*
This research was supported by the Office of Naval Research under
Contract NOOOl4-67-A-032l-0002.
t
.
Deparenent of
Canada.
~~thematics,
University of Montreal, Montreal, Quebec,
Reproduction in whole or in part is permitted
for any purpose of the
United States Government
FUNCTIONS AND A
DENSITY VERSION OF THE CEIITRAL LIMIT THEOREM*
GEI~ERAL r~ONENT
Walter L. Smith and Sujit K. Basut
DepaI'tment of Statistias
University of North Cal'OUna at ChapeZ Bitz
Institute of Statistics Mimeo Series No. 847
October, 1972
.-
GEl1EJAL HOlIENT FUHCTlOiJS AND A
DENSITY VE;U;IOl'l OF THE Cl.:JTRAL LEUT THEOREH
by
Walter L. Smith* and Sujit K. Basu
•
§
1.
INTRODUCTION
Let Xl'
such that
':-:"i
x2 , ..• ,ad info
=0
and
be an infinite sequence of
iid
random variables
r:~~ = 1. It is wEll known that
• zn ,
tends, in distribution, as
n
+ ~
cf1(x)
to the normal distribution with p.d.f.
=
This is the famous Central Limit Theorem.
to be able to claim that the
that is, in general, false.
say,
p.d.f.
In some situations it is desirable
of Z tends to
n
cf1(x), a conclusion
An important theorem on this subject is the
following one to be found in Gnedenko-Kolmogorov (1968, p. 224):
Let {Xn } be a sequence of iid random variabZes with
2
EXn = 0 and EXn = 1. Let Zn = (Xl + X2 +... +Xn) lin. If~ for some
integer m ~ 1~ Zm possesses a density that is integrabZe in the r-th
THEOREM A:
power
(1
< l' S 2)~
then for aZZ Zarge n
is absoZuteZy aontinuou8 with a p.d.f.
*
~
the d.f.
Fn~
say~
of Zn
f n 8uah that
This research was supported by the Office of Naval Research under
Contract N00014-67-A-032l-0002.
1
2
f n (x)
uniformly in
XoJ
< X <
_ClQ
-+
<P (x)
as- n
-+
ClQ
oJ
+ClQ
A rather different theorem directed to the same question was given by
Smith (1953):
THEOREM
Let
B:
{X}
n
be a sequence of iid random variables with
2
Ex = 0 oJ EX = l and Z as before. Suppose that Elx II' <
fora some
n
n
n
n
integera I' ~ 2 and suppose that fora some aonstant a > 0 and large raeal
Ee itXn = O( It I-a ). Then Z has a density f
for aU large n and
n
n
ClQ
,lxl Y
uniformly in
X oJ
-
ClQ
Ifn(x) - <P(x) I
<
X
<
+ClQ
oJ
-+
0
for aU·
oJ
Y
as n
-+
ClQ
t
oJ
suah that
O:s;
Y :s;
r.
Be it noted that Theorem A imposes a weaker condition on the distribution
of the
{X} than does Theorem B.
On the other hand, granted the condition
n
in Theor.em B on the characteristic function, the conclusion is more informative and, possibly, more useful.
In particular, Theorem B has enabled
Cox and Smith (1954) to prove some quite general renewal theorems.
The purpose of this note is to prove a Theorem which embraces the ideas
of both the above theorems and even allows one to introduce quite general
kinds of moments of the
{X} rather than the moments of integral order in
n
Theorem B.
DEFINITION: A ;tunation
11 (x)
Lelon[.' to the· cZaeE
of
i)
ii)
iii)
M(x)
M
is non-deareasing
~ is non-inaraeasing
x
H(O) > 0 •
x, dej'ined for ,. ~. 0 wiZZ be oaid to·'
oj'
oJ
oJ
oJ
3
In Smith (1969) some properties of this class of functions are investigated
in detail.
Typical examples of such functions may be ones which (as x
~ ~)
asymptotically equal
If
M(x)
€
M,
<S
a)
x
b)
a constant.
and v
~
0
log x
is a real number, we shall write
M • Elx IV
v
when this moment exists.
with 0 < 0 < 1 ,
n
M(IX
n
I)
We shall write
for the familiar absolute and ordinary moments respectively.
Our theorem
is then:
{X } be a sequence of iid random val'iabZes with an
n
2
abs.oZuteZy aontinuous d.f. ~ EX = 0 and EX = 1. Suppose M(x) € M. Then if
n
n
THEOREM I.
Let
i)
ii)
it fo Uows that
as n
~ oo~
M <
v
~
fop some
foro some m
2:
2
(so that
A
v
< 00) ~
~ 1
the sum X1+' ,,+X
has a p.d.f.
m
whiah is integroabZe in the :r>-th powero (1 < p S 2)~
Z = n--l- (Xl+"'+ X)
n
n
unifomZy in x
v
~
_00 <
x
has a p.d.f. suah that
fop any
< ~
y~
0 s
y s v •
The usefulness of this theorem lies in the fact that it enables us to
write, when suitable conditions for the theorem apply,
f (x) ... <p(x)
n
r
(x)
+ _ _n
_
;L + Ix IvM( Ix I>
4
where
r (x)
n
~
0 as n
~
uniformly in x,
00,
_00
< x < +00.
This result can
be a useful tool in theoretical investigations, and it is fortunate that
it makes full use of whatever knowledge is available concerning the existence of quite general moments of the random variables.
It might be noted that condition I(ii) can easily be shown to be
equivalent to the following alternative condition:-
_00
* (ii)
= Ee itXn
is the characteristic function of the {Xn }
< t < 00 ~ then for some p > o~ {w (tn P is in L (_oo~ +(0).
1
In presenting Theorem I we have supposed the {X } have a p.d.f. This
I
If w(t)
n
is not assumed in Theorems A and B; their conditions merely ensure that
Zn has a p.d.f. for all sufficiently large
n.
We have chosen to assume
a p.d.f. for the {X } to simplify in one or two places an already complicated
n
proof.
Condition l(ii) will, as in Theorem A, ensure that Z has a p.d.f.
n
for all large n, we leave it to the reader to note the places in our proof
where, at a slight increase in complications, we can avoid our 'lsimplifying"
assumption.
Be it noted that considerable ingenuity must be expended to
produce an example in which the {X } do not have an absolutely continuous
n
distribution and Z , for some
n
§2.
n, does.
HOTATION AND PRELIi\'jHlARY LEI-MAS
Throughout this paper {X } will be a sequence of iid random variables
n
itXn
with a p.d.f. f(x), and c.f. w(t) = Ee
, - 00 < t < +00. We shall set
Zn
and write
Q (t)
n
(2.1)
as is well known.
= Ee
itZ n
•
n (t)
n
=
X +X 2+·· ,+X
n
l
10
Thus
= {w(t/In)}n
,
5
If
r
~
is a positive real we write
= fix n Ir
moment (when it exists) and ~
For each integer
n
r
and real
x
= fx nr
r
for the familiar ordinary
for the absolute moment.
we define
(2.2)
(2.3)
(2.4)
A (t,x)
= {a n (t/~,x)}n
(2.5)
B (t,x)
= nn (t)
t~enever
set
n
n
- A (t,x)
n
the following inversion integrals are absolutely convergent we
t
f
(2.6)
n
(u)
= 21T1 f~
_co
a
(2.8)
1
b (u,x) = 21r
n
for each integer
n
~
(u,x) =...1.
n
21T
(2.7)
and real valued
f-co
e- itu
u
dt ,
e- itu A (t,x) dt
n
+00
f-co
nn (t)
e
and
-itu
B (t,x) dt
n
x.
If
integer and the following derivatives with respect to
r
,
,
is a positive
t
are known to
exist, we denote them thus:
a (r) (t x)
n
'
a (t,x) ,
n
A~r)(t,X) =
t These absolutely convergent integrals provide the aontinuous p.d.fis that
we shall use, and which are the subjects of our theorem.
6
and so on.
We also write
p(x)
= p{IXn I
> x} and q(x)
=1
- p(x) for real x > O.
We write i'h(x) e· L (r) 11 to mean
I
+00
I~(x) ,r dx <
00
•
_00
~(x)
FinallYt
and N(t) will denote the p.d.f. and the c.f. t respectively,
of the standard normal distribution.
We are now ready to prove some basic lemmas.
LElvi,1A 2. 1 For any integer
l'
=
L
j=l
l'
s n,
L~
J
••• ex
where the summation
of the integer
l'
LJ
(1' .)
n
J
(t,x) [ex (t,x)]
n
n-j
extends over the j-part partitions (r1,r2,···,rj )
and the coefficients
C (r ,r , ... ,rj ) depend onZy on
r 1 2
r 1,r 2, .•. ,rj and are independent of n.
Note that n[j]
= n(n-l) ••• (n-j+l)
and that rltr2t ••• trj may not be all
distinct.
PROOF~
The lemma follows from the r-fold differentiation with respect to
t of (see (2.4»
the equation
An easy combinatorial argument shows that
r!
x
n!
from which the claim made for the coefficients C is seen to be valid.
r
7
LEMI4A 2.2
For eaah fixed x ~ 0, and aZZ reaZ t, if ~2 = 1 and ~l = 0,
2
exp (_Jat ), as n -+ co. Moreover, this aonvergenae is unifol'f1l
A (t,x) -+
n
UYith respeat to x
in I~ I ~ 1.
2
PROOF: By the Central Limit Theorem, nn (t)
-+ exp(-~t)
Thus, from (2.5), we need to show
0, uniformly for Ixl ~ 1.
Since
~
<
2
B (t",x)
n
-+
as
n
-+ co •
co
(2.9)
nx
uniformly in Ix I ~ 1.
2 p(lxllO)
-+
0 , n
-+ co ,
Also
s exp{np(lxllil)} -
1
The right hand member of this inequality tends to zero uniformly in
-Ix I
~ l, by (2.9).
Ixl ~ 1
Thus the lemma is essentially proved (clearly the set
can be replaced by
Ixl ~ 6 for any fixed arbitrarily small
6 > 0).
LEi':U'1A 2.3
If 1J 2
for aZZ reaZ
t
=1
and III
=0
and real- fixed :x
then for eaah fixed integer k
~
o.
PROOF: Fix x 1 0 and real. Since
a (t,x)
n
=J
lulslxllil
e itu feu) du ,
8
it is evident that
4It
~
n
(t,x) is an entire function in the complex t-plane.
Furthermore
~(1)(0 x).
f
i
n'
u f(u) du
lulslx/Iil
= -i f
u f(u) du
lul>lxllil
since
~l
• 0.
Therefore
f
u f(u) du
lul>lxllil
1
Ixllil
if we appeal to the fact that
~2·
1.
One can also obtain the equation
~(2)(t,x)
n
so that, for any
u 2 e itu f(u) du ,
= -f
lulslxllil
K > 0, provided' lIt I s K/ Ix 110
But
~n (t,x) • ~ n (O,x) - t~(l)(O,x)
n
f
t
°
(z-t) ~(2)(z x) dz
n"
where the contour integral can be taken along a straight line joining
to
t.
Thus, provided
Iltl s K/lxl ,
I~n [Iri'x
t
J Sl+.-kL+J..U:.l.
nlGtf 2n2
°
9
'e
It follows that in the intersection C , say, of the circle It I S L,
KL
for some L > 0, and the strip IItl S K/lxl, we have
IVt,x) I
<
S
[1 +nTh + ~:~T
'
D(lxl> , say,
where
Ll
I I
D( x ) • exp(rxr
+ 2'
L
2 K
e) •
Therefore, for each x ; 0, {A (t,x)} is a sequence of entire functions such
n
that~
(i)
(ii)
D(lxl> for all t e CKL and all n;
2
For all real t 'AA (t,x) .... e 4-t , n.;+ a> •
!An(t,x)I
S
n
Hence, by Vitali's convergence theorem,
uniformly with respect to
in CKL • By a familiar theorem on uniformly
convergent sequences of analytic functions (see, e.g. Titehmarsh (1939)
§2.8l) it follows that
t
A(k) (t,x) .... N(k)(t) uniformly with respect to
n
t
in C • Since L can be chosen arbitrarily large it is clear that the
KL
lemma is proved.
LEliuvlA 2.4
let nO
>
Suppose F(x) to be abso'lute ly continuous; fix
0 be a fixed integer.
'A
= def
Then if
sup
Ixl~I.
Itl>cS
it foUows that 0
S
'A
<
1.
cS > 0
~
and
10
PROOF: If the lemma is false there must be sequences of reals {t }
n
such that t n > to' and {Yn} such that Y + ~ , with the property that
n
+y
it u
f n e n f(u) du + 1 ,
-Yn
But, since
w(t) + 1. The Riemann-Lebesgue lemma insists that {t}
n
n
be a bounded sequence. Thus {t } must have a finite limit point t * , say,
n
this implies
and the continuity of
have t *
~6
w(t) then requires
* = 1.
w(t)
But we must then
and are forced to the contradiction that F(x) is lattice.
Thus
the lemma is proved.
Let us write
r
O
= mr(r-l) -1
•
It is well known that condition I(ii)
' Iw(t)l n eL(l) (and hence, since
O
t, fw(t)In eL(2) also). Thus the integral in
in section 1 implies that, for all n
Iw(t)1 S 1
for all real
> r
(2.6) is absolutely convergent so the p.d.f.
L(l) n L(2).
and belongs to
f (u) exists, as claimed,
n
Since the probability measure defining
w(t) dominates the defective one defining a n (t,x) (for fixed x ; 0), the
existence of
f (u)
n
o S a (u,x) S f (u)
n
n
implies that of the defective p.d.f.
almost everywhere in
+
Thus
2
00
f
u.
[an (u,x)]
du
dt •
S
-~
and so, by Plancharel's
theor~J,
~
(2.10)
f lan (t,x)1
00
2n
~
S
J Iw(t)1 2n
~
dt •
a (u,x)
n
and
11
·e
This inequality enables us to prove:
LEHFiA 2.5
fo!' aZZ n >!'o and if ~1 = o~ ~2
If w(t) e: L(n)
<
co
~
then
+co
f
lex n (t,x) In dt
-co
fo!' aZZ Zal'ge
n
whepe C is a finite aonstant independent of x F o.
~
PROOF: Choose a small 0
> 0, then there will be a finite
nt
Iw(t)! s efor all It I S
w(t) about
o.
n > 0
such that
2
This is an easy consequence of a familiar expansion of
~l z
t · 0, when
f
~2
0 and
+0
Iw(t)l
n
<
co
+0
dt
•
Thus
e -nnt
f
S
2
dt
-0
-0
e
....£
Ii
S
C
l
s
(2.11)
Ii
for some finite C •
l
Let
=
Al
then 0
S
Al < 1 since Iw(t) ,2n
sup Iw(t) \ ,
Itl~o
is the c.f. of an absolutely continuous
distribution.
Thus, if we fix
nO > r
f
S
k
A~k-no)
be any integer,
f
k > nO'
Iw(t)\n o dt
Itl>o
k
S C A
2 l
(2.12)
•
and let
Iw(t)\kdt
Itl>o
for some finite C
2
O
~
O•
From (2.12) and (2.11) it is immediate that
12
+co
I
Iw(t)
k
1
dt • 0(-)
I
lit
_00
and then (2.10) shows the existence of a finite C , independent of x ; 0,
3
such that
for all x ; 0, i.e. the lemma is proved for
n even.
The case of
n odd,
say n • 2k+l, follows easily from the observations:
Ia 2k+l (t,x) I
(i)
(2k+l)
~
..
12k
la 2k+l (t,x)
(ii)
where
LEi"o4A 2.6 For some 8uffiaiently small
6 >
lan(t~x)12 ~ e-~t
in the range
ItI
~ ~
0 and all large n
~
2
.
PROOF: Consider the equation
(2.13)
la
(t,x)
n
12 = If..
K.
eit(u-v) f(u)f(v) du dv,
where K. is the (u,v)-set where lui ~ Ixlln and Ivl ~ Ixlln.
(2.14)
•
Ia n (O,x) I2
Plainly
~ 1 •
If we differentiate (2.13) once with respect to
t
we obtain
13
2
ddt1a (t,X)1 1
(2.15)
n
teO
· 0 ,
and a second differentiation yields
d~ la
dt
(t,x) 12 •
-II
n
eit(u-v) (u_v)2 feu) f(v) du dv •
K
Thus
2
d
-2
dt
la
n
(t,x)
I2
•
If
e
-K
it (u-v)
(u-v)
2
feu) f(v) du dv •
But the right-hand double integral will be made small as
in
t.
n
+ ~
uniformly
Further, since Iw(t)\2 is the characteristic function of a random
variable with zero expectation and variance 2, its second derivative will
~
be arbitrarily near 2 for all sufficiently small Itl.
Thus, for all
sufficiently small It I and all large n ,
d
2
-2
dt
la
n
(t,x)
I2
~ -1 •
If we combine this inequality with (2.14) and (2.15) we find
Thus the lemma is proved.
Given
0 > 0 there exists
8uffiaiently large m and n
•
where A is
~(o)~O
< ~ < 1~
~
1 la (t~x) 1mdt
Itl>o n
independent of x~ Ix I ~ 1 .
< A~m ~
suah that for all
14
·e
PROOF:
Fix an integer
q, say, such that
+00
f la q (t,x) ,q
_00
for all Ixl ~ 1.
and such a q.
dt < A ' say,
1
Lemma 2.5 assures us that we can find such a finite A1
nO' A, be as presented in Lemma 2.4.
Let
Then, for m >q,
n > max (q,n )
O
Let Ix* 11Cl· Ix lin and note that _Ix I > 1 implies Ix* I > 1.
Thus the lemma is proved, with A
e
LEivllvlA 2.8
a (1)
n
uniformZy in
and that
and
.
If III = 0 and ll2 = 1, then
In
PROOF:
= A1A- q
Then
I xl
~1,
as n
-+
00
(..J.. ,x;' -+
In
-t ,
•
It is known (Smith(1953), for example) that
w(2)(t/ln)
e~2) (.....L. ,x)
-+
-1.
-+
Thus we need to show In
In w(1) (t/In ) -+ -t
e~l)(~
,x)
-+
0 uniformly. Now
In
In S(1)
n
so that, for
•
Ixl ~ 1 ,
(.-£.. ,x)
In
= iln I
lul>lxllii"
e itu / In u f (u) du
0
15
·e
u 2 f(u)du
Ilne(l) (...L,x)lsf
In
n
lul>1n
The right-hand member of this inequality tends to zero as n
A similar
~~.
t3 (2) (_t_ ,x) •
argument deals with
rn
n
§3 PROOF OF THEOREl4 I •
It is well-known that condition I(ii) implies that Iw(t)l n
for all
n > r
O
defective p.d.f.
€
L(l)
and we have seen that this implies the existence of the
a (u,x)
n
for fixed x
+0
and that
This implies that the Fourier transform A (t,x)
n
€
a (u,x)
n
£
L(l) n L(2)
L(l) n L(2) also (for
n > r O). From (2.5) we can then deduce that Bn(t,x) also belongs to
L(l) n L (2) for n > rOo
In proving the theorem we may plainly assume, with no loss of generality,
that 11(lxl> ~ 1
for all
x and that M(lxl)/lxl is non-increasing for
Ixl ~ 1. In view of Theorem A we need only prove that
Ixl Y MClxl)lfn (x) - ~(x)1 ~ 0
as
n ~ ~ uniformly in Ixl ~ 1.
the case
(3.1)
Y· v.
Indeed we need prove this result only for
Thus our proof will be done if we can show
lim Ix Iv z,1( Ix I> bn (x,x) "" 0
n-+-co
(3.2)
lim Ixl
n~
v
M(lxl)lan (x,x) - ~(x)
uniformly in Ixl ~ 1.
To begin, we note that for Ixl ~ 1,
•
"" Q(Iii) , say.
I ""
0 ,
16
But, since M\) < ~ , Q(IJl)
can find
nO(K)
+
0 as n
+ ~.
Thus, for any fixed
K > 0 we
such that
< e
(3.3)
-K
for all n ~ nO' Ixl ~ 1.
For ease let us write
r'
for
~"
for
~ j"[~]
j-l
L j=n-2k
j=l+[~]
t'
I..
1 ,
for
~ jan
j=n-2k
where
k
is some fixed large integer,
part of y.:'
n > 2k, and [y] means "the integer
Then (2.5) and (2.8) show
dt •
We deal with the three sums separately. First we note that
==
(since
•
18
such that
(-E.,x)
nlil
I
F(1) (x) , say t
n
~ p(lxllil» if l-Je can show there is a finite constant C
l
17
·e
for 1 S j S
[~].
But this is ensured by Lemma 2.5.
Thus
when we bear in mind that Ixl ~ 1 and M(lxl) ~ 1.
If we now appeal to (3.3)
we see that this last expression tends to zero as
n
Ixl ~ 1.
+
=.
uniformly in
Thus
F(l)(x)
n
uniformly in Ixl
~
+
0
as n
+
=
1.
Next, let
Then Lemma 2.5 and (3.3) show that
and, provided
K
is chosen large enough, this last expression also tends
to zero uniformly in Ixl
Since
en (t,x)
a
~
1.
wet) - a (t,x)
n
we can appeal to Lemma 2.5 to see
that a constant C ' say, exists such that
2
C
+00
•
f Ie (t,x) 12k
_= n
dt <
-!
vn
18
·e
for all
n
and all
suppose
n > 6k - 1
x; 0 ,provided
and n - 2k + 1
for some finite constant C •
3
notation,
(3)
Fn
e
~
k
is large enough.
j ~
Thus, if we
n,
Therefore, in an obvious extension of our
V
L
99'
(x). ~ C3 1x I M( Ix I)
= C3 (2k+l) Ixl v
from which it follows easily that
2k
{np ( Ix lin) }
H(lxIHnp(lxlln)}2k
F(3) (x)
n
+
0 as n
Combining the results on F(l) (x) F(2) (x) F(3) (x)
n
'n
'n
+
00
uniformly in Ixl ~ 1.
we see that (3.1) is
proved.
To deal with (3.2) we have to break the argument down into two cases.
CASE 1: v
>
2
Let us define the integer
K
v
•
[v] + L
• M(lxl)/lxl, so that L(lxl) decreases as Ixl
Let us also set L(lxl)
increases. With a nodding
reference to Lemma 2.1 we introduce the following notations •
•
19
·e
(:I<v+l) (t,x) .. C
(1
A3,n
(:~",+l)
, .....'"
V'
x a (.K",)
n
)
n4'('I<;",+l)
n
[2]
(~,x) an(l) (..!.,x) [a (..!.,x) ]n-2 ,
yn
In
n -li\
A(x",+l) (t,x) .. A(~",+l) (t,x) - A(:Kv+l) (t,x) - A3(~+1) (t,x)
l,n
n
,n
2 ,n
We then write
+00
D
j
,n
and
D
l ,n
(x) ..
J e-itx
_~
Aj(.Kvi"l) (t,x) dt ,
(x) .. fe -itx{Al(.~+l) (t,x) _m
,n
We note that, when the "pseudo-densityll
An (t,x) .. 21T1
J e itu
t
2,3,
N(:~+l) (t)}
dt •
a (u,x) exists
n
(x
fixed) ,
a·n(u,x) dn
where the integral is over a finite range.
respect to
j ..
,n
Thus we may differentiate with
under the integral sign to establish that
An (~+1) (t,x)
is the Fourier Transform of
(iu) (:K",+l) a (u,x)
n
Thus, so long as the integral is absolutely convergent,
Tfiis, and similar considerations for the normal distribution, enable us to
•
write
20
·e
~
IxIKV-\l
1
... -2
1T
For each
x
~
and every integer
0
+
D
(x)
1 .n
D
(x)
2 .n
+
D
(x)
3 ,n
I
r > 2. we deduce from dominated
convergence that
(r)(
t
)
n -%(r-2) a
-x
n
;;:
(3.4)
-+
0,
A mean value theorem gives us that
a (1) (...!. x) ... a (1) (0 x) + -!. ci (2) (it x)
n
In'
n
•
Inn
In'·
(3.5)
for some 0 <
~
~
But In a(l)(O
x)
n
»
< 1.
uniformly in Ixl ~ 1.
Thus there is a constant
Inla~l) (~,x)
(3.6)
n, Ixl ~ 1.
for all
converges to zero as n
I
S
c1 +
c
1
-+
~
> 0 such that
It I
From (3.4) and (3.6) we find that for each x with
Ixl ~ 1 ,
lim
(3.7)
n~
A (Kv+l ) (t x) .. 0 .
j
,n
'
,
j ... 2,3 •
Next we prove:
LEi{~·1A
3.1
For aU
~
t
as n
A (I)v+i)
l~n
uniformty in
PROOF:
Ixl
-+ ~
oJ
(t~x) -+N(Kv+l) (t) ~
~ 1 .
If we refer to the definition of
see it is the sum of terms like
A(KV+1) (t,x) and Lemma 2.1 we
l,n
21
(3.8)
(kj )
an
= KV
with kl + k2 + ••• +k
j
t
t
In
;n
Suppose that l of the numbers
+ 1.
j
(--,x)[an(--,x)ln-,
k ,k , ••• ,k
l 2
j
are 1 and the rest are at least 2.
Then
K
v
+
~ ~
1
+
2(j-~)
so that
v+ 1) ~ ~t
with equality if and only if k ~ 2 for all r
(3.9)
j
- ~(K
r
a)
SUppose
=
1,2, ••• ,j.
'<' holds in (3.9).
Then the absolute value of (3.8) is
~ Constant • nj-J;(Kv+~+l)
lin a(l) (.-!.,x) It
n
In
~ Constant • nj-J;(Kv+~+l)(cl + Itl)~
by (3.6).
in
Ixl
b)
This last expression tends to zero as n
+ 00,
uniformly
~ 1.
SUppose
'=' holds in
(3.9).
In this case (3.8) has the special
form
r a (1) (--,x
t )] ~
n [j]n -J;( Kv+~+l) [rn
n
In
and thus, as n
+
00, converges to
22
·e
(-t)
i
~ 2
x (_l)j-i x e-~t
uniformly in Ixl ~ 1, by Lemmas 2.2 and 2.8.
Thus each term of the finite sum representing
Al(Kv+l)(t,x) converges to a
,n
finite limit as n ~ ~, uniformly in Ixl ~ 1.
In view of Lemma 2.3 and the
limits (3.7), we have proved Lemma 3.1.
R > 0,
Let us set, for some large
=I
IN(Kv+l)(t>I dt ,
Itl>R
o>
and for some small
0 and large
n,
Then we have
An examination of the proof of Lemma 3.1 in conjunction with (3.6)
and the easily proved fact that
a polynomial
p(ltl)
in
It I
,
la~2)(~,X) I ~ 1 , will show that there is
v'i
of degree not exceeding
K + 1
v
and with
non-negative coefficients, such that
(3.10)
In the range It I
~
R, (3.10) shows that the convergence of Lemma 3.1 is
23
bounded convergence, the bound being uniform in Ixl ~ 1.
Thus
(3.11)
uniformly in Ixl
~
1.
Furthermore, given any small
£
>
0
we can choose
R so large that
(3.12)
Next we observe, from Lemma 2.6 and (3.10), that
2
_t (n-Ky-l)
IA~:~+l)(t,x)1 s
P(ltl)e
4n
Thus
t
~
2
(n-K:v-1)
4n
dt
and the right-hand member, by monotone converges, goes to
J
p(ltl)
, as n
-+
ClO
•
RSltl
This can be made as small as we please by choice of sufficiently large
Thus for large fixed
(3.13)
R and all large
1 (n,x)
3
We must now deal with
uniformly in Ixl
< £
1 (n,x).
In
r
1.
for r=O,l, ••• ,K: -1.
v
Thus, referring to (3.8) and Lemma 2.1, we see that
•
~
We note first that
4
10. (r) ("";',x) Is).
n
n
R.
24
·e
IAt~+l)(t,x)1
(3.14)
\I+1
K
1
~~~-.,...
L
n~(K\I+1)
If
P and
j=2
are any integers, we have from Lemma 2.7 that
q
Since 0 < A < 1, the right-hand term tends to zero as n
~ ~.
Reference to
(3.14) then shows that
I (n,x)
4
(3.15)
~
uniformly in x, Ixl
~
~
0,
as n
~ m
,
1.
Combining (3.11), (3.12), (3.13), and (3.15) shows that
ID 1n (X) I
< 4£
for all sufficiently large
Dln(X)
uniformly in
Ixl
~
1.
~
0,
as n
as n
uniformly in
Ixl
For fixed
1 ~ lui ~ Ixl~.
Thus,
~ m ,
It then follows, a fortiori that
(3.16)
•
n, uniformly in Ixl ~ 1.
~
~
,
~ 1.
x
in Ixl
Then, for
~
1,
Ixl
let
I
nx
be the set of rea1s
u
~ 1, we have an obvious argument,
such that
25
·e
+1
la(K v +1)(t,x)I
n
~J lul Kv+1 f(u) du +
-1
~
1 +1 J luI KV -
V
J
lu1 KV+1 f(u) du
1
nx
~
lul
v
f(u) du
nx
(3.17)
Thus, for Ixl ~ 1 ,
= n -~(K,,-1)
v
+
and we note that since
n
~ ~
, for fixed
K
v
>
v > 2, the right-hand side tends to zero as
x.
We use (3.17) in the following steps, in which we suppose Ixl
~
1,
a (-.!.,x)
nl;
~
{
L(l)
+M
L(
1)·
n~ K'Vv
~ n-~(v-l)}
L(lxlln)
x
n-l
dt
I
+J~I a (t
\ n- 1 dt
-,X)
n In
_lXl
26
·e
since M is non-decreasing, and so
=
~-<
Ix IIii)
1 .
H(
But
L(l)
M
v
+
n
as
n -+
00.
~(v-2)
-+ 0
Further, Lemma 2.5 shows (after an easy modification of a kind
we have already demonstrated) that
is bounded uniformly in
(3.18)
Ixl
~ 1.
~
IxI KV - V
uniformly in Ixl
~
Thus
D2n (x) -+ 0 ,
co
,
1 ..
To complete our discussion of the case
D (n) •
3n
n-+
We note that, for
~
ID3n (x) I
IxIKV-V
Ixl
~ 1,
v > 2
we must deal with
27
·e
where, for R> 0 ,
J
1
(n)
la
(1)
=Itl>o~
f
I
=
J (n)
3
f
Itl~R
a
(-!.,x)
~
Ila
(-!.,x)
n ~
In-
II
(t
) an (t
)
~,x
/U,x
(1)
2
dt
I
n-2
By (3.6):
n-~(V-3)J1(n) ~ n-~(v-3) n-~
+
uniformly in Ixl
~
0
r
as n
+
f
Itl~R
00
(It I + c ) dt
1
,
1.
Inequality (3.6) also yields the result:
dt •
28
·e
by Lennna 2. 6.
Thus
n~(v-3) J (n) .... 0
2
uniformly in Ix I
~
n ....
00
,
~L
The fact that
uniformly in Ixl
~
n4"(v-3) J (n) .... 0 , n .... 00 ,
3
1 follows from Lemma 2.7 and the inequality la~l)(t,x)1
The three limit results on J , J , J reveal that
1
2 3
L( x) D () .... 0
IxI KV- V 3n x
uniformly in Ixl
~
,n ....
oo
1. This result, together with (3.16) and (3.18) proves
the desired result that
IxIVM(lxl)lan (x,x) - ~(x)1 .... 0
as n ....
CASE 2:
00
,
uniformly in Ixl
v
= 2:
~
1.
This proves the theorem when
In the previous case the fact that
critically in dealing with D (X) and D (x).
3n
2n
that
x~(lxl)lan (x,x) - ~(x)1
D
where we now write
For
Ixl
n
JL(n3) (t ,x ) dt
e -itx -"2
+00
f
-00
~
v > 2 was used
L(lxl)l(ix)3 a (x,x) - (ix)3~(x)1
+
v > 2.
I
,
~
1 we observe
~
A :-
l
·"e
and
A(3)(t n)
2n
By
Lemm~
'
= n4"
(t
- , X ) {ex (t
- , X ) In-l
10
n/il
ex (3)
n
•
2.2 and 2.8 we see that
(3.19)
,
Moreover this convergence is uniform in Ixl
~
1.
n-+
OCI
•
The arguments used in Case 1
to deal with Dln(x) will now show that
as n
-+
OCI
,
~
uniformly in Ixl
1.
Next, still supposing Ixl
~
1, we see that
3
:;;
L(1)R +
In
for any R
>
0, all sufficiently large
~ f
&(lxllIi
n.
But, as before,
1 ,
u~'f(lul)
R<lul
f(u) du
30
:e
and
2
u l1(lul) f(u) du ..... 0 , as R .....
f
00
•
R<lul
Thus we can make
uniformly small for all real
and then
n
and all Ixl ~ 1, by choosing
t
R large
sufficiently large.
Let us set
E(n)
= sup
Ixl~l
en (t,x)
-oo<t<+oo
Then we have shown that
E(n) ..... 0 as n .....
00.
We now note that
Reference to Lemma
liictated by
2.5~(with
the familiar change from
x
to
* say,
x,
x*/n-l = xvn ) will show that
is uniformly bounded in Ixl
uniformly in Ixl
of the theorem.
~
1.
~
1.
Thus
This is sufficient to complete the proof of Case 2
REFERENCES
Cox, D.R. and Smith, W.L. (1953) Skand. Aktuarietidskr., pp. 139~150.
Gnedenko, B.V. and Kolmogorov, A.N. (1954) Limit distributions for
sums of independent randOm variabZes. (Translated from the Russian
by K.L. Chung), Cambridge, Mass., Addison-Wesley.
Smith, W.L. (1953)
Smith, W.L. (1969)
Titchmarsh, E.C.
~c.
Camb. PhiZ. Soc., 49, 462-472.
J. AustraZ. Math. Soc., X, 429-441.
(1939)
The theory of functions,
31
Oxford University Press.