BIOMATHEMATICS TRAINIHG'PROGRAM
A tJEAK HIV,I\RIANCE PRINCIPLE HITH APPLIClHIONS TO DQr.1AINS OF ATTRACTION
by
r:Qrdon Sirr.ons1
Department of Statistics
university of North CaroZina at Chapel Hill
Chapel HilZ~ North Carolina 27514
and
William Stout 2
Department of Mathematics
univel'sity of IUinois
Champaign-Urbana~ Illinois 61801
Institute of Statistics
~tUneo
Series #1044
November, 1975
1 TIle work of this author was supported b)' the Air FOTCe Office of SCientific
Research under Grant No.' AFOSR-75-27~;6,
?
~
The work of this author was supported by the National Science FOlmdation.
A ~~EAK INVARIAIlCE PRWCIPLE tHTH APPLICATIONS TO DOMAINS OF ATTRACTION
by
Gordon Simons l
Dep~tment
University of North
C1~peZ HiZZ~
of Statistios
C~oZina at Chapel HiZl
North C~olina 27514
and
William Stout 2
Department of Mathematios
University of Illinois
~1amraign-Urbana~ Illinois 61801
ABSTRACT
An
elementary probabilistic argument is given which establishes a I'weak
invariance principle tl
~vhich
in turn implies the sufficiency of the classical
assumptions associated with the weak convergence of normed sums to stable laws.
The argtnnent) 'lThich uses quantile functions (the inverses of distribution
tions)texploits the fact that tl1Q rro1dom variables X = F-lCU)
flIDC-
and Y = G-lCU)
are t in a useful sense ~ close together \vhen F and G are t in a certain
sense, close together.
Here U denotes a uniform variable on
CO~l).
By-
products of the research are two alternative characterizations for a random
variable being in the domain of partial attraction to a nonnal law and some
results concerning the study of 00':":;.115 of partial attraction.
1 The work of this author was supported by the Air Force Office of Scientific
Research under Grant No. AFOSR-75-2796.
2 The work of this author was supported by the National Science FOlmdation.
ii
M:5 Subject Classification SchenID: 60POS, 60GSO.
Key Words & Phrases: invariance principle 1 domin of attraction 1 dov.ain of
partial attraction 1 stable distribution 1 central limit
problem~ domairL of norrnal attraction, slowly varying
function, quarttile fWlction.
1.
Introduction.
111is paper presents a new method for establishing convergence in law of
nom.ed suns of independent identically distributed (iid) random variables.
A
distinctive feature of this method is that no use of transforms is made (except
the distributional fonn of stable laws is taken for granted).
probabilistic.
The method is
It depends completely on the establishment of an appropriate
invariance principle
0
The invariance principle introduced here is analogous to
the almost sure Llvariance principles appearing in the literature (cf., Strassen
(1964, 1965), Csorgo and
R~vesz
l~jor
(1975), Kom16s,
and
Tusn~dy (1975)~
Philipp and Stout (1975)) except almst sure convergence is replaced by convergence in probability or s ':That is equivalents convergence in law.
reason~
the tenllino10f,IY '1veak invariance
principle~:
For this
will be used.
The weak
invariance principle encapsulated in Theorem 2 belm-; is based upon very e1ementary notions of probability.
TItis contrasts sha.rply vrith the typical almost
sure invariance principle, which depends on the existence and properties of
Brmmian Elotion aT'ld frequently on sone fonn of
invariance principle is
desi~led
to
Sl~orol<hod
den~nstrate
erbedding.
Our weak
the sufficiency of the classi-
cal assumptions associated with the Hea.k convergence of nomed SUIns to stable
laws.
Perhaps more sophisticated weak invariaTlce principles can be found which
would apply more \vide1y vdthin the scope of ...he central limit prob1er:l.
Our main result, Them.'en 2, is concerned lid tIl ra'1dorl variables
within the domain of partial attraction of
srn~e
whiw~
are
infinitely divisible law
(ir1f>lied by assumption A2 below) but outside the donain of partial attraction
of a nomal law (Le., satisfyi"'lg assumption AI) .
essential to our approach.
The latter restriction is
It irrrplies that our theory does not concern itself
2
with the convergence of nomed
all other stable laws.
SlL'TIS
to the nonnal la'!.v.
In this regard, it should be pointed out that Root and
Rubin (1973) have already established the
probabilistic methods.
But it does apply to
nOITellil
convergence criterion by
The effect of our \'lork is to bring probabilistic methods
to another area of the general central limit problem.
By-products of our research are two alternative characterizations (see All:
a'1dAl I") of the class of distributions which are within the domain of partial
attraction of a nonnal 1mv.
pendent interest.
We believe these are nevJ.
Tney nay be of inde-
He also obtain some results about domains, (;)f partia.l
attraction.
In order to introduce our method ~ ul1.encumbered by tech.71ical details ~ we
present a special case
fuller generality.
ill
Section 2.
Sections 3 and 4 present the method in
Section 5 discusses the assumptions being
rr~de
and their
significa.nce in more detail them 'what seeDS to be appropriate for Section 3.
2.
Domain of Normal Attraction of the_Cauchy law.
The follov!ing theoren describes a very specialized weak invariance princi-
pIe buts at the saIne time" with a minimum of computations; yields a'1 interesting convergence in 1mV' result.
Let
X be a Cauchy random variable (centered at zero and) scaled so that
Pre Ixl >x)
,.... -1
x
as x-+oo,
and Y be a syr.-rnetric r2J1tlOm variable satisfying
1
Pr( IYI >x) '" -x as
x-+oo.
3
Theorem 1. On some probabiZity
variables
X1,XZ""
space~
Yl~Y2~'"
and
there exist two sequences of iid random
whose
~embers
are distributed as
X and
Y, respec#veZY,and which satisfy
~1: 1 Y.
L1 = 1
(1)
Since n -12n1=
. 1 ..,
A·
1
= ~!!
1 x.
Ll=
1
+ 0 (n)
n-+oo. 1
a:3
P
is distributed as X, the fo11ol.'ling corollary is
immediate.
Corollary 1. If YpY Z""
are iid and distributed as Y" then
L X.
n -lI11
. 1 Y1. ~
1=
Proof of Theorem 1.
Let U1 ,UZ""
be iid wlifonn variables on
(0,1), and
let F and G denote the (right continuous) distribution ftmctions of X
and Y respectively.
-
~lere
F
-1
is defined
Set
fon!~lly
by
(2)
SiIlce
(3)
the distribution function of Xl
z.1=1
y.-x.
1
1 TIle notation Zn
= 0 D (en )
L
F.
is
(i~l)
meai'1S
Let
arK1 Z=Zl'
ZnIen
-+
0
in probability.
4
EC;uation (1) is identical to n-lL~=l Zi
degenerate conver£ence
theorem~
(4)
n Pr(IZI>n)
n
(5)
Since Var
-1
ZI(IZI~n) ~
Var
2
f:
4
0 as n-+<>:>; which, by the classical
is equivalent ('sL"1ce Z is JYIlIUetric) to
+
0 as n-+<>:>,
ZI(IZI~n)
0
+
vPr(lzl>v)dv
as n-+<>:>.
(see (10) belrnq), condition (4)
implies condition (5).
It is almost irrnediate that
-1,
F
_ 1+0(1)
G- 1 (u) :: 1(0(1~
2 l-u
(u) - 2(1-u) ,
as
u+l
'
and
= - l+O(!l
2u
-1
G (1.1)
=-
1+0 (1)
2u
as u+O.
Hence,
G-1 (u)-F -1 eu)
== 0
( (1-1.1) -1)
Z = G- 1 (U1)_p-1(U1), condition (4) follows bi111ediately: For any
fixed e:>0 and sufficiently large n,
In as much as
Pr(lzl>n) ~ P:r(l_-U~ >n)
...
1
It is a siIrrt')le [latter to replace
variable in Theorem 1.
+
Pr(LJ~ >11)
1
= ZEin.
o
X by any non-noITtCll stable (random)
That is, the classical concHtions for a random variable
Y to be in the dom.ai.."'1 of normal attraction of a (non-nonnal) stable variable
ca"1 easily be shohn to be sufficient using a theorem siBilar to Theorem 1.
Rather than do this; we establish an
in Section 3.
evc~
nore general weak invarhmce pril1.cipl(.
It is applicable to the study of domains of (not necessarily
5
nonnal) attraction of stable lm'is ~
If one were to try to prove the classical central limit theoren using the
above approach, one would have to establish
But this would require Val' Z to be zero, which is of course too strong.
It
is fron considerations such as this that we were led to work with random variabIes which are outside the domain of partial attraction of a nonna1 random
variable>
This is
OUT
starting point in Section .3.
3. The Weak Invariance Principle.
The weak invariance principle described in Theorem 2 below basically says
that, under appro2riate assumptions} it is possible on some probability space
to define two nOTI:t8.1ized
SUJ!l.5
of iid random variables
with specified listributions for the X's and Y's, so that the difference
(6)
C'
0
n =
b( )-1,1l
Li=l
n
is virtually non-stochastic.
y
()-l\,n ~T
i-a n Li=l Ai
Specifically, it is
on = hen) + 0 p (1)
for appropriate constan.ts
tions.
h(n).
as
sho\~
tl1at
n-~
He shall begin with a discussion of assurnp-
6
Let
R+
X be an unbounded raIldon variable.
A positive function
(the positive reals) is said to be a norml:ng function for
(7)
lir:1sup xPr(IXI>a(x)) <
a ( 0)
on
X if
00,
X~OO
lirdnf xP"::' (I Zl >a (x)) > 0 ~
(8)
~
(9)
a(x+t)/a(x)
1 as
+
x+oo
Quite clearly, (7) implies that
a(x)+oo
a ( 0)
X> that
is a noming function for
and that
at')
Cl.
a ( 0) Ib ( 0)
C2.
Por each
arid
be")
for each
t>O.
as
In Theorem
x+oo.
b ( 0)
2~
we assume that
is a nonning function for
Y,
are related by the folloWlllg two conditions:
is a s Zow Zy 1)arying function.
BE(O,l)~
as
X+OO
j
xPr(±Y>B-1b(x)) -0(1) :s xPr(±X>a(x))
:S
xPr(±Y>Bb(X))+O(l)
:S
xPr(±}C>Ea(x))+o(l) .
and
xPr(±X>B-la(x))-O(l)
:S
xr~'(±Y>b(x))
Here, in C2 (aiid belm'! in A3 arld J\3 i )
~
the :interpretation is that the statements
in C2 (A3 and A3 i) are true \-Ii. th allpllL';es preceding tile ranc1.on variables and
with all minuses precedi.'1g the rand-on variables.
It is further assur1ed that
X and Y satisfy the follm-ling three assl2nptions which are described for
AI.
l
",iS
lTIUp
x+oo
1:;y2 T
I yXI <-v'\
r
~
2 < o o•
(
x PI' ( IXI>x)
X:
7
A2.
sup
x>l
A3.
Given any
Pr
~Pr~~";:";;:"+- -+-
•
as
0
A~.
B and any subsequence of positive integers
n'Pr(±X>Ba01'))
ni~
for which
i8 bounded away from zero~ there exists for eaoh
a further subsequenoe n"
such that
B'E(O~B)
Pr (±X>.8 1 a (n" ') ) /Pr (±X>Ba (nil))
is
bounded away from one.
The statement of A3 is made sonewhat cumbersone in order to make it applicable in a Hide vario:lty of situations involring unbalanced tails.
when nJ?r (X>a(n))
-+-
For insta."1CC,
0" no conclusion is required, and it ",,'ould be seriously
restrictive to require Pr(~>Bi.a(n))/Pr(X>a(n)) to be bounded allaY frOB one.
(TIle ratio Dieht even aSSlEle the form
iiO/CY
for large n.)
Although somewhat
stronger than A3" assumption A3' below is often true in applications mld
captures much of the spirit of A3:
A3 1 •
Given any
BE(O,l)~
the ratio
PT(±X>Bx)/Pr(±X>x)
bounded away j'rQm one for Zarge
is (defined and)
x.
Hote that both A2 and /\3 imply that, in certain senses,
Pr(lxl>y)/Pr(IXI>x)
is small for
y>x.
AsSt1!'lPtion Al simply means that the random variable
domain of partial attractioD. of a
page 113 1 for a proof.)
(10)
Ex2ICIXI~x)
r("nT~al
variable.
(See Paul Levy (1954),
!',ltegration by parts yields
x
:: 2
fa
~
v Pr(IXI>v)dv - x'"'PrClxl>x.)·
Consequently? Al is equivalent to
A1' •
X is not in the
f~ vPrC Ixl >v)dv
limsup - - - ; : - - - - - - <
2
x~
x Pr( IXl >x)
co.
8
In Section S, it is shown that Al is also equivalent to each of:
Al'; .
Por some
e:>0 )
x-
J:
C
is an increasing function of x
v Pr(IXI>v)dv
on R+ •
Alii'.
For some
e:>0 and some
;ii>1
x 2-e: Pr(lxl>x) :::; f!yZ-e: Pr(IX!>y)
It is useful to
(11)
F01'
that assuf:1ption
kn01';
A~
~
y>x>O.
implies
inf Pr (I XI >Ax) /Pr (I xl >x) >0
each A>1 9
2
x>O
This is immediate froTi1 lU n ~
Theorem 2.
Suppose
X
0
and Y possess norming functions
a(o)
respectiveZy:; and that they each satisfy assumptions Al-A3.
conditions Cl and C2 are satisfied.
(12)
X and
I~=lYi
..,.~ ::
Dln)
y~
Then on some probability space there exist
for some sequence of constants
whose members
respectively, and which satisfY
I~=lXi
()
a n
0
Further$ suppose
two sequences of iid random var'iables
aPe distributed as
and b( L
+ h (n) + 0
p
(1)
,
as
n~
h(n).
If X is a stable variable of index aE:(O,2)
iDmediately follmvs from (12) that
Y is
1.<'1
aDd a(n)
= nl / a
, it
the domain of attraction of X.
2 The fact that Al implies (11) is implicit
in the work of C. Heyde (1969).
He derives from Al ar. inequality like that of Al" 1 but with e:=0. It is
weaker than Al'; I ~ but:: it immediately implies (11). In all earlier paper,
Heyde (1967) assumes both lU and a weakened version of (11). It is now Imo"m
that the latter is red.undant.
9
TIle proof of Theorem 2 depends on Lemma 3? which, in turn ~ depends on
Lemmas 1 a.Tld 2.
We now state these lerlIDlas, for easy reference, delaying their
proofs lIDtil after the proof of Theorem. 2.
Lemma 1
0
Let
')
. lO
B
positive sequence
R+
b e a St.-oWt.-Y
'1"
•
f'.'
varY1.-ng
Junct1.-on
on
xn
t oo~
E
there exists a sequence
n
+
Then", for each
a
sufficiently slowly
that
(13)
for every sequence
Y11 which satis.fies
(14)
Lemma 2.
Let
X and Y be unbounded random variables with norming functions
a(Q) and b(o)~ respectively.
(11).
If a(o)/b(o)
Further~ suppose
X and Y satisfY assumption
is slowly varying~ then
n~
(15)
for every pair of sequences
xntco and yntco for which a(Yn)/a(~) is boundec
away from zero and infinity.
Lerma 3. Assume the hypotheses of Theorem 2.
where
Fix
F,;,n>O and Let
F and G are the distribution functions of X and Y (appearing in
the statement of Theorem 2) respectively.
large integer N such t7w.t for
n~H and
Then for each
UE
(c3~c4)
6>0, there exists a
(possibly empty),
10
Proof of Theorem 2.
Xl y2Cz ~ . , .
Define the sec:uences
of iid lmifom variables
Ul'U2~'"
in tenns
(0,1)) by letting
(on
(i~l)
where
Yl' y 2} . . •
and
,
F and G are the distribution functions of X and
Y~
respectively.
Set
Y.
Zni
=
X.
odb- - a(~r
and
Sn
= L~=l
It rust be shown that for a sequence of constants
Sn - hen)
(16)
4
0 as
hen)
H-tGO.
An intuitive argument suggesting "lhy each
Zni
and hence
Sn
(except
for centering) is small with probability close to one is perhaps helpful before
we plunge into technical details.
p0ssible ranges for
close to
U.
Either
l (u)
(y) ,. . G-ben)
an""
Y/b(n)
ancl$ hence,
hence~
Zn
i.
U is close to 1 or 0,
1 or 0 or U is not close to 1 or
with srnall probability and,
F-~
Sl.rppress the index
Z
must be small and, hence,
O.
causes no problem..
O.
In
til;;
Zn :::: 0"
There are three
U is moderately
1ne first case l1appens
In the second case,
third case, both
X/a(n)
and
Thus, as desired}
The actual argument proceeds by tnmcatioTl and centering.
Although somewhat
easier; our arl;,1.ll'1ent, in essence;; amounts to verifying the conditions of the
classical d.egenerate convergence theorem.
(See Loeve (1963), page 317.)
For each n, partition the unit interval into five subintervals
11
nln,
z = maX(F(-r,a(n)) ,G(-~b(n))),
cl
=
c3
= min(F(t,a(n)),G(t,b(n)))
I
II
C
L,...
I
cl
0
13
tC
1
14
-j
c
z
c4 = I-nih.
c
3
IS
I
4
which will be chosen later, are intended to be
The constants
n and
and positive.
Some of the intervals may be ern:pty.
14
~~
I
1
For instance, if
sn~11
c4~c3;
is empty.
inc first step involves truncation.
5' =
n
Let
Z'.
l.i=l nl and
\,n
where
= Znl.
= 0 othenlise.
The
Zni
I'
s
d 1nence,
a.Tl,
5ni
are bounded random variables.
Fix
::; Pr ( I51 - ES I I>e:) + PI' ( IS1l I>£)
n
n
n
+ Pr(LJ~=l [UiE(Il+I S)])
::; £-2 n Val' Z~l + n Pr(U1E(I1+I S))
+ 2n.
= £ -2 n VaT
::; £-2 Var
S;1
It will be argued tllat
(17)
n E(Z'l)
n
2
+
0 as
n+oo,
e:>0.
Then
12
so that
1imsup PrCIS -ES'I>2£)
n n
n~
(13)
Even though
value of
Sn
~
n>O and £>0 are
ES1~
arbitrary~
$
2n.
this does not prove (16) since the
HOi'lever ~ if hen)
depends on 11.
then (16) follows fron (18) and what Loeve
trization inequality.
is defLled as a median of
(1963~
page 245) calls a symme-
Thus, tnll1cation has reduced the task to showing (17).
-1 (u) _ G-1)2
()) du
(! a(nT
bCn
The integrals over
12~
13 and
1 must be treated separately. The integral
4
over 12 is treated like the integral over 14 and,hence,will not be discussc' .
lVe shall discuss the integral over 1
first.
3
2
F-~()) - G- l cu ))2du $ 2 {(F-~()))2 + (G,-l
t,
a
n
D'fri)
J
1
a
n
J.
l
D
n
JU
1
3
3
J
I (
~
2
U)J
JF(~a(n)) (F-~())]2J1J
F(-t:a(n))
+ 2
a n
IG(~b(n)) (G~l()))~dU.
G(-~b(n)) (n J
He shall only discuss the L'1tegral
I (n) =
the integral involving
ShO\'I
(19)
G
FC:aCn)) r·r.aC£¥l-l)2
-.r'aU ndu"
'
l
F( -~a(n))
f
C2.1'1
be handled the
S8I'.e
that
liTI~up
n~
n1(n)
+
0 as
~fO.
'\A,'UY.
Specifically ~ we shall
13
This will be sufficiently strong (to prove (17)) providing it can be shown that
for each small but fixed
(20)
I;>O?
p-l(U)
(
f 4 aCn)' -
n
G-l(U))2 ".,
I
Een)
cit..
-+
0
as
n~.
recalling (3) and using (IO)?
r (-t;a(n)<~t;a(n))
2
nl (n) = - { : - EX
a (n)
:<.;
~n
a (n)
Since
sa (ll)
f0
v Pre Ixl >v)dv.
X sa.tisfies assumption JU it satisfies
Al'? it follows for sone small
n len)
:<.;
2n
-r:-:a (n)
Since a(n)
y>O
t,:
and lUll.
y fa(n)
.
v Pr(IXI>v)dv
0
X? (19) follows from (7).
Using LeJ7l111a 3 s we have? for each sTII.all 6>0
and all sufficiently large TI,
By (a) and A2,
sup n Pr (I xl >Aa(n))
n
Using Al Ol ? then
and some large II that
is a noming function for
It remains to prove (20).
j\~ i
-+
0 as
A-+oo.
14
In turn, this implies
sup n Pr(IXI~(n))
-+
0
as A-+oo.
n
Thus, for some large A?
c1 = n/n ~ Pr(IXI~(n))
?:
F(-Aa(n))?
and
c4
(22)
=1
- n/n::; I - Pr(IXI~Aa(Il)) :;;; F(Aa(n)).
Consequently, recalling (3),
(23)
:;;; _.~- EX21 (I xl :SAa(n)) ,
a"'(n)
But? by assumption AI and (22), there is a large L such that
(24)
_~n
2
2
2
Ex r(lxl:SAa(n)):;;; oA LnP(l x l>i\a.(n)) s 6A Ln.
a (n)
Since 0>0
c~,
be made arbitrarily snaIl, (20) follows from (21), (23) and
(24).
Proof of Lemma 1.
0
Accord~.nL
to Karamata is representation for slowly varying
functions (see Feller (1966), page 274),
sex)
= c(x)
Sex)
can be expressed as
exp J(X n(vl dv,
o v
15
where c(x)
c
+
0
~
n(x)
~id
+
0 as x+oo.
Thus
Yn
exp
(1+0(1))
Ix n~v)
dv as n+oo.
n
Hence~
it suffices to show
Yn
'n (v) dv -.. 0
fx _._v
as
H-l<lO.
n
Let n*(x)
= sup
In(v) I. Observe that n*(x)+O as x+oo.
~x
C' n~'d.
dv
By (14),
I < n* (x,.) log (Ynhy,)
n
The latter goes to zero as
11+00
if
En
decreases sufficiently slowly to
zero.
0
Proof of Lemma 2.
III (15)) that
Yn ~xn'
It is easy to see (from the symmetry between
Len~a
Set SC')
and Yn
2 is true if it is true with the additional restriction
= a(
0 )
/b (
0 )
and obtain a sequen.ce
En -}O
~
from Lemma 1.
It
suffices to shm'l that (for at least large n)
a(y.)
11
(25)
~
Aa(x
··n) => x·n
~
Yn En •
Set
£ (A)
According to (7)
for each 1\>0.
(8)
l~ow,
a~d
(11),
=C
m'
C ffild, Li
if a (y) :~.Aa (x)
?..iid
f{pr(I~>A(()p~}
- Pr(fX >a x
'
tUIT!;
£
(A)
x >y~l ~ then
A>.
0
are strictly positive
16
E(A)
Thus, for large n,
Proof of Lemma 3.
~
o
En' and hence (25) holds for large n.
Suppose that the le31a is false.
Then there exists a
0>0
such that
p-I(U)
n
(-a-;C.......
n)..-- -
(26)
for each n
1
G- (u
n
ben)
))2 > 0 (F-l(Un ))2
a en)
in some infinite Lfldex set
for infinitely Dany nEH or un>G(t;b(n))
sider only the case where un>F(t;a(n))
LTl the other case being very similar.
necessary~
Since
N.
un>c y either un>F(t,;a(n))
for il1.finitely many ndJ.
lTe con-
for infinitely many nEN~ the argument
By making
I'J
a snaller index set if
we assume
(27)
Un
> F(t,;a(n))
for all nd~.
Thus, recalling (3),
(23)
By deletLlg a finite number of indices froD
F
-1
(un) >a(l)
for all nEH.
~
re~inder
Clearly r n-Ko as
(29)
if necessary, 'We aSSUT11e that
For each ndJ, let
rn = rax{inter-er k:
In the
H
a(k)~P
-1
(un )}.
of the proof all statements concernulg n will suppose l1EN.
n-Ko and
17
TI1US, by (28) and (29),
a(rn·,:;.
*1) > ~a(n).
(30)
Since
a(rn +l)/a(rn )
-+
as
1
a(rn)/a(n)
frko,
is bounded away from zero.
Since un <c 4 it follows for some large A (see
un ~ F(Aa(n)).
Thus, by (29) and (3),
Consequently,
(22))~
that
a(r ) < Aa(n).
n
is bOlmded. aVlaY from infinity as well as from zero.
a (rn)/a(n)
Hence, it follows fron Lel1'JJ1a 2 that
a(rn)b(n)
a(n)b(r )
n
-+
1
as n-+<x.>.
Thus (26) will be contradicted if we can show that
G-1Cu )
n
bern)
(31)
Vie now digress to show that
ing
Thus
the definition of c 4
n (I-Un)
-+
1
r n/n
from infinity.
of p-l
is bounded away from zero.
(see (22)), and that
is bounded away from zero.
By (30)"
a ( c)
is a nonning ftmction.
Tn (I-Un)
Conse-
is bOlUlded away
In view of (11), this follows fror'l the fact that
Thus
Recall··
un <c , we have un < I-n/n.
4
is bounded away from zero if n Pr ( IXI>~a (n) )
nonning function.
Let
n-+<x.>.
r n (l-u)
n
\'Thich is bOlmded mvay from zero since
quently ,
as
a(")
is a
is bounded away from zero, as desired.
U be a uniform variable on
(0,1).
Then, recalling the definitions
and G- l , a'1d using (29) and condition C2, we have, for each
18
BE(O,I),
s r nPr(U>un )
= Tn (l-un )
s rnPr(F-l(U)~F-l(Uu))
= rnPr(X~F-l(un)) s rnPr(x>a(rn))
Since rn(l-un) is bounded away from zero~ (32) implies rnPr(Y>Bb(rn ))
bounded away from zero and that
In turn, this implies that (letting XAy
Pr(Y>G-1(u )ABb(r ))
n
n
Pi (y>Bb(r ))
n
SLTlce rnPr (Y>Bb(Tn ))
1
a
s
)
~
w-.
is bounded mvay from zero and Y satisfies assmnption
A3, we have
That is,
Since BE(O,l)
~
= rxul(x,y)
is
is arbitrary,
19
I t remains to show that
G"·1. (u )
11""lSU""
..
J}
b(r)n <- 1 ,
n-l>CO
' n
and b(rn )jb(rn +1)
Since
+
1, it suffices to prove that
(33)
-1
and some fixed £>0. Let c(n) = G (lln)-£' Then~ recalling
the definitions of p-l and G- l ~ and using (29) and condition C2~ we have~
for each BE(O,l)
for each
BE(O,l)~
~
r Pr(U>u )
n
n
= r n (l-un )
~ rnPr(X>a(rn+l))
r
~ r ~l {(rn +1)Pr(Y>B- 1b(rn +1))-o(1)} as ~.
n
Again, recalling that
that
(35)
If
rnl>r (Y>c (n) )
r (l-un )
n
is bounded away from zero, (34) implies
is bOlIDded away from zero and that
pr(Y>B- 1b(r +1)) ~ Pr(Y>c(n)) (1+~(1))
n
(r +1)Pr (Y>B-1b (r +1))
n
n
as
n-+oo.
is bOl.m.ded away from zero~ then assumption A3 for
Y is applicable and (33) can be proved in the same way we showed
liminf G-1Cu )/b(r ) ~ B.
n
n-l>CO
n
verges to zero~ then (since
follows from (35) that
On the other hand~ if
(rn+l)Pr(Y>c(n))
(rn +1)Pr(Y>B- 1b(r +l))
n
con-
is bounded away from zero) it
20
for all sufficiently large n, which again implies (33).
Finally, we can
assume, without loss of generality, that either (r +1)Pr(Y>B- 1b(r +1)) is
n
n
botmded away from zero or it converges to zero. For if this is not the case,
o
it can be made the case by deleting indices from the index set N.
In some applications, it is helpful to be able to replace the condition
H
a ( )/b( 0)
0
is slowly varyingn (Le., condition Cl), which appears in Theorem
2, by a condition r.J.ore directly verifiable from F and G.
According to the
following corollary, this can be done if we assume a little more about the
norming function of Y, namely
xPr(IYI>b(x))
(36)
is slowly varying in x,
and if we irLIpose the following additional regularity assUliIption for X:
(37)
A positive funotion
xPr(IXI>y(x)a(x))
YCo)
on R+ is slowly varying whenever
is slowly varying in x.
Corollary 2. Theorem 2 remains valid if oondition Cl is repZaoed by
(38)
Sex)
= Pr(IY\>x)/Pr(lxl>x)
is slowZy varying in x,
and the additional assumptions (36) and (37) are valid.
Proof.
It must be shown that (36), (37) and (33) imply condition Cl.
it will be shown that S(b(o))
(39)
First,
is slowly varying, Le., tP.at
B(~(X'1)1 -+ 1 as x~ for each y>O.
f3(DTx)T
From Karamata' s representation of slowly varying ftmctions, appearing in the
21
proof of Lemma 1 1 it is clear that (39) is equivalent to
b(XY) .., (v)
~ dv + 0 as
b(x)
v
f
(40)
where
n(x)+O as
x+oo.
But b(x)
x+oo
for each y>O,
and b(xy)+oo as
X+oo, at"'1d, hence, it
suffices to show
is bounded as
(41)
X+oo, i.e., to show
limsup
x-+oo
for each y>O.
Since
~
b (x)
b(o)
<
(Xl
and
liminf
X-+OO
~((XY) > 0
A
is a norming function for Y,
is bounded away from zero as
x-+ro.
Since Y satisfies assurnption A2, the
first inequality in (41) must hold.
The second inequality in (41) follows from
A2 and the fact that
Pr (IYI > (~f~))b(xy)J /Pr OYI >b(xy))
is bounded away from
0 as
x+oo.
Thus
B(b(o))
is slowly varying.
Since
x pr(lxl>(~t~~)a(x)) = x Pr(IYI>b(x))/S(b(x)),
it follows from (36), (37) and
condition C1 holds.
(39)~
that
a(o)/b(o)
is slowly varying.
I.e.,
0
22
Observe that the conditions (36)
asymmetric in X and Y.
a~d (37)~
entering into Corollary
2~
are
Quite clearly Corollary 2 would remain valid if
(36) and (37) were replaced by
x Pr(IXI>a(x)) is sZowZy varying in
(36')
X1
and
(37')
A
positive funotion
yeo)
on R+ is slmJly varying whenever
x Pr(IYI>y(x)b(x)) is sZowZy varying in x.
4.
Domains of Attraction.
TIleorem 2 is specialized in this section to the stlKly of stable (random)
variables.
Recall that a random variable X is said to be stable if for each n
there are independent randoll1 variables Xl,X2' ... '~ with common distribution
that of X, centering: constants e(n)
\~
IX.
1.. = 1
1
(42)
a(n)
has the distribution of X.
(43)
O<a<2,
O~p~l,
- e(n)
It is well known that, corresponding to each
properly scaled non-normal stable
(a,p) ,
and scaling constants a(n»O such that
variable X, there exists a pair
such that
Pr(lxl>x) -
1/i\ Pr(X>x) - p/xa as
x~.
l"breover, for such a random variable, a(n) in (42) must assume the fom
nl / a . Observe 1 by direct substitution v that a(x) = xl / a is a nonning
function for any random variable X satisfying (43).
Any stable variable X
23
whiCh';:(when fpr.9perly··"scaled} satisfies (43) will be said to be of type
Theorem 3.
Let X be a stable (rondom) 1JaIliable of type
(a.,p)
and
(a,p). 3
Y be an
unbounded random variable with a distribution of the form
(44)
Pr(Y>x)/Pr(IYI>x)
(45)
where
O$p$l, and Sex)
O<a.<2~
P as
+
x~,
is a slowly varying function.
Then, on some
probability space 3 there exist two sequences of iid random variabZes
and Y1 ,Y 2 , •••
whose members are distributed as
X and
Xl ,X2 , ...
Y, respectiveZy,and
which satisfy
~~1
1.
IX.
1. 1 = 1
=lY'1 = ~~
l/a
ben)
(46)
+ hen) +
n
b(x)
for some positive function
hen).
~10reover, (46)
0pCl)
as n~
and some sequence of centering constants
hoZds for eaoh choice of b(x)
satisfying
(47)
Theorem 3 yields an obvious but well
Corollary 3.
kno~Jrl
corollary.
Under the hypothesis of Theorem 3,
(48)
where Yl,YZ""
~~ lY'1
£1=
ben)
L
- g(n)
-+
x,
are iid random variables distributed as Y, for some positive
3 If the word "typel! were to be used as Lo~ve does (1963, page 202), then each
pair (a,p) would correspond to exactly one type of stable law. This fact
explains our tenninology.
24
function
sex)
g(n) .
and some sequence of centering constants
(48) hoZds for each positive function
b(x)
Moreover:J
satisfying (47).
Corollary 3 is the strongest possible result in the direction it is
stated.
That is, a random variable Y is in the domain of attraction of a
stable random variable X of type
(a,p)
(0<a<2,
the assumptions stated in the corollary.
iff Y satisfies
O~p:s;l)
We have tried unsuccessfully to
obtain the converse of Corollary 3 by probabilistic methods.
Of course, a
proof of the converse based on Fourier transfonns is well known.
Proof of Theorem 3. 11e are going to apply Corollary 2.
(37) and (38) hold.
It is immediate that
Now (44) and (47) combine to give
x Pr(IYI>b(x)) ~ 1 as x~.
Thus,
b(x)-+oo as x-+oo.
b(x+t)/b(x)
~
1 as x-+oo,
I'Ioreover, condition (36) holds and, providing
b(o)
is a norming function for Y.
But (47) implies
An easy application of Karamata's representation (given in the proof of
1) for
Sex) now implies b(x+t)/b(x)
Clearly a(x)
~
1
as x-+oo.
= xl/a is a norming function for
X.
Thus it renains to
verify assumptions AI - A3 for X and Y and condition C2.
only non-trivial conditions to check are AI - A3 for Y.
assugption All than i ts equivalent iU.
Le~a
Of these ~ the
It is easier to chec1c
Showing Al' for Y ammm.ts to showing
JX v1-0'e,Cv)dv
limsup , _0.,..__
<
x~
x2-o:S(x)
co.
25
But this is inunediate from a theorem appearing in Feller (1966 ~ page 273).
Concerning assumptions A2 and A3, (44) yields
>') =
Pr ~~
Pr Y >x
From this and the fact that
13 (x)
MAx)
Act [3 (x) •
is slowly varying. A2 is iImnediate for Y.
Using (45)) the proof of A3 I (and hence A3) for
5.
o
Y is similar.
Discussion of the As.;;umptions.
A major objective of this section is to e:x-plain the significance and to
the extent possible the raison dFetre of the assumptions made in Theorem 2
(our main theorem).
Recall that these assumptions restrict attention to random
variables which have noming functions and satisfy l1 assumptions'; AI - A3 and
n
conditions l1 Cl and C2.
We will try to make a strong case for all of these
requirements with the exception of assumption A3.
Roughly speaking, assumption
A3 requires a distribution function not to have locally nearly flat spots arbitrarily far out in its tails.
Feller
(1968)~
in a
somel~lat
has fOlmd it necessary to make a sirailar restriction.
related context,
We have ambivalent
feelings as to the necessity of A3.
The theory developed in Section 3 is predicated on the use of nonning
fuYlctions to nomalize the sums appearing in (12).
be
considered~
To begin
with~
lVhile other scalings might
our decision to work with nonning functions has two good reasons:
our methods are not suited for studying the classical central
limit theorem (see the discussion at the end of Section 2), and consequently
the usual normalization by
l/In
is inappropriate,
Secondly~
nonning functions
provide exactly the right amount of scaling when one is working with random
26
variables which are in the dornain of attraction of anon-nomal stable variable.
I.e. ~ they are the right things to consider if one wants to study domains
of attraction.
The following theorems explain llhy assumptions AI and A2 are quite natural
when norming functions are used to
no~alize
sums.
In both theorems it is
are iid.. that Tn = Xl+XZ+."+Xn and that X has
assumed that X,XI,XZ""
a n0lT11ing function a(x).
Theorem 4. There exists a sequence of numbers cn such that
{(Tn-Cn)/a(n)}~=l is tight iff X sat~sfies assumptions Al and A2.
Theorem 5.
If X satisfies (11) (which is impZied by
sequence of numbers
Cn.ll the sequence of random variabZes
has no degenerate weak limit point.
SpecificaZZy.ll for eaoh
lirunf Pr(ITn-c I>Aa(n)) >
(49)
A1)~
n-+oo
n
Tightness for the sequence {(T~-Cn)/a(n)}~=l
has a weak limit point.
then for every
(Tn-cn)/a(n) (n::::l)
A>O~
o.
guarantees that every subsequenc8
Thus, by Theorenls 4 and 5 ~ AI and A2 guarantee that
such limit points must exist and be non-degenerate (for properly chosen cn's).
Further, T11eorern 4 implies that Al and A2 are necessary for tightness provided
one normalizes by noming functions.
Thus, Theorems 4 aTld 5 show that assump-
tions AI and A2 are well suited to the study of domins of attraction.
There are interesting connections between Theorems 4 and 5 8J."1.d the subject
of dornains of partial attraction.
Doeblin (1940) shows that an unbounded ran-
dam variable X belongs to no domain of partial attraction if
lim liminf Pr(lxl>A.x)/PrCIXI>x) >
A-+oo
x-+oo
o.
27
In contrast, it can be shown that assumption A2, which can be expressed as
lim limsup Pr(IXI>Ax)/Pr(lxl>x) = 0,
A-'t<X)
implies that
X-'t<X)
X is in some dorr.ain of partial attraction.
Theorems 4 and 5
make this clear \vhen, in addition, assumption AI holds and a noming function
exists for
X.
r"breover, they tell how to properly scale Tn under the addi-
tional assurrptions.
Theorem 2 says something about domains of partial attraction.
denote the collection of all (infinitely divisible) randon variables
that
V
x
X is in the domain of partial attraction of Z.
is eITipty,
Let Ox
Z SUC1l
According to the theory,
Ox consists of one type of random variable ~ or
anon-demnnerable §et of types (Doeblin (1940), Gnedenko (1940)).
Ox consists of
The fol1m'ling
result is an immediate corollary of Theorems 2, 4 and 5.
Theorem 6.
If X and Y satisfy the hypotheses of Theorem 2 for some choice
of norming functions
aCe) and
b(o)~
then
VX=Vy •
'DIe proof of Theorem 6 (which will be deferred until later) shows that,
for random variables which satisfy assurnptions AI - A3, nonning ftmctions
provide the normalizations appropriate for the study of domairls of partial
attraction.
Condition Cl, which says that
necessary in Theorem 2.
positive integer.
a ( 0) /b ( 0)
is slowly varying, is all but
To see this, suppose that (12) holds.
Then, by (12),
Let m be any
28
\'~lY'
L 1= 1
= -'b""7(mn--':-)
1Y.+"'+\'~(
L 1 =.1.
1
Ll = m- 1) n+ 1Y1.
b f\.Tl ) \~
- blma)
ben)
I~ x.
:: ~(n)
1 + mh(n) +
o(mn)
arnJ
_
{1-1
Opel)}.
Thus
Ynis will contradict the conclusion of Theorem 5 unless
bfn~b~mn~
an
mn
(50)
This says that a(o)/b(o)
1
+
as n~.
is slowly varying on the positive integers.
the conclusion of Theorem 2 is concerned with a(x)
values of x
on1y~
and b(x)
Since
for integer
this is all the justification for C1 that we could reason-
ably hope for from it.
However? from (9) and (50), we have
(51)
+
1 as
x~
(tlrrough the rea1s).
If m were not restricted to the positive integers, then (51) would say, of
course, tllat a(o)/b(o)
is slowly varying.
Clearly, if a(o)
sufficiently smooth, it Sllould follow from (51) that
varying.
Indeed this is the case.
a(o)/b(~)
For exmm?le, if a(o)/b(o)
is also slowly varying (cf., L. de Haan (1970), page 6).
a(o)
A3
and b(o)
and b(o)
are
is slowly
is monotone it
fibre reasonably, if
are monotone (this can be relaxed somewhat), and if (as (36)
(for Y) imply)
· c
beAt)
11IDSllp
bet)
t~
+
1 as A+1,
alL'
29
then a modification of de Haan IS argtnnent verifies that
varying.
a(
0 )
/b (
0 )
is slowly
Thus, condition Cl is an extremely reasonable assumption, and it is
frequently necessary.
Finally, we claim that C2 is essentially a necessary condition.
In order
that (12) hold, it is necessary (cf., Gnedenko and Kolmogorov (1954), page 128)
that
n pr(l aCn ) for each
£>0.
br.l~)
1>£) +0
as
n~
Thus, for each BE(O,l),
n Pr(X>a(n))
~
n Pr(Y>Bb(n)) + n Pr(laCn) - btn) I>(l-B)),
which implies
n Pr(X>a(n)) ~ n Pr(Y>Bb(l1)) + 0(1)
(52)
as JrKO,
a discrete analog of
x Pr(X>a(x)) ~ x Pr(Y>Bb(x)) + 0(1)
(53)
as
x~.
Discrete analogs for the other inequalities in C2 are derived similarly.
From
the viewpoint of Theorem 2, there would be no loss in generality in asstmrl.ng
that
a(x)
= a([x])
and b(x)
= b((x]),
for
x>l, in which case, (52) implies
(53) and, more generality, the validity of (12) requires C2.
Thus condition
C2 is essentially necessary III Theorem 2.
In spite of the fact that t..he assumptions of Theorem 2 are chosen with an
eye toward the study of domains of attraction, they do not imply Pr(lxl>x)
and Pr(IYI>x)
are regUlarly varying fimctions.
(If X is in the domain of
30
Pr( Ixl >x)
attraction of a non-normal stable vd.riab1e 1
function.)
Examp1e.
This is demonstrated by the following example.
Let
X be symmetric with
= 2+sin(log
PClxl>x)
x
Clearly X satisfies Al i l l with
A3'.
is a regularly varying
o~l
x)
x>3.
1
and M=3.
A2 is immediate.
Consider
We have
p
p
Elementary calculations show that the above ratio is bounded away from one if
BE (0,1) 1 establishing A3'.
Also ~
a(x)=x
is a norming function.
x~,
But as
~+sin(log
xt)
2+sin(log x)
clearly does not approach one for each fixed
regularly varying.
Clearly a
Thus,
P[lxl>x]
Cl and C2 hold, as well as Al - A3, but with
o
not regularly varying.
The following lenn'las are useful for proving Theorems 4 and 5.
is assumed that the sequence
that
a(x)
is not
Y can be constructed with distribution close
enough to that of X such that
P[/Y/>x]
t>O.
X"Xl'X2~'"
is iid, that
In both, it
Tn = Xl+",+Xn ' and
is a norming function of X.
Lemma 4. There exist a sequence E:n+O and constant C<oo such that for each
A~l
and each sequence of numbers
n ,
.c
31
Lemma 5.
Proof of
If Al
Le~na
hoZds~
4.
then faT' some sequence of numbeT's
E.1 = [ITn -X.I::;3Aa(n),
IX·I>6Aa(n)L
1
1
lTn '
> 3Aa(n)
each intersection Ei nE j
that the
Xi~s
are
and some
~t>O.il
For simplicity, the proof will be given for the special case
cn :::0; the general case offers no additional difficulty.
Observe that
.c
n
J
on each E.1
i~j
.
Let
i=l, ... ~n.
and that
ITn -X.1 -X.J I
> 3Aa(n)
on
Thus ~ by a Bonferroni inequality and the fact
iid~
Pr(I Tn l>3Aa(n)) ~ Pr( uni __' lE1·) ~ L~1=1 Pr(E.)
1
II ::;l<JQ}
.. Pr(E.nE.)
1
J
~ n Pr(IXI>6Aa(n))Pr(ITn_ll::;3Aa(n))
- G) Pr2 (I xl >6Aa(n)) Pr (I Tn-ZI >3Aa(n)) .
Let
Y = n Pr(IXI>a(n)).
n
111en~ for
A~1/6~
..,
+ Y~
Now~
(57)
for
Pr(ITn _Z!>3Aa(n)).
A~l,
Pr (ITn - l !>3Aa(n)) ::; Pr(lTn !>2Aa(n)) + Pr (I XI >Aa(n))
and similarly,
32
(53)
Since Yn
is bounded away froIi1 infinity
follows from (56), (57) and (53).
(cf.~ (7))~
the desired conclusion
Observe that the choice of
En and C does
o
not nce4,"to depend on the value of A.
Proof of lemma 5.
x .
-TIl
= x.1
Fix A>O and let
I(IX.I~(n))
1
and -TIl
X'. = X.-X
1 nl. ,
i=l~ ...
,n;
n~l.
Then
Assumption AI says, for some large
C, that
Consequently, by Chebychev I s inequality,
Pr(lr:-1
(60)
1-
From
(X .-EX .) I>Aa(n))
-TIl -TIl
~
2
nEX I(PJ:ffia(n))
A a (n)
~
en
Pr(IXI>Aa(n)).
(S9) and (60), we obtain
This would prove the ler.una were it not for the fact that the centering
I~=l E7'hi depends on the value of A.
Tn
If cn
is chosen to be a median of
then (55) fo11O\'/s from (61) ani a sylnmetrization i.'1.equa1ity (see Loeve
(1963), page 245).
0
33
n Pr (IXI >Aa(n)) ~ D Pr( Ixl >Aa(n)) /FT (I xl >a(n))
where the constant
4
i~diately
D does not depend on A.
follows from Lemr1a
Conversely~ suppose
show A2 (cf.
~
Thus~
the nif" part of Theorem
s.
{(Tn-cn)/a(n)};'=l
is tight.
It is not difficult to
(8)) if
linsup n Pr(IXI>Aa(n))
-+
A~.
0 as
n~
But then A2 follows from Lemma 4.
AI, i. e., that
But when
1/ill in order to obtain tightness.
X has a fii1ite second moment,
(62)
n Pr(IXI>£yfi)
liui:'lf a Pr(IXI>a(n)) > J,
-+
0 as
n~~
a(n)/vn
-+
0
£>0.
E5
..
I M<O
n~
ness for
that tightness implies
X is outside the dona-in of partial attraction of a nonnal
should be scaled by the factor
T -ET
n n
ShO\'1
If X has a finite second I'lonent, the central limit theorem says
variable.
Since
It remains to
{(Ta-cn)/a{n)}
no rnatter how the
en's
are
.
c~losen.
TIle same kind
of contraJictioIl arises if X is any random varia"ble in the d01?.ain of T)artial
A
attraction of a nonnal variable.
convergence criterion.
The ci.etails can be worked out from the nonJ.al
(See Gnederlko a'll: Kolno[';orov (1954)
~
paF;e 123.
li1mt (62) corresponds to t:le first condition of their Theorem 2.
Bn
The
In tum~ if
is a suitable scaling factor (i.e., factor to ;-:;ult.iply by); then
lininf a(n)Bn = O. This implies
n
tig;ltness iJ:1plies Al.
Proof of Theore:-:J 5.
For such
A~
{(~-cn)/a(n)}~=I cannot be tieht.)
Thus,
0
Inequality (49) only needs to be demonstrated for ~l.
(11) implies
34
Pr (I xl >6Aa(n)) ~
(63)
for some
£>0, depending on A.
£
Pr (I xl >a(n)),
n~l)
Since (8) holds 1 (49) fo11O\l[s from (63) and
o
Lerrrna 4.
Proof of Theorem 6.
c
n
and
d(n»O
ZEV '
Suppose
111en there exist sequences of constants
X
such that, on some subsequence
n'-+oo,
Tn,-cn , L
den!) ~ z,
(64)
where
Tn = Xl + •• ,+x
n
consider the ratios
(n~l).
Let
a(nt)/d(n').
it follmvs from Theorem 4 that
a( 0)
Since
be a nanning function for
Z is not a degenerate random variable
a(n')/d(n')
turn, (64) implies that {(Tn,-cn,)/a(n')}
variables.
in
{n'},
But
a=O
Thus, for some
and real
is impossible by Theoren 5.
nil
I·1--lY1.
• ( ") _
b 11"
TIlUS ,
a~O
Z~VY'
hen'1 )
-
and we have shown that
X and
is bOlD1.ded away from zero.
In
is a tight sequence of random
B
Thus
and for some subsequence
a>O.
I''breover, by (12),
n l '-+00
\,{e
have
Cnll L
Z+(3 .
a\.n '.I --+ a
.~
VXcVy.
By the same argument,
VycV X '
o
It remains to prove the equivalence of AI' ,
Ar'
and AIm.
Proposition 1. Assumptions Al' and All/are equivaZent.
35
Proof.
AI' is equivalent to
h(x)::
for some
O<x~xO
c:>0
~vPr(IXI >v)dv
xO~O.
and some
- £> 0,
x>x
x-
The value
X
o
O
'
can be taken to be zero.
For if
' then
hex)
£:
when
xPr(~.I>x)
~
xPr ( IXI>X )
--_. O
J6vdv
is small enough.
x
Since
HCx) :: log{x-C:
whenever
2Pr(l xl>x o)-C:
£
--=------~o
x
hex)
j:
is the derivative of
v PrCIXI>v)dv}
PrClxl=x) = 0, it follows that Al' is equiV'alent to saying that
is increasing.
Thus AI' is equivalent to Al H •
Proposition 2.
Asswnptions Ai' and All" ape equivalent.
Assume AI i .
H(x)
o
Then Al" holds as well, aJ1U, for some large 1'1 and for each
y>x>0,
~
y
Ja v PrClxl>v)dv
~ (~)£
J:
~ (Z)£: JX
x
= ~£
Thus Al' implies Al YlI
•
0
v PrCIXI>v)dv
v dv PrCIXI>x)
x 2 -£: Prc\xl>x).
Conversely, if Al'll holds, then for each Y>O,
36
y2Pr ~ ~
y2Pr (1.!Ie..->L.<y)'--__
fbxPr(IXI>x)dx fbxE-InDw2-Epr(1xl >y)dx
=~
ill
Thus Ali?, implies Al I
•
It is clear from the way that
> O.
0
E arises Dl the proof of Proposition 1,
that the M appearing in assumption Al 1i 1 can be taken to be as small as 2/f..
Also, it is easily seen that the E appearing in asstunptions Al if and Al'" has
to be smaller than two.
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