No. 1796
8/1989
A Probabilistic Approach to Domains of Partial Attraction·
SANDOR CSORGO
Bolyai Institute, University of Szeged, Szeged, Hungary
1. INTRODUCTION AND RESULTS
A new unified approach to one of the most classical problems of probability theory,
the problem of the asymptotic distribution of sums of independent and identically distributed random variables, was recently presented in 16]. This approach, based upon the
asymptotic behavior of the uniform empirical distribution function in conjunction with the
tail properties of the underlying quantile function, identifies the portions of the sums that
contrubute the ingredients of the limiting infinitely divisible law, lihows clearly how these
ingredients arise, delineates the effect of extreme values on the limiting distribution, and
leads to a probabilistic representation for an arbitrary infinitely divisible real random variable. Since in the meantime Sa to's bounds for the tail probabilities of infinitely divisible
distributions, used in the proof of Corollary 1 in
--
161, have heen derived among other things
in [9] from exactly this represen.tation, our approach is now fully self-contained and uses
Fourier analysis only to ensure the uniqueness of this probabilistic representation, givefl in
Theorem 3 in 16], through the uniqueness of the Levy canonical form of the characteristic
function of an infinitely divisible law.
Even though the main theorems in
16]
were strong enough to derive the basic results
on domains of attraction and stochastic compactness in Corallaries 1, 3, 4, 10, 11 afld 12,
and some results concerning domains of partial attraction in Corollaries 2, 5, 7, 8 and 9,
there are a number of important classical or potentially new results on domains of partial
attraction which do not follow from them as they stand in 16]. The main reason for this is
that the construction of the sequence {Tn} in Theorem 1 in
161 appears
to be complicated
and hence this theorem does not contain an analytic criterion for the choice of the variance
(J2
of the normal component of the limiting law. Also, some classical results of Doeblin
and Gnedenko were felt to be "out of the reach" of this approach altogether (cf. the second
paragraph on p. 328 in [61).
The present paper is an organic continuation of [6], completing the theory of our
'probabilistic approach'. We begin with augmenting Theorem 1 in 16] by showing that the
• Work partially supported by the Hungarian National Fo'undation for Scientific Rez;earch,
Grants 1808/86 and 457/88.
1
sequence {r n } can be constructed, or in fact is already constructed in such a way that allows
an analytic description of u 2 • Also, st.arting out from thc probabilistic represcntation of
an infinitely divisible random variable obtained in [6], we show that not only all of the
mentioned classical results of Doeblin and Gnedenko are wcll within the prescnt approach,
but their present formulations and purely probabilistic proofs shed more light on their
essence and the most interesting of them can considerably be improved. A number of
seemingly new results are also derived.
Considering that the constructional problems
mentioned on p. 328 in [6] are also solved in [10], the present paper virtually completes
the theory in its essential lines.
This theory has been designed primarily for sums of independent and idcntically
distributed random variables, or, more generally, for various sums of order statistics of such
variables (d. [6], [7], and [8]). However, the formulation and the purely probabilistic proof
of Theorem 11 below for the convergence of an arbitrary sequence of infinitely divisible
distributions opens the door for the problem of the asymptotic distribution of row sums of
row-wise independent but not identically distributed infinitesimal random variables in an
arbitrary triangular array to be included into the theory. Indeed, this theorem is a variant
of the purely Fourier-analytic Theorem 2 on pp. 88-92 in [14], one of the core results in
that book, and using then accompanying infinitely divisible laws (p. 112 and Chapters 4-6
in 1J4]) this inclusion becomes feasible.
First we review the basic notation from
161
and then state the results.
Th(~
proofs are
in Section 2, and the results and their place in the literutere are discussed in Section
a.
Theorem 1* and Corollary 5* below are completed or improved forms of Theorem 1 and
Corollary 5 in [6]. Emphasizing that this paper is a continuation of [6], the numbering of
the theorems and corollaries here continues that of in [6]. Whenever there is a reference to
anyone of Theorems 1-5 or Corollaries 1-12 without a reference number, we refer to the
corresponding result in [61.
Let Xl, X 2 , •• , be a sequence of independent random variables with a common (rightcontinuous) non-degenerate distribution function F and quantile function Q(s) = inf{x :
~
s}, 0 < s
Q(O) = Q(O+), for each integer n 2: 1, let X I,n ~ .••
the order statistics based on X}, ... , X n, and for (} < S < 1/2, introduce
F(x)
~ 1,
~ X n,n denote
(1.1)
where u
1\
v = min(u,v). Then for all n large enough the quantity
a(n)
=
n l / 2 u(l/n)
2
(1.2)
1 0 and nan
is positive. Let an > 0 be any sequence such that an
-t
0 as n
00,
-t
and
consider the functions
Q(s/n+)/a(n),
tPI(n,s) = {
Q((I - a n )+ )ja(n),
and
nan < s <
rt -
00,
0 < s ~ n - nan,
-Q(I - s/n)/a(n),
tP2(n,s) = {
-Q(an)/a(n),
n-
7la n
<s
<:
00.
Let E~j), E~i), ... ; j = 1,2, be two independent sequences of independent exponentially
distributed random variables with mean 1 and with their partial slims s~j> = E~j>
+ ... +
E!.i), n ~ I, j = 1,2, as jump points, consider the standard left-continuous independent
Poisson processes
00
Nj(u) = L.I(S~j> < u),
0~u
<
00,
j
= ],2,
(1.3)
n=l
where 1(·) is the indicator function. Considering also two non-decreasing, non-positive,
right-continuous functions tPl and tP2 on (0,00) such that
1
00
tP}(u)du <
00,
for all
e> 0,
j = 1,2,
(1.4)
define the random variables
and
00
vp> = -
f
lsI:!)
(u - N 2(u))dtP2(U) -
"+1
where m
~
o and
-
j
k
~
1
f
1
Sl:l)
k+l
UdtP2(U) + ktP2(S~~1)
k+I
tP2(u)du - tP2(1),
0 are arbitrary integers. These variables are non-degenerate if
tPl 1- 0 and tP2 1- 0, respectively. Let N (Il, 0 2 ) denote a normal random variable with
mean Il and variance 0 2 , understood to be the constant Il if 0 = 0, let
-t D
convergence in distribution and in probability, respectively, and finally, let
and
~
-t
p
denote
denote weak
convergence of functions, that is, pointwise convergence on (0,00) at each continuity point
of the limiting function.
The following is,a completed form of Theorem I, where the additional results appear
in (1.6)-(1.10) holding along the original given subsequence {n'}.
3
THEOREM 1'. Assume that there exists a subsequence {n'} of the positive integers
{n} such that for two non-decreasing, non-positive, right-continuous fUlJctions tPI and tP'l
defined on (0,00) we have
tPj(n',.) => tPj('), j = 1,2, as n'
(i) If tPI = tP2 == 0, then for all fixed m ~
1 {n'-k
-(')
L Xj,n' - n' /
a n
j=m+1
°and k
-t
00.
~ 0, as
1-(k+I)/n'}
Q(u)du
n'
00,
-t
--tv
N(O, 1).
(m-t-I)/rt'
(ii) If the limits tPI and tP2 are arbitrary, tl,ey necessarily satisfy (1.4) and there
exist two sequences {In'} and {r n,} of positive integers such tlJat, as n'
r n, In' - t 0, In,lr n, --t 0, and for any pair of fixed m ~ and k ~ 0,
°
1
a(n')
{
-t
--t
00,
/(l"dl)/n'}
L
Xj,n' - n'
Q(u)du -tv V~l),
j=m+1
(m+
I",
l)/n'
.
1
{
r..,
/(r",+I)/n'}
L
Xj,n,-'n'
Q(u)du -tpO,
j=I..,+1
(l",+I)/n'
1
{
rl-(l",+I)/n'}
Xj,R,-n'J1
I, Q(u)du -tpO,
"=n'-r ,+1
I-(r...-t-I); n
"
'
a(n')
00, In'
(1.5)
a(n')
1
where
°
L
n'-k
L
{
a n'
( )
and
n'-l",
j=n'-I",+1
rl-(k+I)/n'}
Q(u)du -tv
I-(I..,+I)/n'
Xj,n' - n' J1
· . f a((rn, + 1)/n')
I1m In
n' ..... oo
o(l/n')
~ Q. = Q.{n'} ~ U{n'} =
0=
-
If Q. = U =
0,
(j
=0
-
1
a(n')
{
-t
n -r,,'
,
d
.
a((rn' + 1)/n')
=
IImsup
n' ..... oo
o(l/n')
and
(j
= lim lim sup
(1.6)
h ..... o.:> n' ..... oo
u(h/ll)
,
(1.7)
aZ ,
(1.8)
u(l/n')
00
1-(r",+I)/n'
L Xj,n' -- n' / (r",+I)/n'
"=r ,+1
'
(j
~ 1 are defined as
, I'ImIn
' f u(h/n')
I1m
h ..... oo n' ..... oo a(l/n')
then, as n'
an
VP),
..
4
Q(u)du
}
--tv
where Z is a standard normal random variable such that Nd·), Z, and N 2 (-) are independent, and hence
1
a(n'}
{n'-k
L
j=m+l
jl-(k+l}/rl'
Xj,n' - n'
}
Q(u)du
Vm,k(th,tP2'0)
-0
:=
(m+l}/n'
V~l) + oZ + VP}·
(1.9)
J..1oreover, whenever 0
> 0,
we ha\'e, as n'
o((ln'
If 0 ::;
Q.
<
(j ::;
--t
00,
+ 1}/n'}/o((r n, + l)/n'}
1, then for some limit point
E
0
--t
(1.10)
1.
IQ,O'I
of the sequence o((r n ,
+
1)/n')/o(l/n') and for some subsequence{n"} c {n/} we have
o((rn"
" } ---+ 0
+ 1)/n"
}/o(l/n
as
n" -
00,
and the conclusions (1.8), (l,9) and (l.lO) holding along {rt}.
--
We know from Theorem 5 that if n=j~lXj':"" Cnk}/A nk converges in distribution
along some subsequence {nd~l of the positive integers, where AnI. > 0 and C nk are
some constants, then what we have is
as
k
---+ 00,
(1.11)
where a > 0, {3 E IR., and VO,O(tPb tP2' 0), with some tPl and tP2 described before and in
(1.4) and 0::; 0::; 1, is defined in (1.9). In this case, we write F E Dl'(tPl,tP2'0) to denote
that F is in the domain of partial attraction of the infinitely divisible law determined by
the triple (tPl' tP2'0). The next theorem gives a thorough characterization of this relation.
THEOREM 6. (i) If 0 > 0, then (1.11) holds along some {ndr=l if and only if
there exist a subsequence
{nklk=l
and a 6 2:
0.0>
0 such that
(1.12)
a(n~)/An'II;
l-l/n~
{n~ jl/n~
Q(u)du 5
--t
(1.13)
6,
Cn~
}
/
An~
{3
---t
h
(1.14)
as k
-+
00, and
.
. . a(hjn~)
hm hm mf a (j
1 n ')
k
h--oo k--oo
a
..
a(hjn~)
= ca
= h--.oo
hm hm
sup (/ I)'
v
k--oo a 1 n
(1.15)
k
If (1.11) holds with tPl = tP2 == 0 (that is, if we talk about the domain of partial attraction
of a normal law) tllen, necessarily, {) = aa and (1.12)-(1.15) all hold with aaj6 = 1 along
the original {ndk=l for which (1.11) is true. If one OftPl and tP2 is not identically zero,
then, having (1.11), every subsequence of {n k}~l contains a further subsequence {nUb 1
for which (1.12)-(1.15) all hold but
may in general depend on the chosen subsequence
{j
{nU~l'
(ii)]fa = 0 and at least one OftPI and 1/-12 is not idelltically zero on 11,00), then (l.11)
holds along some {nd~l if and only if there exist a subsequelJce {nU~l and a
{j
> 0
such that we llave (1.12)-(1.14) and
lim Iimsupa(hjnk)ja(lj71k) -'..: O.
(1.16)
h-+oo k--oo
]f (1.11) holds, then (1.12)-(1.14) and (1.16) generally take place only along subsequences
e-
of tlle given {nk}k= 1 again, with {) depending on the chosen subsequence.
(iii) ]f a = 0 and both tP 1 and
l/J2
are identically zero on [], 00), then (l.ll) holds
along some {nk}~] if and only if either there exists a subsequence {nk}k=l such that we
ha"'e (l.12)-{l.14) and (1.16) wiUJ some {)
> 0, or
(1.] 7)
a(nk)jAni:
and
{n~ /.;:;l
/n;
Q(u)du -
-+
(1.18)
0,
en; } / An;
~ {i.
(1.]9)
If (1.11) holds then again the conclusions take place only along subsequenes of the given
{nd~l'
On the other hand, if any of these sets of conditions is satisfied along some {nk} k=]'
then we have (1.11) along the same {nk}k=l and, besides, in cases (i), (ii) and the first
subcase of case (iii) we also have (l.5) and (1.8)-(1./0) along {nk}' while in the second
6
subcase of case (iii) we also have (l.13)-(l.15) of Theorem 2 in [6/, along {nU, with the
appropriate limiting functions taken from (1.12) or (l.17), respectively.
We note that it is possible to replace Iimsup by Iiminf in (1.16), but then we generally
have to go down to a subsequence also in the sufficiency direction.
The normalizing and centering constants are fixed in the above theorem. If we do not
require this, combine cases (i) and (ii), and are not interested in particular subsequences,
the following simplified form is perhaps more transparent.
THEOREM 7. (a) If a > 0, or a
= 0 but at least one of,tPl
and
t/)'}.
is not identically
zero on [1,00}, then F E D p (tPI,tlI 2,a} ifand only if there exist a subsequence in'} of the
positive integers and a constant c > 0 such that ca
~
1 and
(1.20)
as n'
-+
00 and
· ). . f a(h/n'}
I1m 1m III ( / ' )
a 1 n
h-oo n'-oo
I·
= ca = h-oo
1m
I·
a(h/n'}
1m
sup ( / ').
n'-oo a 1 n
(1.21)
(b) If both till and tP2 are identically zero on 11,00), then F E D,.(tI'l, tI'2' O} if and
only if there exists an in'} such that either we have (1.20) and (1.21) with some c > 0 and
-e
a = 0, or we have
a(n')
- A tPi(n',·)=}tPi(·},
j=I,2,
n'
and a(n')/A n ,
-+
0 as n'
-+
00.
Theorem 6 or 7 readily implies the following improved form of Corollary 5 for the
characterization of the domain of partial attraction Dp(a} of a stable distribution with
exponent a E (0,2).
COROLLARY 5*. FE Dp(a) if and only if there exist a subsequence in'} of {n}
and constants
CI,C2
2: 0,
Cl
+ C2 > 0, such that
and
lim Iimsupa(h/n')/a(I/n') =
h-oo n'-oo
In this case, for each fixed m 2:
as n'
°
and k 2: 0 we have
-+ 00.
7
o.
(1.22)
In order to further illustrate Theorems 6 and 7, we consider two more examples. For
a > 0 and ..\ > 0, let YQ,A denote a random variable with the gamma distribution of order
a and parameter ..\:
~ fX
F u ,>. (x) = P{YU,A :S x} =
r(Q) Jo
e - >.tlUe-Idt , x >
_ 0,
{
0,
x < 0,
where r(.) is the usual gamma function. Consider the function hu,>.(s) > 0,
S
> 0, whic:h
is defined to be the unique solution x > 0 of the equation
roc> e-trldt = ~,
l>.x
and set tPu,>. (s) = -hQ,A(S),
G
u
A
= a
,
a
> O. Introduce also the constant
S
{1°O
e- dt _ !I/J.. ~(I) _e_ dt + r
~dt}
ol+t 2
1+t 2
It/J.,.~(J)1+t2
.
At
O
t
_
-00
For a different purpose it has been checked in [2] that Vm,k(O,tPQ,>',O) = VO,k(O,tPu,>',O) =
YQ,>.(k) for any integers m
~
0 and k ~ 0, where
00
L
YQ,>.(k) = -
tPo,>.(S,) - a(l - el/J.. ·~(k+I»)
l=k+ I
where S, is obtained by dropping the superscript
tional equality
i = 2 in
+ Gu,A'
(1.23)
(1.3), and we have the distribu-
(J .24)
(Here we use the occasion to correct a misprint in 12]: in the rate formula for the approximation of FO,A we should have -hu,>.(k
+ 1)
instead of -ho,>.(k) in the exponent.) Using
now (1.23) and (1.24), case (ii) of Theorem 6 at once gives the following.
COROLLARY 13. FE D p (FUe ,>.) jf and only jf there exists a subsequence {n'} c
{n} such that for each s > 0,
Q(s/n')/a(n') - 0 and
as n' -
00,
where
P>
0 is some constant, and (1.22) holds. In this case there exists a
sequence {In,}OE integers such tllat In' 1
Q(l - s/n')/a(n') - PtPQ,A(S)
{n'-l ..,
-(
') L
an.
00,
In' In'
-
0,
jl-(l ..dl)/n'
Xj,n' - n'
)=1
I/n'
}
Q(u)du
-p
0
and
1
{
a(n')
for each fixed k
n'-k
L
j=n'-I..
~ 0 as
n' -
Xj,n' - n'
,+1
jl-<k+I)/n'}
Q(u)du
1-(l.. ,+I)/n'
00.
8
-D
pYo,>.(k)
.
(1.25)
e-
The second example is the Geometric (p) distribution
P{YJ, = n} = (1 - p)pn,
n
= 0,1,2, ...
wi th parameter 0 < p < 1. Set
_
.1
( ) _
'f'I' S
,,00
L...-n=1
{
nl(log
_1_ _
I-p
,,~
~ < s < log
LJ}=I}
-
_1_ _
I-p
,,~- J ~)
LJ}=I}
o
,0<s<logJ~p'
, s 2: log 1 ~ p'
Again, it has been checked in 12] that V m,k (0, tP", 0)
integers m ~ 0 and k ~ 0, where,
VO,k(O, tPl"0) = Y,,(k) for allY
00
L
Yp(k) = -
V'J' (Sj) and Y, (O) =v Yp.
i
j=k+J
Using these relations, since - 10g(l- p) > 1, case (ii) of Theorem 6 again gives the following.
COROLLARY 14. FE D ,. (Geometric (p)) if and only if there exist a subsequence
{n'} c {n} and a constant
Q
> 0 sudl that Q(sln')j'a(n')
-e
---+
0 for each s
> 0,
Q(1 -- ~)Ia(n') => QtP,,('),
n
and (l.22) holds. In this case there exists a sequence {In'} of integers such that In' ~
In' In'
~ 00, we have
1
a(n')
{
00,
(1.25), and
n'-k
L
jl-<k+l)/n'}
Xj,n' - n'
.
}"=n'-I,+1
for each fixed k ~ 0 as n' ~
Q(u)du
l-(I",+I)/n'
~v oJ',.(k)
00.
Perhaps- the most interesting example is the domain of partial attraction of a Poisson
law with mean- A > O. This example presents special intricate problems which are solved
in [31. When A > 1, we are in case (ii) of Theorem 6. However, as a complete surprise it
turns out that when A < 1, we are always in the second subcase of case (iii). The latter
situation provides the simplest possible example to show that Theorem 2 is not empty.
The situation when A = 1 turns out to be a real borderline case in that both alternatives
of case (iii) of Theorem 6 may occur. For details we refer to [3].
Now we turn to qualitative and deeper results on domains of partial attraction. All
of these will depend upon the probabilistic representation of an infinitely divisible random
9
variable in Theorem 3. For the Poisson processes in (1.3) and a t/J function satisfying the
conditions above and in (1.4), introduce
Wj(t/J) =
~oo (Nj(s) -
+
1
I
+ t/J2(s) ds - 1
1
S)d1J1{S)
Nj(s)dt/J(s)
where
1
and consider
JOO
t/J(s)
1
E>(t/J) = -t/J(l) + 0
+ 8(1,/1), i = 1,2,
(1.26)
t/J:i(s)
I
+ t/J2(s) ds,
(1.27)
where 1/.'1, t/J2 are two such functions, (J 2: 0, and Z is a standard normal random variable
such that N 1 (.), Z, and N 2 (-) are independent. Note that with the notation in (1.9) we
have
(1.28)
by elementary computation, and hence by Theorem 3,
EeitV(tPl'tPZ,U)
= exp
/0
2
{
_~t2 +
2
(e itz _
I _
(e tiZ
I -
-00
+
1
00
o
.
-
.
ltx 2)dL(x)
l+x
itx
}
2 )dH(x)
l+x
for all t E IR., where L(x) = inf{s > 0 : 1f1J(S) 2: x}, x < 0, and R(x) = - inf{s >
o:
t/J2(S) ~ -x}, X > 0, L(·) is left-continuous and non-decreasing on (- 00,0) with
L(-oo) = 0 and R(·) is right-continuous and non-decreasing on (0,00) with R(oo) = 0
and by (1.4),
0
/ -£
x2dL(x)
+
r x dR(x) <
10
2
00
for any
E
>
o.
Thus, modulo an additive constant, V (11'1, t/J2' (J) represents an arbitrary infinitely divisible
random variable by p. 84 in
/141.
A basic role will be played by the rescaled functions
t/J (T) (s)
= t/J ( ~ ),
s > 0,
(1.29)
T
is an arbitrary positive number. This is because if (Nfl}(.),
NJI}(.)), ,
(N~r)(-), z(r}, NJT)(.)) are independent copies of(Nd·), Z, N 2(·)) and Vdt/JI,t/J2,(J), ,
Vr (t/Jl' t/J2, (J) are the corresponding independent copies of \' (t/JI, t/J2, (J) determined through
(1.27), then by Lemma 2.1 below,
where
T
Z(l},
r
LVI(t/JI,t/J2,(J)
=D
V(1/J~r},t/J~r),vr(J).
1=1
A key result is then the following.
10
(1.30)
e·
THEOREM 8. If F E D1.(1/.'h tP2' a), then F E D,.(tP~T), 1/1~T), Jia) for any
and if, furthermore,
T
> 0,
(1.31)
for some sequences
Tm
>
0, am
>
0, and em E IR, as m
--?
00,
where a. 2':
°and the
functions tPi and tP2 satisfy the conditions above and in (1.4), then also F E D,. (tPi ,tP2' a.).
Let F1/!1 ,1/!2 ,cr (.) denote the distribution function of F (iPI, 1/.'2, a). Although the following result is just a simple corollary to Theorem 8, we call it a theorem in view of its special
interest.
THEOREM 9. (Gnedenko's transitivity theorem).
F1/!I,1/!2'cr E D p (iPi,tP2,a.), then FE D1.(tPj,tP:i,a.).
If F E Dr.(tPI, 1/.'2, a) and
Introd ucing the class
-e
of infinitely divisible distributioJ1s, or types rather, that partially attract F and denoting
by F·T the r th convolution power of F, another consequence of Theorem 8 is the following
result.
COROLLARY 15. We have D;) (F)
= D; I (F·r)
for any r
= 1,2, ....
As usual ([14], p. 40) we say that the distribution functions G I and G 2 are of the same
type if G 2 (x)
= G I (ax + b), x
E IR, for some constants a > 0 and bE IR. Since the property
of being of the same type is symmetric and transitive, the set of all distribution functions
(of the cardinality of the continiuum) can be decomposed into mutually disjoint families
of distribution functions, each family consisting of distribution functions that are of the
same type, that:is, into mutually disjoint types. Of course the same can be said about the
subset of all infinitely divisible distribution functions, a type being determined by a triple
(tPI, tP2' a). Since if F belongs to the domain of partial attraction of an infinitely divisible
dist.ribution function t.hen it also belongs t.o the domain of partial attraction of any other
distribution function of the same type, the property of partially attracting an F is, as
tacitly used throughout above, the property of the corresponding infinitely divisible type.
By the remark on p. 270 in 16], the non-normal stable type with exponent 0 < 0: < 2 is
given by the triple ((Q)tPI,(u) tP2,0), defined in Corollary 5' above, while the normal type,
usually called the st.able type with exponent 2, is given by the triple (0,0, a), a > 0.
11
THEOREM 10. (i) If F belongs to the domain of partial attraction of only Olle
type, then this type must be a stable type witlJ some eXpOnelJt 0 <
0: ~
2.
(ii) If F belongs to the domain of partial attraction of a non-stable type, then it
belongs to the domain of partial attraction of continuum many different types.
The only reason to include the constants E>(tPd and E>(tP:t), defined after (1.26), into
the difinition of V (tPl, tP2, a) in (1.27) was to have the equality (1.30) in the given nice
form. The follwing result, the general convergence theorem for infinitely divisible laws, is
more transparent if we leave these constants aside, and hence introduce (d. (1.28))
lI(tPl,tP2,a) =V(tt'J,tP2,a)
+ 0(tPd - E>(1P2)
=VO,O(tPI,tP2,a) - tPdl)
= -
+ tP2(1)
/00 (Nds) _ s)dtt'J (s) - Jot Nds)dti'I (s)
(1.32)
I
+aZ
+
/00 (N (s) _ s)dtlJi(s) + JIJ(I N (s)dtl':ds)
2
2
I
As so far above, we assume in what follows, unless the contrary is evident from the text,
that all occuring tP functions (with subscripts and/or special marks such as
tV or ¢) satisfy
the usual conditions above and in (1.4). Let tPh tP2, {tPlk}~1 and {thk}~J all be such
tP functions, let a ~ 0, ak ~ 0, C E IR., and Ck E IR. be constants, and consider the functions
sHh) =
loo tP~ds)ds + a~ + loo tP~ds)ds,
h > OJ k = 1,2, ....
(1.33)
Then convergence is characterized as follows.
THEOREM 11. We have
(1.34)
if and only if
(1.35)
Ck --+
as k
--+ 00,
(1.36)
c,
and
S2:=
2=
lim lim inf s%(h) = a
lim lim sups%(h) =: 8
h-oo
k-oo
h-oo
k-oo
12
2
.
(1.37)
e-
The last theorem, based on Theorem 11, details convergence in distribution of sums of
independent and identically distributed infinitely divisible random variables and hence is a
counterpart of Theorem 6. Of course, the general theory based on the quantile function is
applicable to this problem in principle. However, it is more natural to base here everything
on the behavior of the corresponding t/J functions.
Let {n'} be any given sequence of positive integers tending to infinity, possibly the
whole {n}, and consider the infinitely divisible random variahle
an arbitrary triple (t/JI, t/J2' (1)
ai(n';h) =
1- (0,0,0). For a number
V (t/J], "1'2, (1)
belonging to
It ? 1, illlrodu<:e
[00 t/J~(ujn')du + n'o2 + J~OO tj,~(ujn')du
= n' {
r t/J~(.s)dS}'
lQ
roo
1h/n'
+ (12 +
1f1i(s)ds
(1.38)
1hln'
which is non-zero for all n' large enough. The role of the" natural" normalizing sequence
a(n') in (1.2) will now be taken over either by
al (n') = al (n';
1),
(1.39)
or by
a2(n')
= (n'{
+
roo roo (u
A
v)dtjJ du) d7h (v)
/00
roo (u
l/n,1 /n'
1/2
A
1
(1.-10)
v)dtl'Au)dt/J;!(v) } )
(ai(n') + t/Ji(ljn')
=
+ (12
11/n,11/nl
+ t/J~(] /n'))1 / 2,
where the last equality follows from Lemma 2.6 below. Furthermore, corresponding to the
first sequence, define
v~/ (h) = ai(n'; h)jai{n'),
h;::: 1.
(1.41)
Then the quantities
!L
2
= !L {n'} = lim liminfv~/(h)
2
h-oo
and
n'-oo
-2
= v{n'}
= I'1m I'Imsupv n2 , (h)
h-.oo n'-oo
we have 0 ~ !L ~ v ~ 1. While al (n') or a2(n')
-2
v
are welt defined and
will not always be natural, it will turn out that
#In'
(t/J], t/J2)
=!
= n'
n'
Udt/J2(ujn') -
{t
11/n'
J.
n'
udt/Jdujn')
sdt/J.As) _ {I
11/n'
13
Sd~l (S)}
in (1.39) and (1.40)
is always a correct centering sequence.
As before (L30), consider now an infinite sequence
of independent copies of (N I (.), Z, N 2 (·)) and let {V 1(1/1J, 'h, lJ)}~l be the corresponding
infinite sequence of independent copies of \' (1/11,1/'2, lJ) defined through (1.32). The convergence problem for partial sums of these is apparently determined by the behavior of 1/11
and 1/12 near zero. Note in this connection that if
then !!.2
= v2 = 1 for any sequence {n'}.
THEOREM 12. (i) If, as
n'·~ 00,
1/1j(s/n')/a.(n')
~ 0,
S
> 0, j =
1,2,
then
e(ii) If, as n'
~ 00,
1/1j(·/n')/a.(n')
~
1/1;(.),
j
= 1,2,
(1.42)
for some non-positive, non-decreasing, right-continuous functions 1/1i and 1/.'2' then, necessarily, both 1/1i and tPi satisfy (1.4) and there exists a subsequence {n"} C {n'} such
that
as n "
~ 00,
where!!.
~
lJ.
~
v.
If 1l
= v,
then this last convergence takes place along the
original {n'}.
(iii) If, as n'
~ 00,
(1.43)
14
for some non-positive, non-decreasing, right-continuous functions
V'i and 1/'2 and
(1.44a)
or, what is the same by (1.40),
(1.44b)
then again
tPi
and
1/'2 satisfy (1.4) and
(iv) If there is a sequence of numbers An' > 0 such tlJat
tPj(·/n')/A n ,
tPj(·),
=;.
(] .45)
j = ],2,
for some non-positive, non-decreasing, right-continuous functions
tPi
and
1/'2 and
-e
(1.46)
as n'
--+ 00,
then, necessarily, tPj(s) = 0 for all
S ~
1, j = ],2, and, as n'
-t
00,
(v) Conversely, suppose that for some constants An' > 0 and en' E lR,
(1.47)
where W is a non-degenerate random variable. TheIl with some constant c, IV is necessarily
of the form V(tPj,tP2,u.. )
+ c,
and we have
(1.48)
tPj(-!n')/A n ,
=;.
tI';.(.),
]5
j = ],2,
(1.49)
as n'
-+ 00,
and
(1.50)
Furthermore, if
JimsuPIt/Ji(s/n')I/adn') <
~ =
tlJen in the case when
n'
-+ 00,
v > 0,
° such that adn'}/A
there exists a finite 6 >
and wIlen V = 0 or !!. <
v,
(1.5] )
j = ],2,
00,
n' -+CX)
n • .. ~
6 as
there exist a subsequence {n"} C {n'} and a finite
6 = 6{n"} > 0, possibly depending on {n"}, sudJ that adn")/An"
-+
6 as n" -~
If, on
00.
the otller hand,
limsuPI,pds/n')I/adn')
~-oo
= 00
limsuplt/'~(s/n') l/aJ(n') =
or
(1.52)
00
~-oo
for some s > 0, then
adn')/A n •
-+
as
0
n'
(1.53)
-+ 00,
and lJence, necessarily, u. = O. Moreover, if (1.52) holds for some s > 0, but
limsupl,pi(s/n')I/a2(n') <
= 1,2,
j
00,
(1.54)
n'-+oo
tlJell t}Jere exist a subsequence {n"} c {n'} and a 6.=
a2 (nil) / An" -+ h as n" -+ 00. Finally, if
lim sup It/Jds/n')I/a2(n')
= 00
6{n"}
> 0 such that, besides (l.53),
lim sup 1t/J:l(s/n'}I/a:dn')
or
n'-~
n'-cx>
= 00
(1.55)
for some s > 0, which implies (l.52) for the same s, then
a2(n')/A n •
-+
0
as
n'
(1.56)
-+ 00.
We demonstrate Theorem 12 by way of some examples included in the next two
corollaries. In the first of these, for the sake of simplicity, we only deal with one-sided
examples without a normal component that have the form
/00 (N(s) - s)d,p(s) +1 N(s)d,p(s).
1
V (0, t/J, 0) =
In fact, since we always assume that 1t'{S} = 0 for all s 2: I, we will have
1
1
V,(O,t/J,O) =
N(l) (s)dt/J(s);
1= 1,2, ... ,
for independent copies N(J)(-), N(2)(.), ... of the standard Poisson process N(·}. Part
(a) below illustrates part (i) of Theorem 12. The two cases of part (b) when a 2: 2 still
illustrate part (i), while the third case when a < 2 is an example for part (ii). The case
of part (c) when
.x >
1 is an example for part (ii) , while the case when
.x
~ 1
is an
illustration of part (iv). The case a < 2 of part (b), used in conjunCtion with Theorem 9,
has an interesting application to be discussed in Section 3.
16
e-
COROLLARY 16. (a) As n
;.. {~1'
(b) As n
Ja:
- t 00,
N(I)(s)d(-y'logljs) -
V;"} ~o
N(O,I).
- t 00,
{~1' N(l)(s)d(-s-'/") - a'~ I } ~o N(O, I), if
2 ;..
1
y'nlogn
{t t
N(I)(S)d(-S-I/2) __
~}
a
> 2,
-to N(O,1),
2
I=JJo
and ifO < 0: < 2, then
V~
--;;--0:- n1
l
/
Q
{nf; Jot
N(l)(s)d(-S-I/u) - Iln(o)
where tJ1~(s) = -((2 - 0)/0)1/2 s -I/Q, s > 0, and
JLn(O:)=
(c) Let
tj
{
}
1<0<2,
nlogn,
0:=-1,
I-Q'
lr(O,tJ1~,O),
-
u~I'
n 1/..
-t[)
°< 0:
<.
1.
= 2- 2i , j = 1,2, ..., and
and define
-h
tJ1(s) =
{ 0,
Then
tk
~
s < tk-I, k
'
s ~ to = 1/2.
1/2
V(O,tJ1,O)
=
1
= 1,2, ... ,
00
N(s)dt/J(s) =
o
L N(tk)(bk+1 - bk),
k=O
r
where bo = 0, and if x 1 denotes the smallest integer not Jess than x, then for each>. > 0,
(1.57)
17
as k
--t
00, where
(l.58)
and where
v(k, A) =-
k- 1
{
'
if A < 1,
if A ? 1.
k,
Furthermore,
a~(f A2 l)
2k
I--t
a~(f A2 2" 1)
{ 1,
1.,
~
if A > 1,
if A $ 1, (1.59)
and if A > 1, then
(1.60)
while if A $ I, then
(1.61 )
as k
--t
00.
Let FVJI'VJ~'u denote the distribution function ofF(tI'hvJ~!,o). Part (v) of Theorem ]2
can be used to construct negative examples.
COROLLARY 17. If at least olle of ¢I and ¢2 decreases to - 00 so fast t/Jat there
exist 0 < s < t < 00 such that ¢J(s/n)/¢j(t/n) --t 00, as n --t 00, for the corresponding
j = 1 or j = 2 or both, then F VJ1 ,VJ~ ,u does not belong to the domain of partial attraction
I':
'
I':
•
FO ... VJ,o ,
. I ar, t h"JS JS t h e case lor
of any I aw. 1n partJcu
any 0 f t h e d'Jstru b
utlOn'
I unctIOns
where
QtJl(s)
and
Q
={
- exp((1 / s)u),
0< s < 1,
0,
s
2: 1,
is an arbitrary fixed positive number.
'
. . to FVJ1,'h,U . M any aut hors wor ke d
unctIOn
pertammg
L et QVJI ,VJ~,U b e t h e quantI'1 e f
on the problem of comparing the tail behavior of FIjJI'VJ~,u and those of the Levy functions
L(·) and R(·) belonging to tPl and ¢2 as determined below (1.28). (Cf. the references in
[91.) Our last corollary, following from a joint application of Theorems 5 and 12, gives a
result of such a flavor.
18
e-
COROLLARY 18. Suppose that F tP .,tP2'CT E D 1,(V>j,1J1:i,a.) for some t/.'j,1J1;., and
a. 2:: 0, where at least one of 1J1j and 1J1 2 is not identically zero, that is, the attracting
infinitely divisible law has a non-degenerate non-normal component. Then there exist a
subsequence {n'} C {n} and finite constants C1, C2 > 0 suc1J that if 1J1j
QtP.,!/J2'CT
and if tP;'
as n'
t
-+ 00,
(;,+) /1J11 (;,)
-t
Cit
0
<
S
t
0, then
< Sit
0, then
where
0<
Sj =
inf{s: 1J1;(s) = o}::;
j
00,
= 1,2.
We close this section by using the opportunity to correct a few misprints and an
oversight in [6]. In lines -10 and -12 on p. 263, the reference should be to Corollary 11
instead of Corollary 10. On p. 270, the minus sign should be deleted from before Q(I - s)
in (1.32). In line -3 on p. 292, R(snl' In., nil should be R(sn.ln., n d. The summation
index k in the definition of N' in line 8 on p. 297" should start with k
-e
k
= o.
=1
rather than
Finally, the bottom-line inequality on p. 314 is wrong, it holds the opposite way.
However, leaving this line out and referring to (1. 7) in the present paper instead, the proof
is correct as it stands.
2. PROOFS
Proof of Theorem 1·. We need only to show (1.6), the validity of all the statements
following this relation follows from the original proof of Theorem 1.
For any real h 2:: 1, define ~(h) = ~{nl} and a(h)
· . f a(h/n')
n'~oo a(l/n')
a (h) = 1ImIn
-
and
_
a(h)
= a{nl}
.
by setting
a(h/n')
= hm
sup ( / ')'
n'_~oo a 1 n
where a 2 (.) is given in (1.1). Since a(s) is a monotone non-incereasing function of s, we see
=1
that both ~(h) and a(h) are monotone non-increasing functions of h, 0:::; ~(h) ::; a(l)
and a(h) :::; a(l) = 1 for any It 2:: 1. Hence the limits in (1.7) are well defined. Also,
whatever is the sequence {Tn'} to be constructed such that Tn'
-t
00
as n'
-t
00,
the
monotonicity of a(·) implies that
liminfa((T n '
n' ---. 00
+ l)/n')/o(l/n') ::;
19
~
(2.1 )
and
IimSUpa((T n ,
n' .... oo
+ 1)/n')/a(l/n')
~
_.-
(') .,)
CT.
In order to prove the opposite inequalities, we have to go into the construction of
{Tn'} at the end of the proof of case (ii) of Theorem
1. Taking up the line in the middle
of p.292 and using the notation developped there with n 1 replaced by n' everywhere one
can construct a strictly increasing sequence {L} of positive integers such that for all s 2 1
we have
p( ma~ w~j)(L) - v~j)l) ~
s-l,
O$h$l.
-
-,
-
p(D.j(l s , sis, n ) - D.j(s, Is))
~
s
-1
,
and
sl/71'<s-1
s
_
,
for all
71'
2 71(S), where the positive integer n(s) is chosen so large that, together with the
e-
above twelve inequalities, we also have
1
Q - ~ ~ Q(sls
and
-
u(sls
+ 1) -
+
1
sls + 1
1
1) - ~ ~ a( n' )/a(n')'
1
sl d + 1
1
- ~ a(
, )/a( -)
s
n
n'
,
n 2 n(s),
for infinitely many
n'.
Clearly, we can choose these threshold numbers lI(S), s ~ 1, so that n(l) < 71(2) < ....
For each n' ~ n(l) from the sequence {n'} there exists an integer s = s(n') such that
71(S) ~ n' < n(s + 1). Now we simply set Sn' = s(n'), In' = ls(n')' and en' = l/s(71') for
any n' 2 n(I), and for the finitely many n' < n(l) we define these three sequences in an
appropriate but otherwise arbitrary fashion. Obviously, Sn'
as n'
---+ 00,
---+ 00,
In'
---+ 00,
and en'
---+
0
and together with the twelve inequalities in the last seven lines of p. 292 (with
the misprint corrected in the third line from below as noted above and with n' standing
everywhere in place of nil we also have
(2.3)
for each n' in {n'} and
(2.4)
20
for infinitely many n' in {n'}. Hence, with Tn' = sn,L n" we have all the conclusions of page
293 and, moreover, from (2.3),
Q ~
•
+ 1)/n')/o(l/n'),
Iiminfo((T n '
n'--+oo
and, since a(sn,L n, + 1) = a(T n ' + 1) is just a sequence in the set of the values of the
function a(h), h ~ 1, and a(lt) ! a as h - t 00, from (2.4),
a~
limsupo((r n ,
+ 1)/n')/o(1/u').
n'--.oo
These inequalities, together with those in (2.1) and (2.2), prove (1.6).
•
Proof of Theroem 6. In all theree case (i)-(iii), the results in the sufficiency direction
as stated at the end of the theorem follow directly from respective applications of Theorem
l' and Theorem 2, and hence we only have to deal with necessity.
Assume, therefore, (1.11) and that we are either ill case (i) or ill (ii). Then by Theorem
5 there exists an {n"} C
{ndr;l
such that
(2.5)
--
and
a(n " )/A n "
as n "
~ 00.
-t
6,
0~6<
(2.6)
00,
If {) were zero, then by Theorem 2 we would have
1
An"
{n"f; Xi - n"Jto,o(n)"}
where
Jto,o(n) =
j
-t f) '-'o,o(t/Ji, t/J 2, 0)
as
n
"~
00,
l-l/n
Q(s)ds,
(2.7)
n = 1,2 ... ,
lin
and Vo,o(-,·,·)-is defined in (1.9), and we would know that t/Jj(s) = 0 for 1 ~
j = 1,2.
S
<
00,
However, (1.11) also holds along {n"} and hence the convergence of types
theorem ([141, pp. 40-42) implies that, as n" -t
II
00,
II
{n Jto,o (n ) - en" }/ An"
- t "(
with some "( E IR and, therefore,
1
A-::n
{
n
"
"
n"
Jto,o(n)
?=Xi-
}
.
-tf)o:VO,O(t/Jl,tP2,O)+,B-"(.
)=1
21
Thus, if fJ were zero, we would have
which is impossible in either case in view of the uniqueness of the representation of an
infinitely divisible random variable in Theorem 3(ii), already mentioned in the introduction
in Section 1. Whence
fJ
> 0 in (2.6).
From (2.5) and (2.6), with 0 > 0, we obtain
II
1
tPj(n ,·)=>;StP;(·),
(2.8)
j=1,2,
and from (1.11), holding along {nil}, and (2.6),
(2.9)
as nil
-t
00.
Using (2.8), for any subsequence {7ta} C {nil} there exist by Theorem]' a
further subsequence {n4} C {n3} and some 0 ~
(J. ;;
I such that
(2.10)
as
n4 - t 00.
Now (2.9) and (2.10) together imply, again by convergence of types, the
existence of some "1 E IR such that
(2.11)
and hence
1
a(n4)
as
n4 - t 00.
{n.f;Xi - n4Jlo,o(n4) }
-tf)
13
~V;,,1I(tr"I,tt'2,(J) + 6
-
"1
(2.12)
Form (2.10) and (2.12), by uniqueness,
1/Jj/o = o.1/Ji/o,
j
= ],2,
and
(J.
=
o.(J/o,
"1
= {i/o.
Since {n3} C {nil} was arbitrary, by (2.6), (2.8), and (2.11) this means that (1.12), (1.13)
and (1.14) all hold along {nil} = {n~}k~l' and as nil
-t
00,
we have
(2.13)
22
e'
If tPl =
tP2 == 0,
then by part (i) of Theorem l' the limit in (2.13) must be standard
normal and hence in this special case we necessarily have 6
= o.a.
Also, {nil} can be
chosen as a subsequence of an arbitrary subsequence {n/} of the original {ndr=l along
which (1.11) holds and, since the limits in (1.12), (1.13), and (1.14) are the same for all
such {nil} in this special case, we conclude that (1.12), (1.]3), and (1.14) hold true along
the original
{ndk=l for which (1.11) is valid.
Let {n3} be again any subsequence of {nil}. Then by one more application of Theorem
1 * one can find a sequence {ns} C {n3} and au,
a(~s)
{t.
x; -
nSl'o,o(n s )}
°
~
u :s 1, such that
~D Vo,u(i .... ~"2,i1)
and
as ns
-t
00.
By (2.13) it follows that
Pn" := a((Tn"
--
u = o.a/fJ, and hence we see that
+ 1)/n " ) - t
o.a/6,
as
n"
-t
00,
since any subsequence of {Pn" } contains a further subsequence with the same limit. Thus
by (1.6) in Theorem I' we obtain (1.15) with {nklk:l replaced by {nil} = {n~}k:l' Hence
we have (1.12), (1.]3), (1.14), and (1.15) along {nk}k:l := {n~}k=l in both cases (i) and
(ii) .
Suppose now (1.11) and that we are in case (iii). Then we have (2.5) and (2.6). If
fJ > 0, then we can follow the above proof beginning from (2.8) and conclude that (1.12)-
(1.15) all hold along the same {nil} chosen at (2.5) and (2.6).
If we have (2.5) and (2.6) with 6 = 0, then by Theorem 2, for any subsequence
{n3} c {nil} we can choose a further subsequence {n6} C {n3} such that
Al {tXj - n6JlO,O(n6)}
n6
-f)
j=l
Vo,O(tPi,tP2'0),
as
71 6 --.. 00,
and of course we have (1.11) along {n6}. By convergence of types, as n6
and
23
-t
00,
Whence t/J; = ot/Jj, j = 1,2, and 'Y = /3. Therefore, along {n~}f=d := {n~}r=1 we have
(1.17), (1.18), and (1.19), and the theorem is completely proved. •
The proof of Theorem 8 requires four lemmas.
Let N (.) denote any of the two
standard Poisson processes N) (.) or N 2 (-) in (1.26) and <:ollsider n' (ti,) defined in (1.26)
by accordingly dropping the subscript j. Let N(I)(-), ... , N(r)(.) be independent copies of
N (.), consider the corresponding independent copies W (I )(l/J), . .. ,nl(r) (t/J) of ,"" (¢), where
r is an arbitrary integer, and recall the notation in (1.29).
Proof. Using the fact that the Poisson process has independent increments,
t.
w (I)(IjI) = J.~ (t.(NI/) (8) -
8)) dljl(8) + [
1.00
+ 10
=0
=
(N(rs) - rs)dt/J(s)
/00 (N(u) _ u)dt/J(r)(u) +
1
00
=
(N(u) -
i
(t.
Nil) (8)) dljl(8)t T6(IjI)
1
N(rs)dt/,(s) + r8(t/J)
r
N(u)dtP(r)(u) -+ r8(tP)
u)d~(r)(u) + fa) N(u)d¢(r)(u)
+
I
r
udt/J(r)(u)
+ r8(tll).
(2.14)
But a very elementary calculation, involving integration by parts, shows that
and hence the lemma.
•
Recalling now the notation in (1.27), we see that Lemma 2.1 implies (1.30) indeed.
LEMMA 2.2. If for some subsequence {n'} c {n}, constants An' > 0 and en' E IR,
(2.15)
then for any integer r
~
1, we have
as
24
n'
-+ 00.
(2.16)
Proof. Let {Xim)}~l be independent copies of {Xi}~l' m = 1, ... , T. Then by
(2.15) and (1.30) we obtain
{t, Ct. xt»
Aln'
•
as n ' _
00.
nTl't'lng
\1.7
- TCn.} ~v V(1/·l'>,1/,ld,y'ro)
Y I, Y2, .. ·.ort
l'
hesequence.l\l
v(I)
v(r) v(I)
""'-"1
,.1\2, •..
(2.17)
v(r)
,.1\2
I·
, ... ,tJatIS,
writing
Y
(j-l)r+m --
we have
X(rrt)
i
l:Sm:::r,
'
t (t Xi »)
m
}=I
=
m=1
(2.18)
j-=I,2, ... ,
:tY
i,
}=J
and since V}, Y 2 , • •• are independent with the common distribution function P, (2.17) IS
nothing but (2.16).
•
Let [.] denote the usual int.eger part function.
LEMMA 2.3. If (2.15) holds, then for any integer T 2: I,
--
-
1{I~l I}
L- Xj - -Cnl
r
AnI
(1/r)
-[) V{¢l
(1/r)
r
'¢2
,a/\'r)
as
n'
(2.19)
---+ 00.
j=1
Proof. If V}, Y2 , ••• are independent with the same distribution func.tion F, then
(2.15) can be written as
A1n {rIEr' Yj - C~} + Aln'
t
l
j=1
and Theorem 5 clearly implies that AnI -
j=rln ' /rl+ J
00 as n' -
Y ~v V('h,1/'2,O),
(2.20)
j
00. Therefore, since the second
sum here has at most r - 1 terms, the second term on the left side converges to zero
in probability as n' -
00.
Thus, breaking up the sequence {Yj}~J into the union of r
independent sequences {XJm)}~J of independent variables, m
=
1, ... , r, according to
the rule in (2.18), (2.20) can in fact be written as
t.
as n' -
(Aln.
{'~I Xj') - ;Cn.}) ~v t.v/(",I,/,>,,,,l'/rl,o/v'T)
(2.21 )
00, where, with appropriate independent copies of V (¢~l/r), ¢~l/r), a / JT), we also
used (1.30) on the right side in conjunction with the trivial fact that
this convergence obviously implies that in (2.19) . •
25
¢(J/r)(r)
=
tI'. Now
LEMMA 2.4. If (2.15) holds, then for any t",o integers T 2: 1 and I 2: 1,
1
-A
lin'
/1' I X
~
L.J
n{' .
IC
i - -
T
]=1
}
n'
-+D
V(.I (111')
'f'1
.1
(111')
,'f'2
,
VI
I}
-0
as
n'
-+ 00.
T
Proof. This follows by applying first. Lemma 2.3 and then Lemma 2.2, upon noting
that tP~l/1')(I)
= tPj'lr),j
= 1,2. •
Proof of Theorem 8. For two distribution functions G and H let
L(G, H}
= inf{e > 0: G(x -
e} - e
:s
H(x} S; G(x
+ e} + e
for all
x € Ul}
denote their Levy distance. It is well known that this distance metrizes the weak convergence of distribution functions on the line.
To prove the first statement, let
Tm
T T as
1ft -+ 00,
where
T
T rn
be rational numbers such that 0 . .::.
> 0 is any given number. Then, as m in each . s > 0, j
Tm
<
T
and
00,
= ],2.
Consequently,
e'
and, a fortiori, with G m and G denoting the distribution functions of the two sides
respectively, L(Gm,G}
-+
0 as m
-+ 00.
By Lemma 2.4, F E DJ)(tP1r"'),tPr"'),~0} for
each m 2: 1, and hence we can pick a subsequence {nk(m)}k:l and constants Adm} :> 0
and Ck(m} E IR such that for
x E IR.,
we have
lim L(FnA;(m), G m } = 0,
k--oo
m = ] ,2, ....
Hence we can choose a subsequence
such that
L (Fnk
1
(m),
fit
G m) ::; -,
m
26
m
= ], 2, ....
(2.22)
Using now the triangle inequality for the Levy distance ([141, p.33), we get·
I
L(Fnk ... (m)' G) S m + L(G m , G),
m = 1,2, ... ,
and hence the first statement.
In order to prove the second statement, we only have to redifinc G m and G to denote
the distribution functions of the left and right sides of (1.31), respectively, and note that
> 0, and Ck(m) E Ill, m
by Lemma 2.4 one can pick now {nk(m)}~l' Ak(m)
such that (2.22) holds again. Hence the same proof works.
= 1,2, ... ,
•
Proof of Theorem 9. By (1.30), the second condition means that for a subsequence
{nm}~::=1
of the positive integers and some constants am> 0 and
-
1 {V(."(n,,,)
.I,(n ... ) . ~ )
'f'1
,'f'2
,vnmo
am
as m
~ 00.
-Cm
}
~f)
Cm
E
IR.,
V(o',.1.+
)
'f'1,'f'2'0.
Hence the statement is a special case of Theorem 8.
•
Proof of Corollary 15. Let r 2' 1 be any integer. Then by Lemma 2.2 or Theorem
--
8, F·r E D p (t/J},t/.'2,0} implies that F·r E D,,(t/Jlr),t,l·~r), JT0), which in turn, as in
the proof of Lemma 2.3 (d. (2.21)), implies that F E D,,(t/JI, t,l'2, 0). This means that
D; 1 (F·r) C D; 1 (F). Conversely, by Lemma 2.3 or Theorem 8, F E D,,(lI'l, 11'2, 0) implies
that F E D p (t/Jp/r), t/J~I/r), 01 JT}, which, as in the proof of Lemma 2.4 (d. (2.17)), implies
For E: D,.(t/Jl ,t/J2, o}, that is, we have D; I(F) C D;I(F·r) . •
Proof of Theorem 10 (i). Let the triple ('IPI, tP2' o) represent the attracting type.
If tPl =
tP2 == 0, and
hence, necessarily,
° > 0, then there is nothing to prove for this means
that the attracting type is the normal type.
Suppose that at least one of
tPl
and
tP2
is not identically zero.
By Theorem 8,
(t/.,~T), t/J~T), Jio} also attracts F for any T > O. But by assumption all these belong to
the same given attracting type (tPl, tP2' 0). This implies that for any T > 0 there is a
constant
CT
> Osuch that
or what is the same by setting
tPj(US)
= autPj(s},
T
= l/u
and au
S> OJ au > OJ j
= 1,2,
and
U '
ol(auv'u}
= 0,
U>
o.
(2.23)
= auvtPj(s) and tP,(uvs) = autPj(vs} = auavV'j(s}, S > 0, j = 1,2, and hence
for any u, v > o. This is the multiplicative form of the Cauchy functional
Thus tPj(uvs)
a UII = aua ll
= l/cl/
27
equation ([1/, p.I7) and from (2.23) we also see that au is a non-increasing function of
> O. Hence au = u-P, U > 0, for some constant p ~ O. Setting dj = -t/'j{so) 2 0,
i = 1,2, where So > 0 is chosen so small that d. + d 2 :.- 0, we obtain from (2.23) that
tPj(sou) = -d j u-I',u > O,i = ],2, and putting finally u ~ sis,), we obtain
U
(2.24)
where Cj
=
djsol'. So far the side condition concerning the (J in (2.23) did not play any
role, nor did the square-integrability condition (].4) for the tf'j s.
Now if (J were positive, then from (2.23) we would get p
(] .4). Hence
(J
= O. Also, condition
(].4) implies that
p
= 1/2 which
= 1/0. for
would contradict.
some 0 < a < 2. Hence
tP2, 0), which, as noted before the formulation
of Theorem 10, represents the stable type with exponent 0.. •
the attracting type is the triple
(a)tPh (a)
The proof of part (ii) of Theorem 10 requires a lemma. Let 1/.'. and tf'2 be two functions
on (0,00) satisfying the regularity conditions above (1.4), hut not necessarily (1.4) itself,
such that at least one of them is not identically zero and wnsider the following condition:
There exist a set I with cardinality less than continuum and positive numbers
{Tk : k E I}such that for each T > 0 one can find a Tk, k E I, and a constant
(2.25)
c{T,k) such that c(T,k)tPj(TS) = tPj(TkS) for all S > 0, j = 1,2.
If for a given T >
° it is T
m
E {Tk : k E
I} that is given by this condition, then we
write T >- Tm . Setting Hk = {T > 0: T >- Tk}, condition (2.25) implies
UHk = (0,00),
(2.26)
kEf
and the required lemma is the following.
LEMMA 2.5. If condition (2.25) holds, then for each u >
au
° there exists a constant
> 0 such that
tPj(US)
Proof.
= autPJ(s)
for all
s > 0,
i = 1,2.
(2.27)
Condition (2.25) implies via (2.26) that there exists a kn E I such that
Hk u is uncountable. (Otherwise the cardinality of the half line (0,00) would he less than
28
continuum.) Now Tk.. mayor may not be in lJ k", but we never the less can fix a to E Ilk"
such that to
i' Tk".
Since the set
{(log t - log TkJ / (log to - log Tk,,) : t E
Ih,J
is uncountable, it cannot be a subset of or coincide with the set of rational numbers. Hence
we can find a tiE H k.. such that with
VI
= t,/Tk", 1 = 1,2,
10gvI/logvo = (Iogt l -logTk,,)/(loglo -logTko)
(2.28)
is irrational.
Now with the fixed constant c = c(t J, k u) > 0,
sIs
tP'(VIS) = t/J'(tl-)
}
} Tko
1
= -tP(Tk,,-)
= -1,'1(8),
C}
Tko
C}
S > 0; j = 1,2,
from which by ind uction
--
Similarly, with the fixed constant d
I
1
= c(to, ko),
tPj(vus) = dltPj(s),
S> 0;
j
= 1,2;
I:::. 1,2, ....
Hence for any k,1 = 1,2, ... ,
s > 0;
j
= 1,2.
Using now Kronecker's well-known theorem, it follows form (2.28) that the set {k log VI
Ilogvo : k,l
positive integers} is dense in IR, and hence the set {v~/v~J
:
-
k,l positive
integers} is dense in (0,00).
Let u > o-'be arbitrary. Then we can find two sequences of integers km,lm
~
1 such
that
Choosing now a sufficintly small So > 0 for which tPj(min(so, uSo)) =I 0 for at least one of
j = 1 or j = 2, for that j we have by right continuity that
29
Hence the limit
0< a u := lim d l ... /c k ... < 00
1n--00
necessarily exists, and we have (2.27). Since u > 0 was arbitrary, the lemma is proved.
•
We point out that the monot.onicity of 1PI and '1/'"2 is lIot used in the above proof.
Proof of Theorem 10 (ii). The condition means that F E D 1,(V 1,1/J2, a) for some
'
a 2: 0 and functions V'I and 1/J2 satisfying the usual conditions above and in (1.4) such that.
at least one of 1/Jl and
o<
0:
< 2, if a
V'2 is
not identically zero but (2.24) is not true for any p
=
I/o,
= O.
Suppose the conclusion is not true. This implies by Theorem 8 that, in particular, the
set of types given by {(1/.1~I/T),1/J~l/T),a/JT) : r > O} has <:ardinality less then contilluum,
which in turn implies that condition (2.25) holds in such a way that for the constant c(r, k)
found for r > 0 also satisfies (:(r,k)r/JT
S
> OJ
au
= a for each k E
> OJ
j = 1,2,
I. Lemma 2.5 then implies that
aa
-7,
and
QuV U
= 0,
U
:> 0,
for some constant a > 0, which is a version of (2.23).. Hence the argument following (2.2:1)
in the proof of part (i) implies that
o<
0:
0
=0
and that (2.24) holds with p
.=
I/o for some
< 2, and we have a contradiction. •
Aiming at the proof of Theorem 11, the following simple formula is very important.
LEMMA 2.6. If 1/1 is a function on (0,00) satisfyitlg tlle conditions above and in
(1.4), then for any c > 0,
11
00
00
1
00
(u /\ v)d¢(u)d1/J(v) =
tjJ2(u)du + et;J"2(e}.
Proof. By a somewhat lengthy but elementary computation we obt.ain that for any
e < t < 00,
.1 l
t
t
(u /\ v)d1/J(u)d1/J(v) =
l
t
1/J2(u)du 1 cflJ2(c) + ttjJ2(t}
- 2e1/.1(c)ljl(t) - 2t/l(t)
it
(2.29)
tjJ(u)du.
By (1.4), 1/J(t} ~ 0 and t1/J2(t) ~ 0 as t ~ 00, and also
O:S ¢(l) [
¢(u)du
:s I¢(tll
(I.'
:s (l¢'(ll [
30
du [
¢'(U)dU) 1/'
¢'(u)du) 1/' _ 0
e-
as t
-+ 00.
Hence, letting t ~
00
in (2.29), the lemma follows. •
Proof of Theorem 11. First we consider sufficiency. Let h > 1 be any common
continuity point of 1/.'1 and t/J'J. and introduce the decomposition
(2.30)
where
Vk(h) = -- [h(Nds) - S)dt/Jlk(S) -
JI
t
Nds)dt/Jlk(S)
Jo
1
1
+ /h(N 2 (S) - s)d1hds) +
+ (Ndh) and
1
N 2(s)dt/'2k(S)
h)t/Jlk(h) -- (N 2 (h) - h)t/J'.lk(h)
00
Zk(h) = -
+
{(Nds) - s) - (Ndh) - h)}dtPlds)
roo {(N
Jit
2 (s)
- s) - (lV 2(h) --
h)}d~J2d~)
+ (JkZ + Ck.
The basic motivation for such a decomposition is that, due to t he fact that a Poisson
-e
process has independent increments, \l k(h) and Z A:(h) are independwt for each k.
A very elementary probabilistic reasoning, based on the fact that. a Poisson process
has a fixed discontinuity at the point 1 with probability zero, one can show that by (1.35),
as k
~ 00,
Vk(h)
~p l'(h):= -
-1 Nd~)d1/.)ds)
1
fh(Nds) - s)dt/Jds)
+ fh (N 2(s) - S)dt/J2(S) +
I
+ (N I (h)
t
In
N'.l(S)llt/J 2(S)
(2.3] )
- h)t/JI (h) - (Ndh) - h)t/J'.l(h).
Furthermore, since (Ni(h) - h)/Vh has a limiting (standard normal) distrubution and
fJ-
V
ht/Ji(h)
-
-+
0 as h
-+ 00,
j = 1,2, by (1.4), we see that
(2.32)
On the other hand, since a Poisson process has stationary increments,
1
00
Zk(h)
=[)
Zk(h) := -
1
{Nds - h) - (s - h)}dt/Jlk(S)
+ (JkZ
(2.33)
00
+
{N 2 (s -- h) - (s - h)}dt/'2k(S) + Ck,
3]
where
Furthermore, if (N
(N} ('), Z, N2 (·)),
p)(-), Z(j), N~j) (')), j
= 1,2, ... , k, are independent copies of
then, substituting first s
= ku,
(2.3-t)
that is, Zk(h) is distributed as a sum of k independent and identically distributed random
variables for each k. Also, EZk(h) = Ck
~
c, as k
~ 00,
by (1.36) for each h > 1, and
using (2.34) and Lemma 2.6,
e(2.35)
Now exacly as in the proof of Theorem I' above, llsing (2.31) and (2.32), we can
consruct a sequence {hk} of positive numbers such that hk ~
00,
hk/k ~ 0, and
(2.36)
and, with
§.
and
s given in
(1.37) but otherwise not using condition (1.37),
(2.37)
Since by (2.30) we also have
(2.38)
32
with the two terms on the right side being independent for each k = 1,2, ... , and
(2.39)
the above considerations, (2.37), and condition (1.37) now applied yield
where the standard normal variable Z is independent of the two independent. Poisson
processes through which the limiting random variable in (2.36) is defined.
The last
convergence, (2.36), and (2.38) now give
as k
--t
00 and hence the sufficiency half of the theorem.
Now we t.urn to necessity, starting out again from (2.30) and knowing (2.33), (2.34),
and (2.35), where presently h 2:" 1 is arbitrary.
Suppose first that the sequence {sHh)} is unbounded for some h 2: 1 and/or {cd is
unbounded. Then, since Zk(h) = l'k(h)Sk(h)
-e
+ Ck,
~\'here
it follows by elementary probabilistic considerations that {Z k, (h)} is stochastically
bounded for this h along some subsequence {k'}
has a limiting distribution as k'
--t
c {k}.
UII-
By assumption V k,(h) I Zk,(h)
00, as it follows from condition (1.34) and (2.30),
and hence it is stochastically bounded. As another set of elementary probabilistic considerations shows, this can only happen if {V k,(h)} is also stochastically unbounded.
Hence both sequences {Vk,(h)} and {Zk,(h)} are stochastically unbounded and Vk,(h)
and Z k' (h) are independent for each k'. However, this easily implies that the sum sequence {V k' (h)
+ Z k' (h)}
is stochastically unbounded and this, via (2.30), contradicts the
assumption in (1.34). Thus we conclude that both the sequence {sHIl),h 2: I} of functions
and the sequence {Ck} of constants are bounded.
Since
siO
is a non-increasing continuous function on 11,00) for each k, for each
subsequence {k'}
c {k} we can choose a further su bsequence {k"} c {k'} such that
(2.40)
for some non-increasing function s~O on [1,00) and a constant c~ E IlL Then by (2.33),
(2.34), and (2.35),
as
33
k"
~
00,
for each continuity point h ~ 1 of s:(-). This, the independence of the summands in (2.30),
and (1.34) now imply that
as
k " -,
00,
where W(h) is a proper, possibly degenerate random variable for each continuity point
h ~ 1 of s:(-). This could not happen if we had tPjk" (s)
j = 1 or j
= 2,
because this limit would then be
-00
-00 for some s = Sli > 0 and
for all 0 < s ::S So and then, ~;ince
again the sum of the three terms in V k" (h) that involve
three terms that involue
tP2k"
-t
~)1k"
and the sum of the other
are independent, the sequence {V k" (h)} could not even be
stochastically bounded if h is large enough. Therefore,
IimsuPltl'ik,,(s)1 <
00,
S
> 0,
.J:;:: ],2.
k" ...... 00
Then along a further subsequence {k"'} C {k"},
tPjk"'(')
for some
tPi
and
tPi.
=?
l,b;(.),
j.= ],2,
(2.4 I )
on (0,00). Since- we can clearly assume without loss of gen('rality that
{k"} above has been chosen in such a way that all three terms in S~II (.), given in (1.33)
or (2.35) converge separately, that is, O~II -~ iJ2,
for some non-increasing functions 'PI and 'P:I. on (0,00) as k"
lemma that both
tPi
and
tPi
-> 00,
it follows by Fatou's
satisfy (1.4).
Now, by (2.40), (2.41), and the already proved sufficiency part of the theorem we have
as
k
,"
->
00,
where
o· = lim s. (h) < 00.
h-oo
Then by (1.34) and uniqueness,
tP; = tPj,
j = ],2, and c' = c, and since {k'}
c {k}
was
arbitrary, we conclude that (1.35) and (1.36) hold true.
Starting out finally from these conditions (1.35) and (1.36), the sufficiency proof
provides a sequence {hk} such that hk
-4
00
34
and hk/k
->
0 as k
->
00,
and (2.36)
e-
(2.37), and (2.38) are all in force. Now if we had §.2 < 8 2 , then along one subsequence {k'}
for which s~, (hk') -
we would have
§.2
U
•
and along another subsequence {k } for which
siu (h u)
k
----+
which, in view of condition (1.36), is impossible. Hence ~2
common value must be
0"2,
s2
we would have
= 82,
and by uniqueness this
that is, (] .37) is also satisfied . •
Proof of Theorem 12. Everything is based on the following distributional equality,
following from (1.30) and (2.14) in the proof of Lemma 2.1:
n'
~
L
V 1(.,..,
)
'1-'1,,/-'2,0" =v
(.1.
1T
(TI'
t'
'1-']
)
1(TI') ,Vr,)
n 0"
,"P2
+ I1n' (.1,/-'I,tI'2 ) .
(2.42)
1= 1
(i) Setting tPjk(')
-e
= tPtU(-)/adn~)
and
= 0,
Ck
k
=
],2 ... , j =--01,2, where we
obviously denote the k-th element of the sequence {n'} by n~, the st.at.ement follows from
Theorem II if we note that, with the present !/Jjk,
lim (h tP;ds)ds
k--oo
11
=0
for each
h 2: ],
j
=.
1,2,
the constant SUP{t/J;k(l) : k ~ I} < 00 being the common integrable rnajorant, and hence
since h can be replaced by 1 in the latter sequence in the curly braces and then we have]
for each k . •
(ii) First we have to show that
vJi
and
tP2
satisfy (1.4). Let] < s < t < 00 be
arbitrary and choose s' and t' to be continuity points of both
1
vJi
and
V'2
such that
< s' < s < t < t' < 00. Then for a standard left-continuous Poisson process N(·),
using (1.42),
1,
t'
-1,
t'
(N(u) - u)dtPj(u/n')/ad n')
35
(1V(u) - u)dtPj(u)
almost surely as n'
11
t
-7
Hence by Fatou's lemma and Lemma 2.6,
00.
1, 1,
1,
t'
t
(u
1\
v)dtJ1j(u)dtJ1j(v) S;
E
=
s;
t'
(u
t'
(
1\
v)dtl'j (u)dtl'j (v)
(N(u) -
l~~~ E (
1,
u)d~J; (u)
) 2
•
t'
) 2
(N(u) - u)dtJ1J (u/n')/a.(n')
.t'ln' t'ln'
..
J~'ln' f~'ln' (u 1\ v)dtl'} (u)dtl'j(v)
= hm mf fOO
.
fOO
.
n'-oo lin' 1;..r(u)du + 0- 2 + lin' 1/J~(u)du
..
fl7n' tJ1J(u)du + 1/}(I/n')/n'
00
lin' ti't(u)du + 0- 2 + fl/n' 1/Ji(u)du
< hm mf fOO
- ,n'_oo
S; 1 + (tJ1j (1- ))2
for j = 1,2, where in the last step we again used (1.42). Thus for any I < s <
00,
(2.43)
Also, again by (1.42) and Lemma 2.6, for any continuity point s 2: 1 of 1/1; ,
s(tJ1j (s))2 = lim st/J;(s/n')/al (n')
n'--9OO
oo
.
f , f~7n'(u 1\ v)dt/Jj (u)dtPj (v) -- J$O:n' t/I;(u)du
= lun s /n
I
_
2
n'-oo
fl7n' tl'Hu)du + 0- + fl7n' t/J~(u)du
.
< hmsup fOO
-
n'-oo
S; 1
fl7n' t/J;(u)du
+ tP;(1/n')/n'
2
+ 0- 2 + foo
I/n,1/J 2(u)du
2
l/n,tP l (u)du
+ (tPj(I-))2
and hence
tP;(s)
-+
0
as
s
-7
00,
j = 1,2.
By routine manipulation based on (2.29), this and (2.43) together imply that Loth
tPi and
tP 2 satisfy (1.4).
Using now (2.42), (1.42), and the fact that the s~(h) of Theorem 11 belonging to the
present tP}'k(')
= tP~nU(-)/a.(n'k)'
j = 1,2,
J
is the same as v n2, (h) of the present theorem,
1
the second and main statement follows from Theorem 11.
36
•
(iii) If s > 1, then by (1.43) and (1.44)
1sr
...
XJ
1
\\lC
obtain similarly as in case (ii) above that
00
.:l
(u /\ v)dt,l,;. (u)d¢j (tI) S (¢j (I - )r~ < 00
and
•
Hence (1.4) follows again for both t,I'i and
s~(h) =
t/.'2.
Using (2.42), (1.43), and the fad that
a!(n k)v 2 , (h),
a:i (uk) n k
h:::- I,
the main statement follows from Theorem 11 on account of (1.44).
•
(iv) For any s 2 ], using the mOIJotonicity of the t,I,), the formula (1.40), and condition
(1.46) ,
ItPj(s/n')1 _ l¢j(s/n')1 a2(n')
An'
a2(n')
A fI'
< ItPj(s/n')1 a;!(n')
- ItPj(]jn')1 An'
::; aAn')jA n , - 0,
as n'
-+
j
= ],2,
00, so that the first statement follows. Using (2.42), (] .45), the realation
h?::
and the fact that (1.46) clearly implies that adnk)/AfI~
statement follows again from Theorem 11.
I,
~ 0 as k
--0
00,
the second
•
(v) That the limit is of the stated form follows from Theorem 5, and then (1.48),
(1.49), and (1.50) all follow from Theorem 11 via (2.42).
Suppose now (1.51).
Then for any subsequence {n3} C {n'} there is a further
subsequence {n4} C {n3} such that
(2.44)
as n4
00, where tPl and tP2 are some functions with the usual properties above (1.4).
Using now part (i) or part (ii), we see that there exists a further subsequence {nil} C {n4}
-+
such that
37
has a non-degenerate limit as nil ~
Hence by the convergence of types theorem ([141,
00.
> 0 such that adn")/An" ~ 6. In the special case when
:!l = v = 1I > 0, it follows from (] .50) that a. = 6v > O. Since a. and v are determined
by the whole original sequence {n'}, tllis implies that /) must be t.he same for all such
subsequences {nil}. Since {n3} was an arbitrary subsequence of {n'}, al (n')/A n • ~ () > 0,
ppAO-42) there is a () =
as n'
--t
00,
{){n ll
}
in this special case.
If we have (1.47) and (1.52), then the same (1.52) must be true for all 0 < t < s, and
hence (1.53) follows from (1.49) easily. Also, (1.47) and (1.55) imply (1.56) in the same
way.
If (1.47), (1.52), and (1.54) hold, then we have (2.44) with a.(n4) replaced by a:l(714),
and our statement concerning the convergence of
a2 (n")
/ An" follows by repeating the
argument below (2.44). The theorem is completely proved.
•
Proof of Corollary 16. All the asymptotic equalities below are eit.her obvious or
obtained by elementary calculations.
(a) Clearly,
ai(n) =
-nJI
logsds .....
7l,
I/n
so that
0<S<71,
and we have the condition of part (i) of Theorem ]2 as n
--t
00,
and t.he statement follows
with the centering sequence
But
as n
--t
00,
and hence the statement.
(b) Presently, for any h ~ 1 and
•
Q
> 0,
_...!L n
2 '
Q
> 2,
n log n,
Q
= 2,
0.-
ai(n;h) = n {I s-2/tJlds .....
lh/n
{
-.lL
2-u
38
(!! )2/tJI
h
'
o '-. 2,
•
and
n
..
•
{
Hence, as n
0=1'
a> 1,
nlogn,
a
nIl'"
1- Lt
0: <.
'
= 1,
1.
-+ 00,
0< s
<.
n,
s > n,
? 2, and we have (1.42) for j = 2 if
limiting function t/J;. Finally, note that if Q < 2,
converges to zero for each s > 0 if
and this implies that
Q
=
Q
Q
<.
2, with
v = o.•
(c) Using the fact that tk/tk- 1 , 0 as k
-+
00, and setting
for each A :> 0, by elementary manipulations we see that for all k large enough,
s < A,
s 2: A.
Since bk/bk-t 1
-~
= bk+ I
0, upon setting Ak
we obtain
S
< A,
S::::: A,
as k
-+ 00.
The formula for
given in (1.58) is valid for all k large enough and follows by simple computation just as
the relations
(A - 1)2:l
ai (nk)
--
A2
2k
1
-
Hl
2 2"
if A > 1,
,
if A < 1,
,
39
and
2
a2 (nk) --
{
A2
2H1
ai(71k),
if A> ],
,
if A :S 1.
..
Hence (1.59), (1.60), and (1.61) all follow, and we obtain (1.58) from part (ii) or part (iv)
of Theorem 12, according to A :> 1 or A ~ ], in view of t.he fact. that by (] .32),
V(O,tP~,O)
= Vt).II(O,tP~,O) + tP~(])
= N(A) - A/(>'
where the second equality follows from equation (2.2) in
131,
,>
•
I),
also obtained by simple
computation.
Proof of Corollary 17. For any subsequence {n'} C {n} we still have
V'j(s/n')/tPj(t/n') ~ 00 as n' ~ 00, and hence it is impossible to find a numerical sequence
{An'} such that (1.49) could hold with finite limiting functions.
3. DISCUSSION
As said in the introduction, the present paper is an "organic" continuation of
161, the
results of which have been discussed in detail in Section 4 of [6]. A<:cordingly, Theorem I"
here is new. It leads to Theorems {) and 7, which in their full generality and detail may
be considered to be new as well, together with Corollaries 5', ]3 and 14. Many similar
corollaries can be worked out routinly for the characterization of the domain of partial
attraction of given concrete infinitely divisible distributions. As mentioned in Section ],
this problem for the Poisson distribution turns out to be intricate, and the surprising
solution is given in [3].
Another application is the following. In a very interesting recent paper 1]5], A. MartinLor proves a limit theorem along the special subsequnce {2 k }k=I' which theorem "clarifies
the Petersburg paradox"" By Theorem ]0 (ii), however, there might exist continuum many
very different clarifications. In [5], we investigate the problem how unique is the one given
by Martin-LoLWe construct all possible subsequences aJld describe all possible limiting
laws, and finally conclude that Martin-Lof's clarification is unique mod ulo a constant
factor. The inclusion of this factor has the practical consequence that the game can be
played arbitrarily and not just in blocks of size 2 k as suggested by Martin-Lof's particular
clarification. This paper 15] has the additional didactic aspect that it also illustrates all
the results from
171
and
[Sl
on the Petersburg game.
A characteristic-function version of Theorem S was first proved by Gnedenko
1]31
in
] 940 by purely Fourier-analytic methods. In contrast, our proof is purely probabilistic,
40
e.
based on the representation of an infinitely divisible random variable. From his theorem
Gnedenko 1131 also deduced what is Corollary 15 here and his transitivity result in Theorem
9 (not called by him as such), the latter of which is also cited without proof in [141, p. 189.
In the proof of Lemma 5 in [61, concrete constructions are given to show that the
domain of partial attraction D,,(a) of a stable law wit h exponent 0 < a < 2 is wider
than its domain of attraction D(a). This was first proved independently by DoeLlin
IJ] I,
published also in ]940, and Gnedenko 1]31. Both proofs an' non-constructive. Doeblin uses
characteristic functions, while Gnedenko's interesting proof is based on the transitivity
theorem. In our setting, Gnedenko's proof is as follows. Consider any F in the domain of
partial attraction of the distribution
y,,~.. ,o
of V (O,:;J;'A' 0), where, for 0 <: a < 2,
_t/Ju(s) '-- {-S-I/U,
0<
S
,.1,
s ? I.
0,
By Khinchin's theorem ([14], p.184, or IlOj) such an F exists. Then by Corollary IU (b),
FJ,~ . ,0
is in the domain of partial attraction D,.(o.) of the distribution of \7 (0, t/J;.., 0), and
hence by transitivity (Theorem 9), F E D,,(a). However, F cannot be in the dO(Jlaill of
attraction of this distribution exactly because it is partially att racted to F.I,tP.. ,Il. \\'hile
-e
this proof is not constructive, it is certainly much simpler than the one given by K. L.
Chung in his footnote on p. 189 in [141, also based on Gnedenko's transitivity.
Theorem ]0 (i) was proved by Gnedenko [131 with the dlaractcrbtic-fuJH:tion method
just as that weaker version of Theorem 10 (ii) where "uncoulltable" stands in place
of our "continuum". This weaker version of Theorem 10 implies what Cnedenko and
Kolmogorov [141 write on p. 189: "Each distribution law F belongs to the domain of
partial attraction of one or a nondenumerable set of types or else does not helong to any
domain of partial attraction at all." This conclusion was also achieved by Doeblin 1111.
(Note that distributions not in the domain of partial attraction of any law do exist by
Corollary] 7 here.) At this point an historical remark is perhaps tolerable. Doeblin and
Gnedenko appear to be compet.ing on these results at the time. Doeblin announced his
results, proved in [111, without proof in the Paris Comptes Rendus in two communications
already in 1938 (Vol. 206, p.306 and p.718).
However, Gnedenko 1]31 states that he
obtained his results in the spring of 1937 and they were part of his dissertation that he
has defended in that year. This "competition", if there was one, is of course not surprising
with Levy standing behind Doeblin and with Khinchin and Kolmogorov behind Gnedenko.
To the best of our knowledge, the present unimprovable version of Theorem 10 with
the "continuum" is new.
The original weaker or "uncountable" version would require
41
Lemma 2.5 under the stronger assumption that condition (2.25) holds with a countable
set I. It is perhaps interesting to note that the first proof that we had for this weaker
version of Lemma 2.5 started out from the Baire category theorem applied to (2.26). The
extension of that proof under the present weaker condition (2.25) is impossible under the
usual Z Faxioms of set theory since the corresponding extension of the Baire category
theorem is known to be an independent axiom just as the continuum hypothesis. Hence
the problem of extending the original Doeblin-Gnedenko result appeared to be 'one of those
set-theoretic problems' at first sight. However, the present proof of Lemma 2.5 wmpletely
bypasses all these problems and is in fact much shorter than the first one was.
The essence of Theorem 11 was already remarked upon in the introduction. We believe
that the present formulation is cleaner than the original characteristic-function version,
first proved by Gnedenko /12], and also that the present proof, based on the fact that a
standard Poisson process has independent and stationary increments, really uncovers the
ultimate reason behind it.
Theorem 12 is new with all its details. The const.ruction in Corollary 16(c) is a
version of a wnstruction, a different one with different bul. similar purposes, given in
A modification of the present construction will be given in
141,
131.
where again the intricacies
of a Poisson limit are treated in more detail and a. case. also illustrates part (iii) Gf
Th~­
orem 12.
Corollary 17 is new in its generality. The special case of the concrete example when
o
= 1 corresponds to a half-sided version of the example of Gnedenko and Kolmogorov 1141,
pp. 186-189, showing by a rather complicated
characteri~t.ic-function proof
the existen(:e
of a distribution not in the domain of partial attraction of any law at all. Examples similar
to theirs were constructed independently by Levy, Khinchin and Gnedenko in the years
1937-1939.
Corollary 18 is also new.
ACKNOWLEDGEMENTS
This paper has grown out of joint work with Rossitza Dodunekova of the University
of Sofia on [3]. I thank her also for her comments on the manuscript of this paper. I am
also grateful to Vilmos Totik of the University of Szeged for his help in the proof of Lemma
2.5 and to V. V. Vinogradov of Moscow State University for sending me a copy of [131.
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42
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•
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e
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•
43
J. Appl. Probab.