D
Si,n cj (t) −→
Qj cj (t)
j! i=1
j!
j=2
j=2
(n → ∞).
Next we show that
∞
X
zj
P
Qj cj (t) −→ 0
j!
j=M +1
(M → ∞).
(3.19)
According to the strong law of large numbers, there is a random variable k0 such that
E2 + . . . + Ek ≥
k
2
if k ≥ k0 .
Hence,
|Qj | ≤
=
j
∞ X
Zi
i=1
∞ X
E1
E1 + . . . + Ei
j/α
k0 X
E1
E1 + . . . + Ei
j/α
i=1
=
(3.20)
Z1
i=1
+
∞
X
i=k0 +1
E1
E1 + . . . + Ei
16
j/α
≤ k0 +
j/α
E1
∞
X
i=k0 +1
1
E1 + i/2
j/α
≤ k0 + C(2E1 )j/α (2E1 + 1)−j/α+1
≤ k0 + C(2E1 + 1)
with some constant C > 0, where we replaced the last sum with an integral. The series in
Lemma 3.6 is absolutely convergent and thus we obtain
∞
∞
X
j
X
|z|j
z
P
Q
c
(t)
≤
(k
+
C(2E
+
1))
|cj (t)| −→ 0,
(3.21)
j j
0
1
j!
j!
j=M +1
j=M +1
establishing (3.19).
The final step contains the proof of
( ∞
)
!
n
X zj X
j
lim lim sup P S
cj (t) > ε = 0
M →∞ n→∞
j! i=1 i,n
j=M +1
(3.22)
for all ε > 0, t ∈ (0, 1) and |z| < 0.05. To this end, let K be the quantile function of |Xi |. By
(1.3) we have that
K(x) = (1 − x)−1/α U(1 − x)
(3.23)
with some function U slowly varying at zero. Also,
E1 + . . . + Ei
D
{|X|i,n, 1 ≤ i ≤ n} = K
,1 ≤ i ≤ n ,
E1 + . . . + En+1
where |X|1,n ≤ |X|2,n ≤ . . . ≤ |X|n,n denote the order statistics of |X1 |, . . . , |Xn |. For i =
1, . . . , n + 1, let
Si = E1 + . . . + Ei
and
Si∗ = Ei + . . . + En+1 .
Using (3.23), we obtain
1/α
∗ 1/α
∗
∗
∗
U Si+1
/Sn+1
Sn+1
/Sn+1
U Si+1
/Sn+1
Sn+1
K (Si /Sn+1 )
=
.
=
1/α
∗
∗
∗
∗
K (Sn /Sn+1 )
Si+1
U Sn+1
/Sn+1
U Sn+1
/Sn+1
Si+1
/Sn+1
It is easy to see that
)
(
) ( 1/α
1/α
∗
∗
U Si+1
/Sn+1
Sn+1
U
(S
/S
)
S
D
i
n+1
1
, 1 ≤ i ≤ n =
,1≤i≤n .
∗
∗
Si+1
S
U
(S
/S
U Sn+1 /Sn+1
i
1
n+1 )
17
Let δ > 0 and γ > 0. By the Karamata representation theorem [cf. Bingham et al. (1987)],
there are functions c(t) and u(t) such that c(t) → c0 , u(t) → 0 as t → 0 and
!
Z Si /Sn+1
c(Si /Sn+1 )
U(Si /Sn+1 )
u(t)
=
exp
dt .
U(S1 /Sn+1 )
c(S1 /Sn+1 )
t
S1 /Sn+1
There is ǫ > 0 such that
sup |u(t)| ≤ γ
0≤t≤2ǫ
and
c(t)
≤ 1 + δ.
0≤s,t≤2ǫ c(s)
sup
So by the law of the large numbers we have
γ
U(Si /Sn+1 )
Si
lim P
≤ (1 + δ)
for all 1 ≤ i ≤ ⌊nǫ⌋ = 1.
n→∞
U(S1 /Sn+1 )
S1
Let C = ((2/ǫ)γ supǫ/2≤x≤1 U(x))−1 . Then there is x0 > 0 such that
1
≤ Cx−γ
U(x)
if 0 ≤ x ≤ x0 .
On the event {S1 /Sn+1 ≤ x0 , S⌊nǫ⌋ /Sn+1 ≥ ǫ/2} we have, for all ⌊nǫ⌋ ≤ i ≤ n,
γ
Sn+1
U(Si /Sn+1 )
≤ sup U(x)C
U(S1 /Sn+1 ) ǫ/2≤x≤1
S1
γ
γ
Si
Sn+1
=
sup U(x)C
S1 ǫ/2≤x≤1
Si
γ
γ
Sn+1
Si
sup U(x)C
≤
S1
S⌊nǫ⌋
ǫ/2≤x≤1
γ
Si
≤
.
S1
Thus we conclude that for all δ > 0 and γ > 0
γ
U(Si /Sn+1 )
Si
lim P
≤ (1 + δ)
for all 1 ≤ i ≤ n = 1.
n→∞
U(S1 /Sn+1 )
S1
It follows from the estimates in (3.20) and (3.21) that
1/α−γ !j
∞
∞ X
X
zj
S1
P
→ 0,
(1 + δ)
j!
Si
i=1
j=M +1
if 0 < δ < 1/2 and 0 < γ < (2 − α)/(2α).
2
18
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19