A NNALES DE L ’I. H. P., SECTION B
M IGUEL A. A RCONES
E VARIST G INÉ
The bootstrap of the mean with arbitrary bootstrap sample size
Annales de l’I. H. P., section B, tome 25, n
o4 (1989), p. 457-481
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The bootstrap of the
meanwith arbitrary bootstrap sample size
Miguel
A. ARCONES(1)
C.U.N.Y., Graduate Center, Departement of Mathematics, 33 West 44 Street, New York, New York 10036
Evarist GINÉ
(2)
C.U.N.Y., College of Staten Island, Department of Mathematics,
130 Stuyvesant Place, Staten Island, New York 10301 i
Vol. 25, n° 4, 1989, p. 457-481. Probabilités et Statistiques
ABSTRACT. - We
study
thebootstrap
central limit theorem a. s. and inprobability,
for random variables X with infinite second moment, in the domain of attraction of thenormal law,
or of the other stablelaws,
or inthe
partial
domain of attraction of aninfinitely
divisiblelaw,
and withthe size mn of the
bootstrap sample
notnecessarily equal
to(and
in somecases
necessarily
differentfrom)
the size n of theoriginal sample.
Main results:
(1)
If X is in the domain of attraction of the normal law then it satisfies thebootstrap
CLT inprobability
for all mn oo;(2)
If is aregular
sequence such that mn(log log n)/n
--~0,
and X is in the domain of attraction of a stable law then thebootstrap
CLT holdsa. s.; but it does not hold a. s. if
EX~
= oo andinf mn (log log n)/n
> 0.Classification A.M.S. : 62 E 20, 62 F 12, 60 F 05.
(~)
Supported by a scholarship from the US-Spanish Joint Committee.(~)
Partially supported by NSF Grant No. DMS-8619411. On leave from Texas A & MUniversity.
Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques - 0246-0203
Key words : Bootstrap of the mean, a. s. bootstrap, bootstrap in probability, central limit
theorem.
Nous étudions le théorème central limite «
bootstrap
»presque sûr et en
probabilité,
pour des variables aléatoires X de second momentinfini,
dans le domaine d’attraction de la loi normale ou des autres loisstables,
ou dans le domainepartiel
d’attraction d’une loi infiniment divisible. La taille m" de l’échantillon «bootstrap »
n’est pas nécessairementégale (et
meme dans certains cas elle est nécessairementdifferente)
a la taille n de l’échantillonoriginal.
Les résultats
principaux
sont :(i)
Si X est dans le domaine d’attraction de la loinormale,
il vérifie le théorème central limite «bootstrap »
enprobabilité, quelle
que soit mn tendant vers oo .(2)
Si est une suiterégulière
telle que mn(log log n)/n
-0,
et si Xest dans le domaine d’attraction d’une loi
stable,
alors le théorème central limite« bootstrap »
a lieu presque sûrement. Mais il n’est pas exact presque sûrement siEX2 =
oo et inf mn(log log n)/n
> 0.1. INTRODUCTION
In this article we
study
thebootstrapped
central limit theorem(the bootstrap
of themean)
for sizes mn of thebootstrapped sample
different from the size n of the
original sample
Our work is motivatedby
results of Bickel and Freedman[5], Athreya ([3], [4]),
andCsorgo
and Mason[6].
To describe their results as well as ours werequire
some definitions and notation that will be in force for the rest of the paper. Let
X, Xl, X2,
... beindependent identically
distributed(i. i. d.)
real random variables
and,
for each and let(r~’), j
E ben
i. i. d. random variables with law
P" (c~) = n -1 ~ ~xi
t~~, defined in anotheri=1
probability
space Q’. Thebootstrap
variables can be realized onthe
product probability
space Q x Q’ as follows: let ..., bedisjoint independent partitions
of Q’ withP’ Ai~ =1 /n
for alli, j;
thenn
X ~ (c~’) _ ~
In what follows wedrop
reference to 00’ andi= 1
~ .
write
P
for P’and
for conditional distributiongiven
Let d be anyAnnales de l’Institut Henri Poincaré - Probabilités et Statistiques
distance
metrizing
weak convergence in [R. We writeto mean
If
(1.1)’
holds for somesuch d,
it holds forall,
and even for the sup norm of distribution functionsif Jl
is continuous[12].
The notation*M
~- J -*- 2014
is self
explanatory,
and isobviously equivalent
toso that in
particular
the "a. s.boostrap
CLT"(1.2) implies
the"bootstrap
CLT in
probability" (1.1). Ideally an
is not a constant, but a function of thesample Xl,
...,Xn:
this will end upbeing
the case herewhenever p
is the normal law
N(0, a2).
If either(1.1)
or(1.2)
hold and alsothen,
as is wellknown,
thebootstrap
limit theorems(1.1)
and(1.2)
canbe used to estimate
asymptotic probabilities
of sets definedthrough
u
I (e. g. [7], [5];
see also[11]).
It iscustomary
to take mn = n ini=1 i
(1.1)
and(1.2),
but this is not necessaryand,
if ~, in(1.3)
is notnormal,
then it is evenimpossible ([3], [12]).
Bickel and Freedman
[5]
show thatif EX2
oo then thebootstrap
CLT(1.2)
holds a. s. aslong
as mn ~ ~. But forEX2
= oo the situation is different:Athreya [4]
andCsorgo
and Mason[6]
prove that that ifEX2 = 00,
u. is normal and X satisfies(1.3)
then thebootstrap
CLT inprobability (1.1)
holds for0 C 1 mn/n C2 oo (it
isproved
in[12]
thatthe a. s.
bootstrap
CLT does not hold ifEX2
= oo and mn =n). Athreya
also shows in
[4]
that if X is in the domain of attraction of ap-stable
law , 0
p _ 2,
withnorming constants bn
then(1.1)
holds forand
(it
does not hold if mn = n[12]).
These resultsobviously
suggest the
following questions:
oo be taken to bearbitrary
ifEX2
= ooand
is normal? Under what conditions on does the a. s.bootstrap
CLT holdif Jl
isp-stable,
0 p 2? If(1.3)
holdsonly along
asubsequence ~n’~
m t~j(i.
e. X is in the domain ofpartial
attraction ofu)
does
(1.1)
or(1.2)
also holdalong ~n’~?
These are thequestions
weconsider in this article.
In Section
2,
we determineessentially
all the situations for which thebootstrap
CLT inprobability (1.1)
holds. We show inparticular
thatif y
is
normal, EX2=~
and(1.3) holds,
then(1.1)
holds too withan = bmn
ifmn n,
and an =bn (mn/n) 1 ~2
if mn > n. In the normal case,using
the factn
that L 62
inprobability, an
can be estimated from thesample
i= 1
without
changing
the limit in(1.1).
Results for X in the domain ofpartial
attraction of an
infinitely
divisible law are alsogiven;
theseimply,
inparticular, Athreya’s
resultfor Jl p-stable
andmn/n -+-
0.In Section
3,
we consider the a. s.bootstrap
CLT. We prove that ifEX2
= oo and X is in the domain of attraction of ap-stable law Jl
thenthe a. s.
bootstrap
CLT does not hold for inf mn(log log n)/n
> 0. On then
other
hand,
at least underregularity (~mn~ non-decreasing
andc > 0 for all n E
I~)
if mn(log log n)/n --~
0 then thebootstrap
CLTdoes hold a. s.
The methods of
proof
are not new: werepeatedly
check the usual conditions for the central limit theorem inR,
very much as in[12].
Wecan do this
because,
as observed in[12],
if an - oo then thesystem
~X ~/an : j = 1, ...,
is a. s. infinitesimalby
the law oflarge
numbers:just
noteH
The results on the a. s.
bootstrap rely
on the usualtechniques
for the law of the iteratedlogarithm, suitably
modified. This work owes much to[12]
on
technique, particularly
for Section 2.2. The
Bootstrap
CLT inProbability
The
following
theoremgives
necessary conditions for thebootstrap
inprobability.
Itsproof closely
follows[12].
We use the notation Pois x for ageneralized
Poisson measure withLevy
measure x, as in[2].
If asequence is
non-decreasing
and cn ~ ~, we write oo .2.1. THEOREM. - Let X be a random variable
for
which there exist asequence
of positive integers
~, a sequenceof positive
real numbersAnnales de l’Institut Henri Poincaré - Probabilités et Statistiques
~, random variables en
(03C9),
n E and a randomprobability
measureé
(c,~), non-degenerate
withpositive probability,
such thatThen:
(a)
There are aLévy
measure x and a real number~2 >__
0 such thatfor
all I
1) = 0,
(b) If
c is apositive
number andthen, for
all i witha~ ~
- i,i~
=0,
r__ _ -
(c) If lim
supmn/n
> 0 then ~ = 0 and62 ~
0 in(a)
and(b).
K -* 00
(d) If lim
infmn/n
> 0 then X is in the domainof
attractionof
the normalH -~ 00
law with
norming
constantsbn
= an(n~mn)1/2,
that ism n -i
(e) If (2.1)
holdsonly along
asubsequence ~n’~
c (~ with~mn,~
c~n’~,
then
(a)-(d)
hold but with n’ insteadof
n[in (d) (2.4)
holdsfor
n’ but X isnot
necessarily
in the domainof
attractionof
N(0, a2)].
Proof. -
IfEX2
oo this theorem is a consequence of Theorem 2.1 in[5]. So,
we assumeEX2 =
oo . As shown in theintroduction,
is an infinitesimalsystem,
a fact that will be usedthroughout
without further mention.(2.1 )
holds if andonly
if everysubsequence
has a furthersubsequ-
ence
along
which the limit in(2.1)
holds a. s.Hence, ~, (~)
is a. s.infinitely
divisible. Let
~n’~
be such asubsequence.
The fact that an. --~ ooreadily gives, by
thecorresponding argument
in[12],
that theLevy
measure x(c~)
of
~. (~)
is a. s. a fixedLevy
measure x.Moreover,
if D is a countable set ofpoints 5
dense inR +
such~~
=0,
then thefollowing
limitshold for all 03B4~D almost
surely (by
thegeneral
CLT inR,
e. g.[2]
Chapter 2):
and
where
~2 (c~)
is the variance of the normalcomponent
of SinceEX2 = oo ,
the law oflarge
numbersimplies
n
L X2
I(I Xi I
-~ oo a. s. and this is all that is needed to prove,i= 1
exactly
as in Lemma 3 of[12],
thatThen
(2.6)
becomesIn
particular,
--~ 0(even
-~ 0by
the law oflarge numbers)
and therefore~2 (c~)
is a tail randomvariable,
hence a. s. constant, saya2.
We have thus shown that p(~)
is a. s. a(possibly random)
shift of~. = N (o, ~2)
* Moreover(2.5)
and(2.6) show, by
thegeneral
CLT in
R,
that(2.1)’
holds a. s.along
n’. Therefore(2.1)’
holds.We will prove
(b)
forc =1,
the case ofgeneral
cbeing entirely
similar.Set
bn
= and rn = mn A n. Consider asubsequence
for which(2. I )
converges a. s.
Then,
eithermn/n
1infinitely
oftenalong
thissubsequ-
ence, or
mn/n
> 1 i. o. or both.So,
we mayspecialize
to twotypes
ofsubsequences n’
for which(2.1)
holds a. s.: thosesatisfying 1,
andthose for which > 1. In the first case the summands in
(2.5)
and(2.8)
are bounded and therefore theirexpected
values converge to theexpected
values of theirrespective
limits(by e.
g.[I],
Theorem 3.2 or Annales de l’Institut Henri Poincaré - Probabilités et Statistiquesexercises 13 and
14,
p.69,
70 in[2]). Thus,
we obtain for all6 e D,
and
These three limits
imply [recall EX2
= oo and(2.7)] by
thegeneral
CLT inR
(e. g. [2], Chapter 2)
that the limit(2.3)
holdsalong ~n’~ (bn,
= an, in thiscase). Suppose
nowmn,/n’
> 1 and that(2.1)
holds a. s.along
n’.Then, by taking
a furthersubsequence
if necessary, we may assumen’
mn./n’ -~ oo].
If c = oo then(2.5) implies L I (~ XL > ~an.)
- 0 a. s. Ifi= 1
ce(0, oo )
then thecorresponding argument (on
binomiallimits)
in then’
proof
of Theorem 1 in[12] gives
alsothat L
I(I Xi
>~an,)
- 0 a. s. Theni= 1 n’
L
I( Xi > ~an,)
= 0eventually
a. s.(since
this sequence isinteger
valued~=1 i
and tends to
0),
whichimplies
that the limits in(2.5)
are 0 a. s. thatis,
a~ = o. This
argument already
proves(c). Then,
sincen’
L Xt
I(I Xi ( >
= 0eventually
a. s.,(2.8)
becomesi= 1
n’
We can
apply
the converse CLT(e.
g.[2],
p.61)
to(2.9)
and obtain[recall bn,
= an’{n’/mn’)1/2
in thiscase]
that for all 03B4 >0,
But
(2.10) implies by
the CLT in R(recall EX2 = oo),
[where
EX can bereplaced by
EXI( X I bb,~.)]
i. e.(2.3)
for the sequence~n’ ~ .
Thisargument already
proves(d)
since itgives (2.4) along
asubsequ-
ence of every
subsequence.
We have thusproved
that everysubsequence
has a furthersubsequence along
which the limit(2.3)
holds.Hence, (2.3)
holds.
(a)-(d)
areproved.
Theproof
of(e)
followsexactly along
the samelines since
nothing
in the abovearguments depends
on the sequence~h}
being
all of[ ]
Before
proving
the converse of Theorem 2.1 in thegeneral
case, we will consider the case of X in the domain of attraction of the normal law.There are two reasons for
thjs:
one is itsimportance,
and the other is that in this case we haveregular
variation as an additional convenient tool.Theorem 2.2 was obtained
by Athreya [4]
in thespecial
casemn = nand by Csorgo
and Mason[6]
for 0 c2 oo. We extend their results toarbitrary
2.2 THEOREM. - Let X be in the domain
of
attractionof
the normal law.Concretely,
assume there are constants ao such thatLet m~ -~ 00. For
CE(0, oo } fixed,
let, ~ . ,. -
Then,
Proof -
The caseEX~
oo isproved
in[5];
see thebeginning
ofSection 3 below for an alternative
simple proof
that in this case(2.13)
holds a.s.
So,
we can(and do)
assume Todispense
with thecentering
for the rest of theproof
note that once shifttightness
of the~
law
of 03A3 X03C9nj/an’
has been established for asubsequence {n’}
of N thenj= i
the
centering
variablesprescribed by
the usualtheory
arehence, an, 1
mn,Xn,
is allowed ascentering
ifAnnales de l’Institut Henri Poincaré - Probabilités et Statistiques
for
mn./n’
c. DefineU (t) = EX2 I(|X|~ t)
and note thatE X I X > t
~
d LJ s s since E X
~
f (s)
dLT s in
>
t) = ~t dU(s)/s / since E/( Xj)= dU(s)
ingeneral . Integrating by parts
andusing
Theorem1,
p. 281 in[8]
weobtain
; =
> _E|X|I(|X|I > bm , n = U {bm n ,)
En,where En. ~ 0.
Then,
since mn U ~ 1(by (2.12)
and the classicalb mn
ndomains of attraction theorem - e. g.
[2],
p.86), (2.14)
follows.For every
subsequence
there is a futhersubsequence ~n’~
such that eithermn,/n’
-~ 0 ormn,/n’ ~ ~, E (o, oo].
In the first case we may assume an,By (2.12)
and the domains of attractiontheorem,
normal case, andTherefore,
and
Therefore,
and
Then,
for asubsequence ~n"~
of~n’~
these limits hold a. s. for all 8 ED,
and the
CLT,
normal convergence case(e.
g.[2],
p.63), together
with(2.7), gives
Suppose
nowmn,/n’ ~ ~, E (0, oo].
We can assume an, =bn. (m",/n’)1~2
sincethis holds at least from some n’ on if À= oo, and if À oo then
~1~2 by regular
variation. We observe firstthat, by (2.12),
This is
just
Raikov’s theorem(e.
g.[9]),
but it followstrivially
from> bn ~ -~ U,
the first two limits are necessary
(and sufficient)
for(2.12)
tohold,
andthe last one is a consequence of
and theorem
1,
p. 281 in[8],
whichgive
Hence we
have, by
the definition of an,,Since in the case we are
considering
a". > for some constant d >0,
we have n’X
>03B4an’}
n’X >
~ 0 for all 03B4 > 0. Thisimplies
Annales de l’Institut Henri Poincare Probabilités et Statistiques
n’
~
I(~ Xj
>8~.)
-~ 0 inprobability,
hencet=i 1
hence,
even if À = oo,Moreover,
this limit and(2.17) give
So,
these two limits hold a. s.along
somesubsequence ~n"~
of~n’~
and allb E D. For this
subsequence
the usual CLT and(2.7) implies (2.16).
Wehave thus
proved
that(2.16)
holdsalong
somesubsequence
of~n"~
of anysubsequence
of Hence(2.13)
holds.[ ]
2.3. Remark
(estimating an
from thesample). -
In Theorem2.2,
letm m
with
an (ro’)/an
--~ 1 inprobability (as
a random variable defined on Q xQ’).
Then it is easy to prove
(by e.
g.passing
tosubsequences)
thatn
Since £
1 inprobability (2.18)
showsthat an
can bereplaced
ini= 1
(2.13) by
thefollowing
functionan
of thesample:
To
check that(2.19) satisfies
--~ 0 in pr.just
note. thatn
L
1 inLp
for every p 1(e.
g. Theorem 3.2 in[1]);
then the casei= i
is obvious and the case mn n follows
by Chebyshev’s
inequality.
The choice(2.19) of 03C9n
may not be toopractical
for mn n.Another
possible choice,
for all n, isActually, by Raykov’s theorem for every subsequence
there is another
subsequence {n’}
such that ~ 1 in0)’ -probability;
then dominated convergence onfi 03C9n’-1| >~}~0
shows that ~1 inan, J
(co, (03C9’)-probability. ]
2.4. Remark
(On
thecentering
in Theorem2.2.). -
Thecentering Xn
in
(2.13)
cannot ingeneral
bereplaced by X.
Forinstance,
if the law of X hasdensity
andmn = 2n, then bn~ n1/2 log n
and
Finally
we consider thegeneral
case of X in the domain ofpartial
attraction of an
infinitely
divisible law. Withoutregular
variation for U(t)
we do not know how to treat the case
mn./n’
- oo in thefollowing
theorem.2.5. THEOREM. - Let X
satisfy
for
some n’ ~ ~, an, ~ oo, 03C32>__
0 andLevy
measure 03C0{possibly 0),
with isuch
i~ = 0.
Letm~. ~’
oo,~mn,~
c~n’~. Then, if
- 0 orif
x = 0 and sup oo ,in
probability.
Proof -
Hereagain,
we can assumeEX2
=00. Let us first consider the casemn,/n’
--~ 0.By
the usual arguments(with subsequences
and theAnnales de l’Institut Henri Poincaré - Probabilités et Statistiques
CLT)
it suffices to show that forevery 8
inD,
and
where op denotes a sequence of random variables
tending
to zero inprobability. [The equal sign
in(2.25)
follows from(2.7).] (2.23)
followsfrom the
following
two limits which holdby
the converse CLTapplied
to(2.21 )
andmn,/n’
- 0 :and
(2.24)
isproved
inexactly
the same way. As for(2.25)
wesimilarly
notethat
[by (2.7)]
and
._ 2014" ’"n
Let us now assume sup
mn./n’ _
c oo and 03C0 = 0.By passing
to subse-quences and
applying
the firstpart
of theproof
if necessary, we assume infmn,/n’
> 0. In this case werequire
arelatively
more refined method forproving (2.22).
Letfd( -v)|:03A3~f(i)~~~1}
where
f(O)
=f,
is the i-th derivative off and ~f~~
is the sup normof f : R - R.
Thend3
metrizes weak convergence([2], Chapter 1).
TheLindeberg
method ofproof
of the CLT(e. g. [2],
Theorem2.1.3)
andthe
triangle inequality together
with a few trivialestimates,
as in[2],
Theorem
2.3.2, give
where
K = { 1 + (S/~) ~~2)/6
andSince x = 0 and inf
mn,/n’
>0,
it follows that for all -À >0, E
>0,
Annales de l’Institut Henri Poincaré - Probabilités et Statistiques
and
therefore,
By (2.27),
s=
( 1
+op) 6n,,
s wheren’
Hence,
aslong
asan, s
is bounded inprobability [as
is the case: see(2.28) below] 6n, s
can bereplaced by
in(I3’ ° s)
and(I4 ~ s) . By
the converseCLT in the infinite variance case,
(2.21) implies
_
"’n’
This and the
previous
observation showthat,
for all E >0,
Analogously,
since supmn,/n’
_ c oo ,lim sup Var
(8f,
~03B4)
limsup m2n’ 4 EX4
I(I X I _
(2.28)
and(2.30)
thengive, by
theprevious observation,
Now, (2.27), (2.29)
and(2.31) yield
for all E > 0. This is
just (2.22). [ ]
This
previous
theorem(together
with easy considerations oncentering
that we
omit) immediately gives:
2.6. COROLLARY
(Athreya, [4]).
- Let 8 be anon-degenerate p-stable
random
variable,
0 p2,
and let X be in its domainof
attraction, withnorming
constantsbn
that iswhere i can be zero
f
p 1 and + ooif
p > 1. Thenif 0,
2.7. Remark. - It is easy to
check,
with the methods used in thissection,
that in Theorems 2.5 and 2.6 the centers in thebootstrap
limitresults
(2.22)
and(2.33)
arestochastically equivalent
to the centers in therespective
limits(2.21)
and(2.32)
for theoriginal sample.
3. THE a.s. BOOTSTRAP CLT
Bickel and Freedman
[5]
showed that ifEX2
oo then thebootstrap
CLT holds
a. s. for
anybootstrap sample
size mn -+ oo . Theirproof
usesdistances in a way somewhat similar to the last
part
of theproof
ofTheorem 2.5 above. Here is a somewhat more natural
proof
of theirtheorem:
3.1. Short
proof of
Theorem 2.1 in[5].
- In order to showby
the usual CLT(e.
g.[2],
Cor.2.4.8,
p.63)
it suffices to provefor all 8 > 0
(then
one makes the set of measure one where convergence takesplace independent
of 8just by taking
a countable set of8’s,
asusual),
and
Now,
sinceEX2
oo, we have that for anyp _ 2,
Annales de l’Institut Henri Poincaré - Probabilités et Statistiques
by
thestrong
law oflarge
numbers(replace ~mn ~2 by
c rational and take limlim ).
Then(3.4)
followsimmediately,
and so do(3.2)
and(3.3):
and
The a. s.
bootstrap
CLT forEX2
= oo is somewhat morecomplicated.
We know from
[12]
that ifEX2
=00 and mn = n the CLT cannot bebootstrapped
a. s. The same is true if cn for some c > 0 : .’3.2. PROPOSITION. -
If for
some sequence an ~ ~, random variables cn(t,~)
and random measure ~(cc~) non-degenerate
withpositive probability
there exist oo such that inf > 0
for
whichthen
EXZ
oo .Proof - By
Theorem 2.1 we can takeH
and then = N
(0, a2)
a. s. for some ~E (o, oo).
Then(3.5) implies (converse CLT)
for all 5 > 0. This and inf
mn/n
> 0gives
hence,
alsofor all p.
So,
ifEX2
= oo the truncated variance condition of the CLT forX J
becomes[recall (2.7)]
If we let then
(3.7) gives,
as in the lastpart
of theproof
of Theorem
2.1,
that X is in the domain of attraction of the normal law withnorming
constantsbn.
SinceEX2
= oo thisimplies
inparticular
that(in
the sense thatbn/dn-+ 1)
withL (n),
whereL (n) ~
00.Hence oo , so that we can
apply a
result of Feller(e. g. [15],
Theorem
3.2.5,
p.132)
to conclude that the limit in(3.7)
is either 0 or+ oo, a contradiction.
Therefore, EX2
oo .[ ]
We believe that
Proposition
3.2 is not bestpossible.
Infact,
in view of thefollowing result,
it ispossible
thatProposition
3.2 holds true for allsequences
m" ~
oo such thatliminfmnLLn > 0,
whereL r = log (r v e),
n
and
LLr=L(Lr).
3.3 THEOREM. -
If EX2=~, f X
is in the domainof
attractionof
anon-degenerate p-stable law,
0 p _2,
andif A :
= inf mn(LL n)/n
>0,
thenthe a. s.
bootstrap
CLT does not holdfor
X .Proof -
We can assumemn/n -+
0by Proposition
3.2. If 0 is the p- stable limit and are thenorming
constants then the results of Section 2give
where 8
is a shift of 8. Let us assume 0 p 2. If this limit held a. s.then we would
necessarily
haven
for all 8 >
0,
where 03C003B4=03C0{(- 03B4, 03B4)c}, 03C0 being
theLevy
measure of L(8).
Hence
0,
S > 0. We will show that(3.9)
does not hold under thehypotheses
of the theorem. If suffices to consider 8=1. LetAnnales de l’lnstitut Henri Poincaré - Probabilités et Statistiques
Since
(3.9)
with 8= 1 isequivalent
toLet and let
ak = bmk
and pk =pnk.
We claimfor all K > 0. To estimate the
probabilities
in(3.10)
we use Prokorov’sexponential inequality (e. g. [15],
Theorem5.2.2,
p.262)
which states thatn
for 03BEi independent centered, |03BEi| ~
csnwhere sn = L
E03BE2i,
i= 1
We take
§; = I (] X;
where s2k=nk-1pk(1-pk), andc= Since xi and - e we have for 5 > 0 and k
large enough,
Hence we can
replace
arcsinhby (1- 8’) log
in Prohorov’sinequality.
Weobtain
for k
large enough,
since ~.(3.10)
isproved.
And(3 .10) implies
that for all K >
0,
We show next that there exists L > 0 such that
To prove
(3.12)
we will invokeKolmogorov’s exponential
minorization([15],
p.262): for 03BEi
as above there are, for all y >0, E (y)
andx (y)
suchthat if
E >
E(y)
and E(y)
thenNow the variance of the sum is
~:==(~"~-i)~(l"/~)
sothat,
withc==
1/sk
and s==nkL mksk
wehave,
for klarge enough, ~ c~ 2L/7i:i
arid8 ~ 20142014 (~~/~)~~ ~
oo’Hence, taking
y= 1 we canapply Kolmogorov’s inequality
forWe then
have,
for klarge enough,
which is the
general
term of adivergent
series ifHence
(3.12)
holds for all 0L~03C0(1)03C01 2
A2014L .
But(3.12) implies
2
. ~ _
(by disjointness
of the intervals(~-i. ~J
an(Borel-Cantelli)
that(3.11)
for K=L/2
and(3.13)
show that a. s.- nk
(I (| Xi|>
>L/2
> 0infinitely often,
and therefore(3.9)’
hk i = 1 does not hold.
If p = 2 and the
bootstrap
CLT holds a. s.then,
sinceEX2
= oo,by
theconverse CLT and
(2.7)
the limitAnnales de l’Institut Henri Poincaré - Probabilités et Statistiques
must hold. A
proof completely analogous
to the above one shows thatfor some L
> 0,
infinitely often,
which contradicts(3.14)
because~k
U(ak)
~a2. [ ]
ak
Finally
we show that the a. s.bootstrap
CLT for X in the domain of attraction of any stable lawalways
holds if mn(LL n)/n
-0,
at least underregularity
of(mn/m2 n
> c for some c > 0 andoo).
Theorem 3.3 shows that the result issharp. Thus,
the next theoremimproves Athreya’s
result
(Corollary
2.6above)
for these sequences.3.4. THEOREM. - Let 9 be a
non-degenerate p-stable
randomvariable,
0 p__ 2,
and let X be a random variable in its domainof
attraction,concretely,
let Xsatisfy
with ~. Let be a sequence
of positive integers regular
in the sensethat oo and >_ c
for
some c > 0 and all n and such thatThen
Proof. - By
Bickel and Freedman’stheorem, only
the caseEX2
= 00requires proof. Let,
for X >0,
b >0,
As
usual,
it suffices to show that(~~, ~ _ ~ = 0 if p = 2; c~2 = 0 if p 2;
forp = 2,
lim isredundant). Letting
s-~o
pn, ~ = P ~X
> these limits areequivalent
to
The three limits can be
proved using exactly
the sametechnique. So,
wegive
the details of theproof only
for(3.18). By
Borel Cantelli(3.18)
willfollow if we show
for all E > 0. To prove this we
symmetrize, apply Levy’s
maximalinequal- ity,
and then use anexponential inequality
to estimate theresulting probability.
For thesymmetrization
we use an idea ofHoffmann-Jorgensen ([13], proof
ofCorollary 3.4).
Let us consider thefollowing
i valuedvectors
Then
Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques
If
(v)r
denotes the r-th coordinate of v E thensince and
mk/k -+
0. Therefore we canapply
toL
Ev;)
e. g. thesymmetrization
lemmas in[10] (Lemma 2.5)
to obtainthat for each E > 0 there is n
(E)
oo such that for n > n(E),
where is an
independent
sequence of i. i. d. random variables withP ~E~ -=1 ~ = P ~Ei = -1 ~ =1 /2, independent
ofBy (3.22)
and(3.23),
thenroof of f 3.21~ reduces to showing
for all E > 0. In order to
apply
P.Levy’s
maximalinequality
we writeand notice that the sets of indices
and
are both
decreasing
as k increases sincebmk
increases with k.Therefore,
for
Xi,
...,X2n fixed,
the above maxima areactually
maxima ofpartial
sums
respectively
ofL Ei suitably
ordered. So we cani e A2n - i iEB2n-1 i
apply Levy’s
maximalinequality conditionally
on theX/s
and then inte-grate
withrespect
to theXI’s. Taking
into account thatB2n-l
c sothat we can
apply Levy’s inequality
twice to the secondprobability,
weobtain
Since ~ 03C003BB and since
Kn log n, Kn
~ oo, Proho-rov’s inequality (stated
in theproof
of Theorem3.3) applied
to the lastprobability
in(3.25)
showsthat,
for nlarge,
where d is a fixed constant.
(3.26)
is thegeneral
term of aconvergent
series and therefore(3.24),
hence(3.21)
holds. This proves(3.18). [ ]
3.5. Remark. - The
centering
in(3.15)
can be taken to be EX forp > 1 and zero for p 1. We may ask if the
centering
in(3.17)
cananalogously
be taken to beXn (o)
for p > 1 and zero for p 1. In then
case p > 1 the answer is affirmative if and
only
ifmn I (Xi - EX) -
0nbmn
i - ~a. s. Theorem 3.2.5
[15] (Feller’s theorem)
showsthat,
underregularity conditions,
this isequivalent ( X > mn
oo , but(3.16)
doesnot
imply
that this series converges. This holds if e. g.constant for some a > 1. A similar observation can be made
for p
1.ACKNOWLEDGEMENTS
We thank the referee for a careful
reading
of this paper. His comments led us tochange
theregularity
condition in Theorem 3.4 and to correct severalmisprints.
Annales de l’Institut Henri Poincaré - Probabilités et Statistiques
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(Manuscript received December 22nd, 1988) (accepted May 26th, 1989.)