No. 1798
8/1989
THE DOMAIN OF PARTIAL ATTRACTION OF A POISSON LAW
SANDOR CSORGO
1
and ROSSITZA DODUNEKOVA
2
Groshev gave a characterization of the union of domains of partial attraction of all
Poisson laws in 1941. His classical condition is expressed by the underlying distribution
function and disguises the role of the mean A of the attracting distribution.
In the
present paper we start out from results of the recent 'probabilistic approach' and derive
characterizations for any fixed A > 0 in t.erms of the IInderlying quantile function. The
approach identifies the portion ofthe sample that contributes the limiting Poisson behavior
of the sum, delineates the effect of extreme values, and leads to necessary and sufficient
conditions all involving A. It turns out that the limiting Poisson distributions arise ill
two qualitatively different ways depending upon whether A > 1 or A < 1. A concrete
construction, illustrating all the results, also shows that in the boundary case whell A .::.: )
both possibilities may occur.
KEY WORDS: Poisson law; domain of partial attraction; quantiles.
1. INTRODUCTION, THE RESULTS, AND DISCUSSION
Let Xl, X 2, ... be independent random variables with the common distribution function F(x) = PiX ::; x}, x E IR, and corresponding left-continuous inverse distribution or
quantile function
Q(s) = infix : F(x)
~
s}, 0 < s ::; 1, Q(O) = Q(O+).
We say that F is in the domain of partial attraction of the Poisson distribution with mean
A > 0 if there exist a subsequence {n'} of the sequence {n} of the positive integers and
1 Bolyai Institute, University of Szeged, 6720 Szeged, Hungary.
2 Department of Probability and Statistics, University of Sofia, 1090 Sofia, Bulgaria.
Mailing address: Sandor C8org6, Bolyai Institute, University of Szeged, Aradi vertanuk tere I,
H-6720 Szeged, Hungary
1
normalizing and centering constants An' > 0 and en' E IR such that
(1.1 )
as
where
.E..
n
,
-t
00,
denotes convergence in distribution and Y>. is a Poisson random variable with
mean A, that is,
P{l">. = k} =
Ak
i! e->.,
k = 0, ],2, ....
In this case we shall write FED" (Poisson (A)). It is well known that {n'} cannot be the
whole sequence {n} in (1.1) because only stable distributions have non-empty domains of
attraction (cf. Gnedenko and Kolmogorov(4) or Csorgo, IJaeusler, and Mason(2), abbreviated as CsHM from here on). On the other hand, every infinitely divisible distribut.ion has
a non-empty domain of partial attraction by a basic theorem of Khinchin (Gnedenko and
Kolmogorov(-4), p. 184) and hence D J• (Poisson(A)) is not void. Furthermore, the Poisson
distribution is the most important infinitely divisible di~tribution in the sense that the
class of all infinitely divisible distribl:2tions is the closure of the dist.ribut.ions of random
variables of the form c1Y>'1
+... +cmY.\"., where C), ••• ,Cm are constants and
)'>'1"'"
)'>.".
are independent, and closure is meant with respect to weak convergence (Gnedenko and
Kolmogorov(4), pp. 74-75). Therefore, the problem of the characterization of /)J' (Poisson
(A)) has a distinctive theoretical appeal.
Starting out from the classical general criterion of convergence to an infinitely divisible
distribution that involves a condition to set the variance of the normal component and
two conditions to set the Levy measures of the canonical Levy form of the characteristic
function of the limiting distribution (Gnedenko and Kolmogorov(4), p. 124), Groshev(5)
has proved that
FEU D,,(Poisson(A))
>'>0
if and only if
2
liminf (
x 2dF(hX)/ (
dF(hx) = 0
h-oo 1\X-ll>£ 1 + x
1 Ix-ll<£
2
for any
e> O.
While nice lookong, it is not immediate to see in what ways this condition (cited also in
Gnedenko and Kolmogorov(4), p. 190) restricts F.
The recent study CsHM(2) of a 'probababilistic approach' to the problem of convergence of centered and normalized sums of the form in (1.1) revealed that whatever infinitely
divisible random variable we have as a limit in (1.1), there are two basic possibilities concerning the size of the normalizing factor An"
One is when, informally speaking, An' is
comparable to
a(n') = J;;Ia(l/n'),
(1.2)
where the 'truncated variance' function
0 2 (.)
is defined as
(1.3)
o<s<
1
2'
with u /\ v = min(u, v), while the other is when a sequence An' diverging to infinity faster
than a(n') is needed, that is when a(n')/A n , ---.0 as n' ---.
00.
For example, it is shown in
CsHM(2) that for stochastically compact F's the correct normalizing sequence is always
{a(n')}, but a rather complicated construction also shows that the second possibility also
occurs.
While the program of characterizing D p (Poisson (,x)) within the probabilistic
approach of CsHM(2) has some interest in itself, and in fact requires some augmentations
of the original theory given in Cs(J), one of our primary motivations for the research
reported here was to see whether An' = a(n') is always sufficient in (1.1) or not.
Let X 1 ,n ~ ... ~ Xn,n denote the order statistics based on the sample X ..... ,Xn.
The probabilistic approach in CsHM(2) generally allows to see which portions of the sum
,
2:j=1 Xj,n'
.
contribute the ingredients of the limiting infinitely divisible law or do not
contribute anything at all.
At the same time, it also delineates the effect of extreme
values. In order to cover these in the present situation, we need some more notation.
3
Let E I, E 2 , • •• be independent random variables having the exponential distribution
with mean 1 and consider the standard left-continuous Possion process
00
(1.4)
L 1(8} < u),
N(u) =
u
~ 0,
}=I
with jump-points 8} = E 1
+ ... + E}, j = 1,2, ... , where 1(·) is the indicator function.
Also, set
00
(1.5)
l'k(,x) =
L
1(8} <,x) -+ min(k
+ I,,x)
-,x, A> O;k
= 0,1,2, ... ,
;=k+1
so that we have the distributional equality
V(,x)
(1.6)
g YA + min(I,,x)
:= l'u('\)
- ,x.
For a given ,x > 0, we finally introduce
rd,x) = min{r: r integer and r,x > I},
(1.7)
so that
r]
(,x) = 1 whenever,x > 1 and
r]
= 2.
(I)
Our first result, connecting the special Poisson situation with the general theory in
CsHM(2) and Cs(l) , is the following.
THEOREM 1. FED,. (Poisson (,xJ) if and only jf for each r
~
a genuine subsequence {n/} of the positive integers such that
1
s
-~
(1.8)
- - Q( - )
a(rn')
n'
(1.9)
--:-Q(I - -)
a(rn')
n'
as n'
1
-4
(1. lOa)
00,
where
Or
for each s > 0,
0
S
-4
{or'
0,
if 0 < s < ,x,
if s > ,x,
> 0 is some constant, and
. I'
u(hjn')
I 1m 1m sup
h-+oo n' -+00 u(l jrn')
=
lim lim sup
h-oo n' -+00
4
yn;u(hjn') .
= O.
a(rn /)
rdA) tlJere exists
If conditions (l.8), (l.9) and (l.10a) are satisfied,then
_(1
,)
a rn
(1.11)
as n'
{t
Xj _ n'jl-J/rn' Q(U)dU}
j=1
l/rn'
00, and there exists a subsf.>quence {n"}
-+
C
_~ (Y'\ + 1-rrA)
U
r
{1l'} such that
(1.12)
and, furthermore, t here exists a sequence {in"} of positive integers !'iuch that In"
in" /n"
-+
-+ 00,
0 and for each fixed integer k ~ 0,
nil-I""
1
L X·l,n " a(rn") {
(J.13)
j=1
n"
JI-(III+I)/n"
}
..
Q(u)du ~O
1/1£"
and
1
{
-a(-rn-'-')
(1.14)
n" -k
L
._ II I
J-n -
as n "
-+
..
Xj,n" - n"
II
+1
JI-(k+I}/n"}
"Q(u)du
1-(1 ,,+I}/n
~
urVk(A)
U
00.
We note that condition (1.10a) may be replaced by
(1.10b)
· l'1m .In f o(h/n') = I'1m I'1m .In f vn'o(h/n')
I1m
h--oo n'--oo o(l/rn')
h--oo n'--oo
a(rn')
=0
Under (1.8), (1.9) and (1.10b) we generally have (1.11) only along a subsequence {nil} C
{n'} already, and (1.12), (1.13) and (1.14) hold along a further subsequence {n
lll
}
C {nil}.
This will be straightforward to see from the proof given in Section 2, where an analysis
will lead us to the following main result of the paper.
THEOREM 2. There exists a subsequence {n'} of tile positive integers such that
conditions (1.8), (1.9) and (l.lOa) hold, and hence F E D p {Poisson (A)), jf and only jf
for every 0 < e ~
(1.15)
5
,\
'2'
Using the fact that 02(eS)/02((A - e)S) ~ 1 for any 0 < e ~ A/2 and S > 0 for
which (A - e)S < 1/2, and hence that this Hminf condition is equivalent to three Iiminf
conditions as in (1.18)-(1.20) below, the monotonicity of the functions 0(') and Q(.) easily
imply that (1.15) is equivalent to
A
for every 0 < e <
-.
- 2
(1.]6)
Using again the fact that for any n > 3A and 0 < e
~
>./2 this whole ratio is never smaller
than 1, and for each n large enough it is non-increasing function of e on (0, >./2] whenever
Q(.) is negative near enough to 0 (otherwise Q2(e/n)/n
~ 0 as
n
~
00
and hence the third
term in the numerator can completely be neglected) a trivial modificat.ion of the argument
in the proof of Corollary 2 in CsHM(2) shows that condition (1.16) in turn holds if and
only if there exists a subsequence {71 / } of the positive integers sucb that
(1.17)
for every 0 < e S
>.
2'
which holds if and only if the three conditions
(] .18)
(1.19)
(1.20)
lim 02(e/n ' )/02((A - e)/n ' ) = 1,
n'-too
lim 02((A
n'--..oo
0 < e ~ >'/2,
+ e)/n')/o:l((>. - e)/n ' ) = 0,
lim Q2(e/n l )/(n' 0 2 ((A - e)/n ' )) = 0,
n'-too
0 < e ~ >'/2,
0 < e S A/2
hold simultaneously along the same {n'l.
The proofs will show that if we have (1.18)-(1.20) or, equivaltmtly, (1.17) along
some {n'l then (1.8)-(1.10a) and hence (1.11) hold along some subsequence {nil} of the
given {n'l. Conversely, if (1.1) holds along some {n'l, then (1.8)-(1.lOa) hold along a
subsequence {nil} of the given {n'l and (1.17) or, what is the same, (1.18)-(1.20) are
6
satisfied along the same subsequence {nil} C {n'}. Then (1.18) and (1.19), holding along
subsequences,will easily imply the second statement of the following.
COROLLARY. Suppose (1.1). If A > 1, thw for each subsequwce {nil} C {n/}
there exist a further subsequence {nll/} C {nil} and a finite constant 6
that a(nll/)/Anll/
-t
6 as nil'
-t
00.
= 6{ rllI/} >
On the other hand, if A < ], then a(n/}/Anl
0 such
-t
0 as
n ' ......... 00.
The corollary demonstrates that limiting Poisson dbtributions in (1.1) arise in two
qualitatively different ways depending on whether A > ] or ,\ < 1. The latter case provides
what is probably the simplest possible example for showing that Theorem 2 in CsHM(2)
is not empty.
In Section 3, we illustrate the above results by a concret.e construction of a quantile
function in the domain of partial attraction ofa Poisson (A) with particular reference t.o the
Corollary. This construction works for all A > 0 and it turns out that the case when A .:. .:. ]
is a boundary case in the sense that both possibilities in the behavior of the normalizing
sequence may in fact show up.
It follows from the transitivity theorem of Gnedenko(a) (cited also in Gncdenko and
Kolmogorov(4), p. 189; for a new proof see CS(I)) that if F E D p (Poisson (A)) for some
A > 0, then F is also in the domain of partial attraction D,,(2) of the normal law. Thus
our necessary and sufficient conditions should somehow imply that
(1.21)
liminfo 2 (s/n}/o2(]/n)
n-oo
=]
for all 0 < s < 1,
which, according to Corollary 2 in CsHM(2) is necessary and sufficient fot F E D (2). At
"
this point we meet again the principal difference bet ween the two cases A > ) and A :::; 1.
Namely, if A > 1, then (1.2]) follows directly from (2.10) below, that is from conditions
(1.8), (1.9) and (1.lOa) with.T
= 1.
The implication is not so direct if A ::; 1 and can be
seen only by extra work.
7
While conditions expressed in terms of the underlying quantile function are completely
natural in our approach, it would perhaps be interesting to obtain equivalent conditions
expressed through F. These are probably uninformative just as Groshev's condition above
and to prod uce them appears to be a non-trivial analytic problem.
As a final remark we note that, starting out from Theorem 12 in Cs(l), it is possible to
characterize in the manner of the present paper the set. of infinitely divisible laws partially
attracted by a given Poisson law. We will consider this elsewhere.
2. PROOFS
Let 1/.' be a non-positive, non-decreasing, right-continuous function on (0,00) such that
1
00
1j.J2 (s)ds <
00
for any e >
o.
In the notation of CsHM(2) and Cs(1), for each integer k ~ 0 consider the random variable
where N(.) is the Poisson process defined in (1.4). For k = 0, this is a spedrally one-sided
infinitely divisible random variable without a normal component. For>. > 0, set
if 0 < u < >.,
-I
1/.'A(U) = {
'
0,
if u 2: ..\.
8
If A :S 1, then we have
VO,k(O,tP,x,O)
=
t
N(u)dJt!,x(u) + ktP.x(Sk+d
1s
k t I
=I(Sk+l < A)N(A) - I(Sk+l < A)k
=1(SHI < .\)
{t
1(S, < .\)
~ k + j;;
J 1(S;
< .\) }
k
=
L {l(Sj <: A)1(Sk+
I
< A) - 1(Sk+ 1 < A)}
j=l
00
L
+
1(8j < A)I(Sk+l < A)
j.d+ 1
00
L
=
1(5) < A),
j=k+l
and if A > 1, then, using the above lines in the last st.ep, we have
=N(A) - A - N(A)1(Sk+l 2: A) - kI(Sk+l < A) ·t min(k
=I(Sk+l < A)N(A) - I(Sk+l < A)k
+ min(k + J,A)
+ J,A)
- A
00
L
=
1(Sj < A)
+ min(k + J, A)
- A.
j=k+l
Putting the two cases together and recalling (1.5), we see that
(2.1 )
In particular, by (1.6),
(2.2)
Vo,o(O, tP,x, 0) = V (A)
g YA + min(1, A)
- A,
>. > 0.
Now we are ready to use Theorem 6 of Cs(l) to prove Theorem 1 here. If
we are in case (ii) of that theorem, while if
>. > 1,
>. :S 1, we are in case (iii). The ambiguity of
9
y}1), ... , Y?)
the latter case will presently be resolved by the well-known fact that if
are
independent copies of Y.>., then
(2.3)
Proof of Theorem 1. To prove necessity, suppose (1. 1). Take any
{XJm)}~1 be independent copies of the sequence {Xj}i= l ' Tn
and (2.3),
I
A
n,
U7
n
{t(t
)=1
XJm»
m=1
-
TCn,} ~ Yr.>.
(2.4)
=-= 1, ... , T.
as n'
v(l)
X(r)
v(l)
fl·t·mg Z I, Z 2, ... l'lor t he sequence .'\.
1 , ... ,
1 ,.'\ 2 , ...
T
~
~ TI
(.A), and let
Then by (1.1)
00.
",(rl
,.'\ 2 , ... ,
..
th
at·IS, wfltmg
.
- X j(m) , 1 <
T J' - ] .)
Z( )-I)r-t
_ rn <
.....;,
,~"."
m -
and introducing C~, = TC n ,
-
(1 - T.A)A n " the convergence relation can be rewritten as
> 1, and by case (ii) of Theorem 6 in Cs(1) there exist
Therefore, by (2,2), the fact that r.A
an {n"} C {n'} and a constant
Or
> 0 such that
t
rn
"
lim Q(-" )/a(rn )
n"
--00
lim Q(1 -
nil --00
t
-II
rn
)/a(rn " )
=
= 0,
{a
r
0,
t
'
> 0,
0< t < r.A,
t >
r.A,
and
lim Iimsupu( hll)/u(--!,,) =
h--oo nil
-00
rn
rn
o.
However, these three conditions are clearly equivalent to (1.8), (1.9) and (1.10a) with n'
replaced by nil.
10
Now we turn to the sufficiency statements. Assume (1.8), (1.9) and (l.lOa). Then by
(2.2) and the sufficiency part of case (ii) of Theorem 6 in Cs(J) we have
1
a Tn
as n'
-t
00,
{
rn
-(
')
1~·
,
I
L:Zj-rn' (
/
rn '
i 1/rn'
j=1
Q(u)du
}~Qrl'(rA)
where ZI, Z2,'" are independent with the common distribution fuction F.
Breaking up the sequence {Zj}f=1 into the union of r independent sequences {X;m)}t.l
of independent variables, m = 1, ...
,r, according to the rule in
(2.4), this can be rewritten
as
n'
1-I/rn'}]
L [Qr a1rn ' {?=X;fI1) - n'!l/rn ' Q(u)du ~ L
r
r
(
m=I
as n'
-t
00.
)
1=1
TIl =
1
[yt
n
) + 1 urr>..]
This clearly implies (1.11) and hence also that F E D J• (Poissoll (A)).
To prove the further statements, we rewrite (1 ;11) as
n'
-+ 00,
where
Cn'
rn
t' Q(u)du + (1 - rA - min(l, A) -+ >.) ()ra(rn'),
il/rn'
I
= n'
/
r
and recall (2.2) again. If >.. > 1, and hence we are in case (ii) of Theorem 6 in Cs(l) or if
>..
:s 1 but we are in the first subcase of case (iii) of the same theorem, then
there exist a su bsequence {nil} C {n'} and a constant br > Osuch tha t.
1
s
-(
") Q (".)
an
n
1
s
-(
") Q(1 - ,,)
ann
and
1
-(
")
an
> 0,
-+
0,
-+
-i"""tP.x(s),
S
1
s > 0,
S
1-
Cn"
{ iI/nil " }
I-~I/n
nil {
A,
Vr
Q(u)du -
11
-t
°
by necessity
as n
II
-
00,
and
.).
(J(hlnl')
)1m 1m sup - ( -").
I
h~oo"
(J 1 n
n
= 0.
~OO
If, on the other hand, A S 1 and we are in the second subcase of case (iii) of Theorem 6 in
Cs(J) then, along some subsequence {nil}
C
{n'l we have
1
s
S > 0,
a Tn
n
1
S
( ") Q(I - ,,) --. Qrt/JA(s),
a Tn
n
( ") Q( ,,) --. 0,
a(n " )/a(Tn " ) -
S
> 0,
( ")
a Tn
{f
nil
I-lin
lin"
-:f A,
0,
and
1
S
" Q(u)du -
en"
}
--.0.
Using now the sufficiency part of Theorem 6 in CsO) once more, we obtain all the theree
statements (1.12) -(1.13) along the ch.osen {n"} and hence the whole theorem.
•
The proof of Theorem 2 requires some preliminary lemmas, the first of which is of
some independent interest.
LEMMA 1. If 0 < s < t < 1/2, then for an arbitrary quantile function
(J2(S) - (J2{t) st(Q(t) - Q(.o;))2
+ 2t(Q(t)
+ t(Q(1 - .0;) -
Q(I - t))2
- Q(s))(Q(1 - s) - Q(t))
+ 2t(Q(1 - s) - Q(1 - t))(Q(1 - t) - Q(t))
and
(J2(S) - (J2(t) ~~(Q(t) - Q(s))2 + ~(Q(1 - s) - Q{l - t))2
+ 2s 2(Q(t) - Q(s))(Q(1 - s) - QU))
+ 2st(Q(1 - s) - Q(1 - t))(Q(1 - t) - Q(t)).
12
Proof. Using the definition in (1.3), as in identity (2.29) in CsHM(2), we have
(J2(S) - (J2(t) .=
itit
(u /\ v - uv)dQ(u)dQ(v)
}'1-8 (u /\ v _ uv)dQ(u)dQ(v)
+ 2{ t' (I - u) (t t>dQ{t>j) dQ{u)
+ /1-.1
I-t
•
+
t'
=: I I (s,t)
I-t
u
({~' (I - t>jdQ{t») dQ{u} }
+ l:ds,t) + 2{J a(s,t) + J4 (s,t)}.
Crearly, Ids, t) ::; t(Q(t) - Q(s))2 and
II (s, t)
=
it
(1 - u)
~ s(1 -
t)
(ill t'dQ(V)) dQ(u) + it
{it
(Q(u) - Q(s))dQ(u)
U
+
(L
it
=8(1 - t) {Q(t)(Q(t) - Q(s)) - Q(8)(Q(t)
t
(1 - V)dQ(V)) dQ(u)
(Q(t) - Q(U))dQ(U)}
Q(8))}
8
~ 2(Q(t) - Q(8))2.
Exactly the same way, using u /\ v - uv ::; 1 - u /\
s
11
for the upper bound, we olJt.ain
.
2(Q(1 - s) - Q(I- t))2 ::; I 2(s,t) ::; t(Q(1 - s) - Q(1 - t))2.
Similarly,
S2(Q(t) - Q(s))(Q(1 - s) - Q(t)) ::; I 3 (s,t) ::; t(Q(t) - Q(s))(Q(1 - s) - Q(t))
and
8t(Q(I- s) - Q(1 - t))(Q(1 - t) - Q(t)) ::; I 4 (s,t)
::; t(Q(1 - s) - Q(1 - t))(Q(1 - t) - Q(t)).
Collecting the upper and lower bounds, the lemma follows . •
13
Let {n'} be a subsequence of the positive integers tending to infinity, let x 2: 1 be any
fixed number and s
>
0 be an arbitrary number. Replacing s by s/xn' and t by sin' and
using (1.2), it will be advantageous to rewrite the inequalities of Lemma I as
0
2
2
(:!rl,) - 0 ( ;i;-) < s (Q ( ;i;-) - Q(
02(1/n')
a(n'}
*')) + s ( Q(1 - *')a(n'}- Q(1 2
'~,)) 2
+ 2s Q( ~)
- Q( x~' ) Q(J - x~,) - Q( :' )
a(n')
a(n')
Q(l- ~,) - Q(:,)
+ ? s Q(l - x~-;) - Q(l - ~)- ----'-'-------'-"a(n'}
a(n')
(2.5)
--=---~- ---~-----'-'--
and
0 2(_8
) - 02(..§....)
xn'
n'
0 2 (1 In')
>~ (Q(;i;-) - Q(~,))2
- 2x
.
a(n')
+ ~ (Q(l-~) - Q(~
a(n')
2x
(2.6)
~l):!
x~,) - Q(-;;}
+ 2 -s2- Q( :,) - QL~,) Q(J----"-'''-------''''-2
x n'
a(n')
a(n')
x~') - Q(l _
-- ~;} --"-_--'-'-,..:--,_....c..:.:-'Q( 1 - :,) - Q( ~ )
+ 2 82 Q(l - -='--:--:-_
xn'
a(n')
a(n')'
-----'-"-_--=.~
~
which hold true for all n' large enough.
LEMMA 2. Let A > 1 be a fixed number and suppose that
Q(s/n')/a(n') --.0,
(2.7)
Q(l - -s }/a(n'} --.
n'
(2.8)
as n' --.
00,
where
lr
{lr'0
,
jfO < s < A,
jf s > A,
> 0 is some constant, and
lim limsup02(h/n')/02(1/n') = O.
(2.9)
h-oo n'-oo
Then, as n' --.
s > 0,
00,
(2.10)
14
and
(2.11)
.
Proof. Notice first that if both s and six are on one and the same side of A, that is
either s < A or six> A, then by (2.7) and (2.8) the upper bound in (2.5) goes to zero as
n'
~ 00.
Put s = 1 in (2.5). Then we get
for all x 2: 1, which implies (2.10) for all 0 < h
x
= It
~ 1.
If 1 < h < A, then putting s .::.. It and
in (2.5) gives (2.10) for 1 < It < A.
To prove (2.11), let h > A and put s = xh in (2.5), where x 2: 1 as always. Then,
•
since six> A, we get
which implies that
_
.
o2(hln').
o2(xltln')_
v(h) := hmsup 2( I ') = hmsup~( --I-') ~~: v(xh)
n'-oo 0
1 n
n'-oo 0
1 n
for each h > A and x 2: 1. Now for each fixed h > 0, v(xh) - .. 0 as x
~ 00
by (2.9), and
hence v(h) = 0 for each h > A and this is nothing but (2.11) . •
LEMMA 3. Let A > 1 be a fixed number and suppose (2.10), (2.11) and that
(2.12)
Q(l/n')/a(n') -.0 as
rt' -. 00.
Then there exist a subsequence {nil} C {n'} and a constant a > 0 such that (2.7), (2.8)
and (2.9) are satisfied with n' replaced by nil.
15
Proof. Take any s < A. Since also six < A, (2.10) implies that the left-side of (2.6)
goes to zero as n'
-+ 00.
Because all the terms in the lower bound there are non-negative,
they all go to zero separately. In particular,
(2.13)
(Q(~) - Q(~ ))/a(n')
(2.14)
(Q(I-~) - Q(1 -- ~))Ia(n')
as n'
n'
xn'
-+ 00.
-+
xn'
0 < s < A, x ~ 1,
0,
--t
n'
0 < s < A, x? 1,
0,
Putting s = 1 in (2.13), by (2.12) we obtain
1
Q(- }Ia(n')
xn'
-+
as n'
0,
-+ 00,
Now if s > 0 is arbitrary then we can choose x
~
for all x ~ I.
1 so that s > 1 j x and, provided that
Q(.) is negative near enouh to zero, we get
s
IQ( - )lla(n')
n'
1
.
:s: IQ(-)l/a(n')
xn'
-+
o.
If Q(.) is never negative, then of course (2.7) is trivial, and hence we have (2.7) along the
original {n'}.
Consider now an s > A. Then, using (2.11),
Q(l- ~}
a(n')
Q(l - :,) u(sln'}
--==---"'--'- --:-:-:~
R)u(sjn'} u(l In'}
0
-+
as n' - ..
00,
on account of the fact that the first ratio on the right side is bounded by Lemma 2.4 in
CsHM(2). Thus we have (2.8) for s > A, still along the original {71'}.
Now we come to (2.8) for the case s < A. Again by Lemma 2.4 from CsHM(2) we can
choose a subsequence {n"}
c {n'}
Q(I where
0:
2
such that
1
Ii
n
II
}Ia(n )
--+ 0'
as n
"
-+ 00,
O. Setting s = 1 in (2.14), we then have
1
II
Q(l - -,,)ja(n } -+
xn
0:
16
as n
"
-+ 00,
x? I,
e
which implies
S
as n
II
II
Q(I-,,)/a(n )-a,
(2.15)
n
~ 00.
0<s:S1,
If 1 < s < A, then we find an x > 1 so that six :S 1. Using (2.15) in
conjunction with (2.14), we see that (2.15) also holds for 1 < s < A. Thus we have (2.8)
also for all 0
< s < A, along {nil}, with a possibly zero limit
lX.
> O. Suppose, on the contrary, that a = O. Then the already proved
We claim that a
(2.7) and (2.8), holding along {nil}, the latter with
0:
0, imply in accordalKe with case
=:
(i) of Theorem 1 in CsHM(2) or Theorem I' in Cs(J) that
}
L" Xj - n" j" "
Q(u)du
a(n") {
n
1
I-lin
j=l
E.
N (0, ))
as n "
-. 00.
lin
Thus, by an application of the augmented case (ii). of the same theorem (Theorem)' in
•
Cs(J»), we also have
.
. . o·~( h,, n ")
..
0 2( It I n ")
11m 11m mf 2( I ") = 1 = Inn Illn Slip ~( , ")'
h--CX) nil --CX) 0
1 n
h--oo nil. '.X> o· I, n
Since {nil} C {n/}, this obviously contradicts (2.11). Therefore
0:
> 0, and
we have
(2.7)
and (2.8) along {nil}.
Finally, it is trivial that (2.11) implies (2.9) along the original {n/}, which then holds
along {nil} a fortiori. •
LEMMA 4. Suppose that (2.10), (2.11) and (2.12) hold true for some A > 1 and
{n/}. Then we have (l.1B), (1.19) and (1.20), and hence (l.l7), for the same {n/} and with
A replaced by A.
Proof. Consider any 0 < c
~
A/2. Then by (2.10),
/
02(c/n )
= 02(cln ' } (02((A - c)/n / )) - 1_. 1 as n ~
'
02((A - c)/n / ) 02(1/n ' }
02(l/n /}
17
00,
that is, we have (1.18) with
as n ' ~
00,
.x
= A. Further, by (2.10) and (2.11),
that is, we have (1.19) with
.x
= A. Finally, we know from the proof of Lemma
3 that (2.10) and (2.12) imply that (2.7) holds along {n/}. Therefore, by (2.7) and (2.10)
again,
which is (1.20) with
.x =
A. •
LEMMA 5. Suppose that (l.17), and hence (1.18), (l.19) and (1.20) hold true for
some.x = A
>
2 and {n'}. Then we have (2.10), (2.11) and (2.12) for the sam(' {1l / } and
A.
Proof. Since A - I > 1 or A-I> A/2, by (1.18) with
.x =
A we have
and
a:l(1/n' )
a 2 (I/n ' )
<
->I,
- a 2 (c/n /) - a 2 ((A - 1)/n l )
1<
1 <c"S.A/2,
which together give
(2.16)
Also, by (1.18) again and (2.16),
(2.17)
as n '
~ 00,
and now (2.16)" and (2.17) together imply (2.10).
18
Furthermore, using (1.19) with A = A and (2.17),
as n'
•
--+ 00.
Since for any h > A one can find a small e > 0 such that A + e < h, this
clearly implies (2.11).
Finally, putting e
=1 <
A/2 in (1.20), assumed with A = A, we have
which is (2.12) . •
Proof of Theorem 2. First assume (1.8), (1.9) and (l.lOa) for some A ,> 0, a fixed
r 2: rdA) of (1.7) and some {n'}. Then we have
t
rn'
Q( - )/ a (rn')
(2.18)
•
t
Q(1 - -)ja(rn')
rn'
(2.19)
as n'
--+ 00,
--+
--+
0,
> 0,
t
{an
0,
O<t<rA,
t > rA,
and
(2.20)
Using Lemma 2 for the subsequence {rn'} and A = fA > 1, we obtai"
0< h < rA,
(2.21 )
h > rA,
and, as a special case of (2.18),
•
(2.22)
•
as n'
1
Q( -- )/a(rn')
rn'
--+ 00.
--+
0
Lemma 4, also used with {rn'} and A = rA, now gives
rA
O<n<
., - -2 .
(2.23)
19
Setting c = t]IT, this is nothing but (1.17) which implies (1.15).
To prove the converse, assume (1.15). Then we have (1.17) for some {n'}. Let
T = T2(oX) = min{m : m integer and mA > 2} >
and with this r set
t]
rl
(A)
= Tc in (1.17) to obtain (2.23). Applying Lemma 5 \.. · ith A
= roX
:> 2
and {Tn'}, we obtain (2.21) and (2.22). Now Lemma 3, used for {Tn'} and A .;..; TA :> 2,
yields (2.18), (2.19) and (2.20) with n' replaced by n", where {nil} C {n'}. Since (1.lOa)
is equivalent to (2.20), the substitution s = tiT finally gives (1.8), (1.9) and (1.lOa) with
n' replaced by n", and hence the theorem. •
Proof of the Corollary. The first statement follows from Theorem 5 in CsHM (2)
because if A > 1, the function
til).,
in (2.2) is not identically zero on the half-line 11,00).
As to the second statement, assume (1.1) for some {n'} and A < 1. Then it holdlS
along an arbitrary subsequence {n"} C {n'}. As d~scribed before the Corollary, we then
•
have (1.8), (1.9) and (1.lOa) along some {n"'} C {n"}, and hen<:e also (1.18) and (1.19)
along the same {n"'}. Thus, for any c > 0 for which A + e < 1,
o2(I/n"')
o2(I/n"')
(
o2(e/n"') ) - 1
o2(e/n"') = o2((oX - c)/n"') o2((A - c)/n"'j
~
as n'"
-+
o2((A+c)/n"') ( 0 2 (eln"')
o2((A - e)/n"') o2((oX - c)/n'"
)-O
l
-TO
00. Let
1
T
so that, in particular, oX
+ (liT) <
> max( T d oX), 1 _ oX),
1. Since (1.1) and (1.11) now hold jointly along {n"'},
the convergence of types theorem (Gnedenko and Kolomogorov(4), pp. 40-42) implies that
a(rn"') I An'"
-T {)
as n'"
-+
00, where 0 < {) < 00. Therefore,
a(n"')
An'"
a(Tn"')
=
An'"
o(l/n"')
r.: (
v ro
r-
,n
1 I
•
",
"')
-+
0
as n
-+
O.
Since {nil} C {n'} was arbitrary, it follows that a(n')IArtl --70 as n'
20
-+
•
00. •
3. A CONSTRUCTION
Consider a number
°<
~
C
I and set
•
Then we have to = c/2 and ti/tj-l ~
and note that bj/bi-l --.
Q(u) = {
00
as j
°
~ 00.
i:\.::;
j .-.
00.
Also, set
Now we introduce the quantile function
0,
if
°
bk,
if
I - Ik
~
u
:s;
I - to,
< u :s; 1 -
Ik
ii,
k
= 0,1,2, ... ,
or, what is the same,
Q(I - t) =
{
~
0,
if to
bk,
iftk-t
I
t:S; I,
:S t < tk,k= 0,1,2, ...
c
Fix A > 0. The crucial element of the construction is the choice of the subsequence
nk
= nk(-\,c)
=
r~ 1 =
tk
min{l: I integer, I 2
~ },k = 0,1,2, ...
tk
By elementary considerations we obtain that for all k large enough,
tk+ I < A/nk < tk, ifA/tk < P/tk l,
(3.1 )
tk+l < A/nk = tk, ifA/tk
= rA/tk 1
Setting now AnA: = bk , we obviously have
•
(3.2)
and from (3.1),
(3.3)
s
{ 1,
Q(l - -)/AnA: ~
nk
0,
21
s < A,
s> A,
as k -
00.
The next step is to analyse the function o2(h/71k), h > 0, and in doing so, we have
to separate three cases.
We always use first formula (2.58) from CsHM(2), then the
•
corresponding case of (3.1), and the final asymptotic equalities are obtained by simple
computation.
If h < A, or h = A and A/tk < r A/tk
1, then for all k
large enough,
(3.4)
"
If h = A and A/tk = P/tk
1, then for all k
large enough,
•
(3.5)
22
Finally, if h > A, then for all k large enough,
•
(3.6)
Now we are in the position to draw the conclusions, distinguishing four cases.
Case 1 : A > 1. Combining (3.6) and (3.4), for all h
a 2 ( h Ink)
a 2 (I
1 nk )
-- 2
:lk -
I
,">
A we obtain that
.:lk
/'1.
--7 0
and by (3.4),
a 2(nk)
A~k
Ttka2(1/nk)
2
--7 A, as k --7
bk
00,
for all 0 < c S 1. These relations together with (3.2) and (3.3) provide an illustration of
Theorem 1 and the first case of the corollary.
Case 2 : A < 1. Again by (3.4), with h
= l/r <
A, where r 2 rl (-X), we have
a(hlnk)/a(l/rnk) --70 for all h > 1 by (3.4) and (3.6), and, finally,
for all 0 < c S 1, as k --7
00.
Together with (3.2) and (3.3) again, these illustrate Theorem
1 and the second case of the corollary.
Case 3 : -X = 1 and
Ulf
choose 0 < c < 1 so that l/c is not an integer. By (3.4),
23
and by (3.6) and (3.4), for all h > 1,
•
as k
-+ 00.
Together with (3.2) and (3.3), these show that we are in the situation of Case
1.
Case 4: A = 1 and we choose c
= 1.
We have by (3.5) that
2 2k E~=122i-J /(1 _ 2-- 2i -
22l
-+
Furthermore, for each r
~
0, as k
l
)
t J
-+
00.
2, by (3.4)
and for all h > 1, by (3.6) and (3.4),
.
These, taken together with (3.2) and (3.3), show that we are in the situation of Case 2.
ACKNOWLEDGEMENTS
S. Csorgo was partially supported by the Hungarian National Fuondation for Scientific
Res~arch,
Grants 1808/86 and 457/88, while R. Dodunekova was partially supported by
the Ministry of Culture, Science and Education of Bulgaria, Contract No. 1035. We are
grateful to Vilmos Totik of the University of Szeged for his initial help in the construction
and to V. V. Vinogradov of Moscow State University for sending us copies of Gnedenko(3)
and Groshev(5).
24
•
REFERENCES
..
1. CSORCO, S. (1989).
A probabilistic approach to domains of partial attraction.
Adu. itt Appl. Math.. To a.ppea.r.
2. CSORCO, S., HAEUSLER, E. and MASON, D. M. (1988). A probabilistic approach
to the asymptotic distribution of sums of independent, identically distributed random
variables. Adv. in Appl. Math. 9, 259-333.
3. CNEDENKO, B. V. (1940). Some theorems on the powers of distribution functions.
lichen. Zap. Afoskov. Gos. lIniv. Mat. 45, 61-72. [Russianl
4. CNEDENKO, B. V. and KOLMOGOROV, A. N. (1954).
Limit Distributions for
Sums of Independent Random Variables. Addison-Wesley, Reading, "Massachusetts.
,
5. CROSHEV, A. V. (1941). The domain of attraction of the Poisson law. lZtlestlj"a
.4kad. Nauk USSR, Su. Mat. 5, 165-172. [Russian]
25