Iernat. J.
Vol.
,th.
& Math. Sci.
339
(1978)339-372
THE EFFECT OF RANDOM SCALE CHANGES ON LIMITS
OF INFINITESIMAL SYSTEMS
PATRICK L. BROCKETT
Department of Mathematics
The University of Texas
Austin, Texas 78712 U.S.A.
(Received August 25, 1977 and in revised form April 3, 1978)
ABSIRACT.
j=l,2,...,k }}
{{X.,
n
nj
Suppose S
is an infinitesimal system of
random variables whose centered sums converge in law to a (necessarily
infinitely divisible) distribution with Levy representation determined by
the triple
(y,o2,M).
{Yj,
If
are independent indentically
j=l,2
distributed random variables independent of S, then the system
S’
{{YjXnj, j=l,2,.o.,kn }}
is obtained by randomizing the scale parameters
in S according to the distribution of
YI"
We give sufficient conditions on
the distribution of Y in terms of an index of convergence of S, to insure
that centered sums from S’ be convergent.
tribution determined by
(y,o2,M)
and
(y’,(o’)2,A),
(y’,(o’)2,A)
distributions from S and
If such sums converge to a dis-
then the exact relationship between
is established.
Also investigated is when limit
S’ are of the same type, and conditions insuring
P. L. BROCKETT
340
products of random variables belong to the domain of attraction of a stable
law.
KEY WORDS AND PHRASES. General central limit theorem, products of random
vaiabl in the domain of attraction of stable laws, Lvy spect function.
AMS(MOS) SUBJECT CLASSIFICATION (1970) CODES.
i.
60F05, 60F99.
INTRODUCTION AND SUMMARY.
The classical linear model for the relationship between empirical data
Y and theoretical or "true" data X is to assume Y
is a random variable independent of X with
the error tends to depend upon X.
E(e)
X + e where e (the error)
0.
In some cases however
For example if X denotes the measurement
of some random phenomenon we may find the empirical data agrees well with X
for values of X which are small, but the error becomes increasingly greater as X
becomes larger.
Such is the case when a measuring device has a constant per-
centage error.
If we have selected a measuring device at random from a popu-
lation of devices whose constant percentage error Y follows the distribution
function G, then we may model the empirical data as YX where Y and X are
independent and E(Y)
i.
The empirical data can be considered as a random
scale change of the theoretical data X, or equivalently the scale parameter
of X has been subjected to a mixture with mixing distribution function G.
The problem we shall consider is the mathematical problem of limit dis-
tributions for sums when the scale parameter is mixed.
S
{{Xnj j=l,2,...,kn }}
Specifically, if
is an infinitesimal system of random variables
whose centered sums converge in distribution to some (infinitely divisible)
random variable
X, and if
{Yj,
is a sequence of independent iden-
j=l,2
tically distributed random variables which is independent of S, we seek
conditions on the distribution of
Y1
and the system S to insure that centered
sums from the randomly scale changed system
S’
{{YjXnj, j=l,2,...,kn
EFFECT OF R2NDOM SCALE CHANGES ON LIMITS
341
In case S’ does converge, we wish to determine the
converge, say to Z.
exact relationship between X and Z.
Here we take an index of convergence
the weak hypothesis E
S+
IYII
<
S
for some 8
of the system S and under
> O,
we obtain a necessary
and sufficient condition for convergence of the system S’.
When
S’
con-
verges we then obtain the exact relationship between the limit distributions of X and Z.
These results are then applied to a most commonly
occurring system, namely that consisting of normed random variables whose
distribution is attracted to stable laws with exponent
we find
S=.
that if EY
2
<
.
In this case
In the classical central limit theorem (= 2) we find
,
then X is attracted to the normal distribution if and
only if XY is attracted to the normal distribution, and moreover the same
norming constants work.
<
For
attracted to a stable of index
,
2 we find that if E
IY 1
+ <
and X is
then XY is attracted to the same stable
law, the same norming constants work, and the exact scale change between
the two resulting stable laws in calculated.
We are also interested in
when the limit laws of S and S’ are of the same type.
A necessary and
sufficient condition for this to happen is that the limit law be either
of purely stable type, or a mixture of stable and normal type.
2.
PRELI>]
AN INDEX OF CONVERGENCE
NARIES
Let us recall several important facts from [3].
If S
[{Xnj
j
1,2,
...,kn
variables, the functions M
n
is an infinitesimal system of random
are defined by
k
7. n
IFxn,j (x)
for
Zj=l FXn,j (x)-l}
for
j=
M (x)
n
k
n
x
<
0
(2.1)
x
>
0
P. L. BROCKETT
342
where F
X
is the distribution function of
n,j
Xnj
We shall say that the system S is convergent to X if there exists a
k
sequence of real numbers [C n n= 1,2,...} such that E j= n,j -C n converges
+
in distribution to X as n
nIX
.
-
In this case X is infinitely divisible
with characteristic function whose logarithm is of the form
2 2
log
fx(U)
iu
We shall simply write X
characteristic function.
0}.
and b
u
+
f
(,2,M)
[e
iux
iux
-i
l+x
2 }dM(x)
to express the fact that X has such a
The symbol
S
b
d
d
represents
-c
l:n[fa +S-c
a
0
All integrals are taken in the Lebesgue-Stieltjes sense.
The
set of points of continuity of a function g will be denoted Cont(g).
Discont(g) is defined to be the set of points at which g is not continuous.
k
n
and
to denote F
We shall hence forth use % to denote
X
Fnj
%j=l
DEFINITION.
Let S be an infinitesimal system.
nj
S
The index
for the
system S i__s defined by
S
with
S
infiX. >
O" lira
dM (t)
n
(x n) + (0,),J
O}
if no such g above exists.
One should remark here that the index
S
as defined above has some
relation to an index defined in 1961 by Blumenthal and Getoor [2] and
later generalized by Berman (1965).
They define the index
spectral function M by
M= inf[c > 0"
i= inf[c
0:
S
I
IxlCdM(x)
< }’
and proved
-i
lim[
x+0
IxlC(M(-Ixl)-M(Ixl)}
0}
M
for the Levy
EFFECT OF RANDOM SCALE CHANGES ON LIMITS
343
As a similar result in [4] it was shown that then S converges to X~
(,2,M)
we have for a e Cont(M)
inf[[, >
S
with
S
6M"
S Ix I<: Ix[SdMn
(x)
*
!xI%dM(x)]
Ix
(2 2)
<
An easy way of calculating
if the above set is empty.
S
is
then given by noting that
’IxIYdM
lim lim sup
I im in f
Olxi<
n+
(x)= 0
,}xlYdMn,,[x)=
if
for any
">
Y
e
-ES
> 0
(2 3)
if
Y <
>S"
We next recall a variational sum result proved in [4].
THEOREM I.
X
(,2,M),
g(x)
Suppose S
and
0(Ixl ) a__s
is an infinitesimal
suppose that g
x
@.
discont(M)
lim Z
n+
R]D4ARK
I.
is a bounded function which satisfies
>
0 for some
S
and which is either continuous
(-,0) and
is of bounded variation over
system converging t__o
over
o__r
(0,) with discont(g) Q
Then we have
E(g(Xnj))
lim
n+<
(-,)
g(x)dM (x)
n
Some comments about the index
S
(cr,e2,M),
two equalities is possible.
For an example where
it is shown that if
[XI,X2, ...]
g(x)(x).
,")
are in order.
M <_ S <--
system S is convergent to X~
then
f(_
If the
and either of the
S .M
see
[4] where
is a sequence of independent identically
distributed random variables belonging to the domain of attraction of a
stable distribution with exponent
,
and if S
[{Xj/B n,
j
1,2, ...,n}}
344
P. L. BROCKETT
then
S
It is clear that in this case
=aJ.
example is given showing
S
M
can be anything.
gent system S is found such that
S =%"
<
if
0 if
In [3] an
<_
Namely for
To show that
2
2
S
a conver-
measures "how"
the system converges rather than to what it converges, an example in [3]
is given showing that for any Levy spectral function M and for any %
there exists a system S converging to (0,0,M) with
S =%"
The index
>
M
S
has proved to be the appropriate index for studying variational sums of infinitesimal systems
[4], and has been shown to be an extension of the Blum-
enthal-Getoor index for stochastic processes with independent increments
[5] which allows a unified treatment of variational sums of such processes.
Let us also state for reference the general limit theorem as found in
Gnedenko and Kolmogorov (1968) or Tucker (1967).
THEOREM 2.
Suppose S
A_ necessary ___and
system.
[[Xnj
.,
i, 2,
j
sufficient condition
k
n
is an infinitesimal
that. EXnj-Cn+X~
(,2,M)
is that the following three conditions all hold
+ M(x)
Mn(X
B)
Jlim sup [
+01im llim inffn
C)
Ixl<
for any r
3.
>
T
x e Cont(M)
for all
A)
xdM (x)-c
n
n
x
n
ixl< e
+
+
x
ixl<
xdF
(x) -7
Ixl<e
/(I+
x nj (x))
2
2
x/(l+x2)dM(x)
)dM(x)
ixl< r
T
0.
A LIMIT THEOREM FOR SCALE MIXED SYSTEMS AND APPLICATIONS
We shall now proceed to answer the questions posed in the introduction
Namely when S is convergent, the index
S
ing conditions insuring S’ is convergent.
yields the needed tool for obtainThe precise formulation, and the
EFFECT OF RANDOM SCALE CHANGES ON LIMITS
345
S’ is given in the
exact relationship between the limit laws of S and of
Also to be noted is that the convergence properties
following theorem.
of S’ are the same as that of S (i.e.,
limiting Levy functions are the same
S S ’)
and the behavior of the
(M= ).
In particular since an
infinitely divisible distribution is continuous if and only if the corresponding Levy spectral function is unbounded, it follows that the limit
distributions of S and S’ are simultaneous continuous or not continuous.
THEOREM 3.
XN (,
2 ,M)
[[Xnj
Suppose that S
i.e.,
d
EXnj -cn
1,2,
j
>X. Let
...,kn}]
yj J= 1,2,...}
is convergent
be a
t_9_o
sequence of
independent identically distributed random variables with non-degenerate
Fy
common distribution function
Assume also that
necessary
and
n=1,2,...]
[Yj,
and with E
sufficient condition
.th.at S’
lim sup
n+oo
(’,
(")2,A),
n
Z
(3.1)
x dM (x)
bl Remark
i necessarily
fo___[r S’
can be chosen
(x)- -Z
x
x
l_f S’ co___p_n-
a_s
S Ix I<XdFx .Y
l<rx3/(I + 2) dFXnjYj
SI
SI i>Tx/(I + 2) dFXn3"Y’3
x
x
n3 j
+ Z
S <- 2).
then the following are true:
centering constants
d
n}
n+.
exists and be finite (so that
Th_..e
...,k
n
ixl< e
lira llm inf
0
J I, 2,
[YjXnj
Then a
be convergent i_s
+0
I
> O.
for some
j=1,2,...} is independent of the system S.
lim
CO= e+O
verges to Z
IY S+ <
(x)
(x)
P, L. BROCKETT
346
2
0
S ,
(-
A(x)
0)
[l-Fy(X/t)}dM(t)
-S(_,0Y
(x/t)dM(t)
S (0, )Fy(x/t)dM(t)
+S(0,m)[Fy(x/t)-l}dM(t)
for
x
<
0
for
x
>
0
(,)2 CO Var(Y) + 2(E(Y))
30
where C
O is as defined in (3.1)
4
5
+
Ps s’
0
pM.
NDTE:
I. If
S<
2 then (3.1) is automatically satisfied with
by (2.3) and necessarily
this case.
Also when
S
q2
0 (otherwise
S >- 2),
so
30
says ()
CO= 0
2
2 and (3.1) holds we need only assume E(Y
0 in
2)
<
to obtain the conclusion.
2
It should be noted that the mere meaningfulness of formula
not be sufficient for the conclusion of the theorem.
20 may
Examples of this are
given in [3].
PROOF.
I) We first note that S’ is indeed an infinitesimal system.
Now, an easy calculation yields
F
(x)
X .y
n
j
S ,
(-
0)
[l-Fy(x/t)dFnj(t)
+
S(0,)Fy(x/t)dFnj(t)
+
e[Xnj 0]X[0,) (x).
(3.2)
EFFECT OF RANDOM SCALE CHANGES ON LIMITS
Here
I if x e A and 0 if x 4 A.
XA(X
x e A
n
and
4 A.
x
0
if
Z F
xn].y j (x)
(x)
7.
for
x
<
0
for
x
>
0
(3.3)
3
nj
g(x)
Let
Let
IF X .y.(x)-l)
and let
=
-Fy(x)
Fy (x)
1
Then using (3.2) in (3.3) we have for x
(x)
n
o
(- oo, O)
347
x>O
x<0.
<
0
[l-Fy (x/t) ]dMn (t)
+
S(0,)Fy (x/t)dMn
(t)
g (x/t)dMn(t)
while for x
>
n
0 we have
(x)
z{(-
, O) g(x/t)dFnj (t)
S(0,)
g(x/t) dF
nJ
(t)
-I- (-,) g(x/t)dMn(t).
Thus we have
(-
An (x)
’
)
(,)
g (x/t)dM (t)
n
for
g(x/t)dMn(t)
for
x<0
(3.4)
x>O.
2) In this part of the proof we shall show that if
we always have lim A (x)=A(x) for all x e Cont(A).
n
Ejyl
S
<
we have
[l-F,,(t) +Fy(-t) =O(t’S -5)
as t
EIY
<
,
then
Indeed since
+
.
By the defi-
P. L. BROCKETT
348
nition of g, we can thus see that for fixed x,
t
+ O.
gx(t)= g(x/t).
Let us write
lim
n+
S
(- ,)
for every x such that
(3.4) and
..
20 we
discont(M)
have lira
Let
show Cont (A) c
there exists a t
o
/=
[x:
(_ ,)
>_
Y0
Thus using (3.5),
discont(M)
c
(3.5)
}.
discont(A)
discont(gx) N
It remains to
If
gYo and M are both discontinuous
Y0 >
0 and t
Thus we have
so that for any small h
A(Y0+h)-A(Y0-h)
.
for all x such that
Equivalently we show
such that
gx(t)dM(t)
N discont(M)=
discont(gx)
be used in any other case.
> 0,
S
An(X)--A(x)
simplicity let us assume that
dM({t0})
By Theorem i we have
gx(t)dMn(t)
discont(gx)
as
o>
0.
Y0
then
e
at t
For
o
A similar argument can
__o(tO +0)-__o(t0-0) > 0 and
> 0 we have
[Fy( to )-Fy(
Y0
)}dM([t0}
Fy(t--)-Fy(t+-)}dM([tO}) [gYo (tO+)’gyO(tO-)}dM({t0})
>_ [gy0(t0 +
0
0)-gy0(to-0) }riM([to})
b
where
=t0h/(Y0-h)
and
for all h, we must have
x
Cont(A).
N=t0h/YO+h.
YO
Since
e discont(A).
A(Y0 +h) -A(Y0-h) >_ b
Thus A (x)
>
0
A(x) for all
To see that A is a Levy spectral function, noted that A is
monotone increasing on
(-,0) and on (0,) and A(+)--A(-)=0.
must also show that
x
2dA (x)
<
We
The proof of this is accomplished
(-, )
in part 5) below.
3) In this part of the proof we shall determine how to calculate
integrals of the form
S(.,)
f(x)dAn(x) and
S(-oo,)
f(x)dA(x) where A
349
EFFECT OF RANDOM SCALE CHANGES ON LIMITS
is given by
(3.4) and A is given by
20
Now for 0
<
a
<
b
<
let
(a,b]
denote the indicator function of the interval (a,b] and calculate
S(_,)S(_,) Xi(a,b]
An(b)-An(a)
S
S
X
(u) dry (u)dM n (t)
(tu)dFy (u)dMn (t)
so that
S(_ ,) f(x>dA (x) S (_
n
,)
S
(_ ,)
f(ux)dFy(u)dMn(x)
(3.6)
holds when f is the indicator function of an interval in (-,). Standard
approximation techniques now allow us to conclude (3.6) holds for the
function in question.
Also (3.6) holds if we replace A
n
0
4) In this part of the proof we show that 4 holds.
(2.4) to show
S S’
S [xI<e Ix [gdAn(x) S
Now for any 5
(-,1
Ix
g
S
>
(- ,)
We shall use
0 we have
lu [8X (-’) (ux)dFy (u)dMn(x)
SI xl<IIxl Sluxl<EIul gdFY(u)d:M +SI xln(x)
by A and M by M.
n
I
uxl <E
dFy (u)dMn (x).
However concerning the second term above we see that
lira lim sup
Ix
ul dFy(U)dMn(x)
0.
P. L. BROCKETT
350
S S’ we must only
lu
Ix
Thus to show
lim lira sup
lira inf
Ix15 S
S
Suppose then that 5
Now
lulgdFy(U)dMn(X)
<
S
0 if 5
>
S
for any e
>
0 if 5
<
S"
Ix lSdFy(X)
>
0.
dFy(U)dMn(x)
and choose e
f
so that
I
and
Slxl<l fluxl<eluxlSdFy(U)dMn(X) >_Slxl<IIXl5 Slul<elulSdFy(U)dMn(X)
n
--
that
S >- S’
5 "Y
<
<
-
5
"S
n
S Ix I<1 Ix]5
lira inf
n
let y
SIxI<IIXlS"dM
E IY
dFy (u)dMn(x) >_
Ixl<l IXUl<e I
S luI<e I lulSdFy(U)
e
show
>
S
n
(x).
EIYI %’
and thus
<
S<5"<"
(x) is bounded in n.
1
n(x) =,
be such that
i.e., such that
SIxI< Ix 15"cccIM
dM
Now
S <- s’
and pick
Then
>
EIYIS" <
To see
0 such that
and also
Slx]<l Slux]<e]uxlSdFy(U)dMn(X)<
Thus
luxlgdFy(U)dMn(X)
0.
I.e.
S’ <- S
and hence
S’ =S
0
5) In this part of the proof we shall show 5 holds by showing
Ixl<l
Suppose 5
>_
II(x)
<
if
if
M and that oj (-,) -IxlSdM(x) < .
some y >_
S >- M such that E IYI
that there is
generality we may assume 5
<
"y so
that we have
>M
5 < M.
5
Now we know by hypothesis
<
.
Without loss of
EFFECT OF RANDOM SCALE CHANGES ON LIMITS
Ixl 5
IxI<l
Sluxl<llUlSdFy(U)dM(x)
5
dFy(U)dM(x) <_ EIYI
i.e.
A <__ M.
then choosing 1
Thus
c
>
0 such that P[
A >_ M and hence A= M.
<
Slxl<l IxldM(x)
M so that S
<
On the other hand, if
>_
351
1
-]
>
+M(-I)-M(1) <
Ix 15 d(x)
oo,
0 we have
In particular, choosing 5
2 we see A is
indeed a Levy spectral function.
6) An elementary calculation using the Helly Bray theorem and Theorem
2 shows that the centering can be taken as in I 0 when (3.1) holds.
7) In this part of the proof we shall show that the finiteness of
the limit in (3.1) is necessary.
We shall show that if
x
lim sup
n+
Ixl<e
2dMn (x)
(3.7)
then
xdF
x
.y.(X))2}
=oo.
(3.8)
P. L. BROCKETT
352
Hence by the general limit theorem S’ cannot be convergent.
Let e>l
and for simplicity of notation let us write
o"
n
(e)=
x
xl<
n
e
(x)-Y.
xdF
xl<e
X y (x))
2
nj j
Then
n
x2 S
(e)>
ux
l<e
x
E[;
S
x
(3.9)
dFnj
udFy(U)dFnj (x)}[S
x
2
S lux I<UdFy (u)
dF
UdFy (u)dFnj (x)2.
S
lira sup
PROOF.
dM (x)
n
luxl<
ixi<e2
2Y.[S
u2dFY (u)
Y’[S
x
S l<eudFy(U)dFnj(x)]2=
_
_
O.
By the Schwartz inequality,
;
{:x
<-<
S
UdFy (u) ]dFnj (x))
(S
max
Ix
P[
S
IXnj
UdFy(U)}2dFnj(X)>P[ IXnj
e2]
S
Ix
2
x2 S
e
2]
u2dFy(U)dFnj(X).
nj
(x)]
EFFECT OF RANDOM SCALE CHANGES ON LIMITS
353
Using the infinltesimality of the system, we have the first term converging
to zero.
The sum of the second terms is bounded, since by the Helly-Bray
theorem
lim sup
n ->oo
x
2
ix i>_e2
S
u
ux
I<
< {E(Y 2)
x
2
2dFy (u)dMn (x)
(x) +
dM(x)].
IXl>e
<II<
This completes the proof of Claim I.
Note that using the above we actually
have
x
lim lim sup
2
e+O
2
S lux l<e
u
2
dFy(U)dMn(x)
Oo
(3.10)
CLAIM 2.
UdFy (u)dFnj (x)
{:S
x
Ixl>--e
< [E (Y2) }1/2o(I)
PROOF.
S
x
2
S lux l<e UdFy
(u)dF
nj
(x):
2dMn (x)} %
By the Schwarz inequality for sums, the sum in the claim is
no larger than
P. L. BROCKETT
354
Applying Claim I to the second term and the Schwarz inequality to the
first term in the above product we have that the left hand side does not
exceed
{:S
x
l<e2x2(S
lU
x
I<udFY (u))2dMn (x)
1/20 (:) <-
[E 0f2)
1/2 (I)
IS Ix i< Cx2d:Mn
which completes the proof of Claim 2.
Now using the inequality
x
<-- S
Slux
Ixl<e
2
l<eudFy (u)dFnj (x)
2
x2:S luxl<udFy (u)]2dM (x)
n
and using Claims i and 2 in (3.9) yields
UdFy(U))2}dMn (x)
u2dFy (u) (S
X2[S
2
(3.11)
2(E(y2))o(:)[S
Since (3.7) implies
that War(Y)-
>
S >- 2,
(x)} +o(1).
n
x
2
we know E(Y
0 and choose
2
eO >
2)
<
=.
Let
0 so small that if
u2dFy (u) (S
UdFy (u))2
>
0 be chosen such
xl
<
e,
<- Var(Y)+.
then
EFFECT OF RANDOM SCALE CHANGES ON LIMITS
<_ 0
Then for
2
()
n
in (3.11) we have
> [Var(Y)-5}
S ixl<e2
x
S
x
ixI<2
2dMn (x) 1/2[ (Var (Y) -5
By (3.7) the lim sup as n +
i.e., lira sup
2dMn (x)-2E(Y 2 )o(I)[f
2
Ixl<e
2
n
(x)} 1/2 +o(i)
(3.12)
2dMn (x)) 1/2 -2E(Y 2 )o(i)} +o(I)
of the right hand side of (3.12) is
=, and S’
n(e)
(S ixl<
x
2
x dM
+
,
cannot be convergent.
8) In this part of the proof we show that if (3.1) holds, then S’
(’,
is convergent to X
(’)2,)
Since we have already shown that
lim A (x)=A(x) for all x e Cont(A), to obtain this part of the proof,
n
we need only show that with
lim lim sup
Now,
if
#S
0
X
0
2
n(e)
< 2, then by (2.3)
part 5 of the proof
so that 3
.2(:)
n
<_
#S’ =#s
(,)2
holds with
(’, 0,A).
(3.1) implies
<_
as given we have
lim lim inf
we must have
2()
n
2
(,)2.
0, and
CO= 0.
so that
x
lim lim sup
(0’)2=0,
2dA
and consequently
(x)
Then we assume E(Y
2)
<
0
S’ converges
We are thus left to consider the case when
S <- 2).
However by
S
to
2 (recall
and as before we have
P. L. BROCKETT
356
o"
2
(,)
S 2S 2dFy (u)dM
+S II>_ x2 S lul<u2dFy(U)dMn(X S II< S lul<UdFy(U)dFn (x))
u
x
n
(x)
(3.13)
-Z(
-Z
(S
x
SIUX l<eudFy (u)dFnj
ux
x
(x))
l<l::UdFy (u)dF
nj
(x))
(S
x
it>_2
S luxl< UdFy (u) dFnj (x)).
Applying (3.10) to the second term, Claim 2 to the last term and Claim i
to the 4th term on the right hand side of the last equality in (3.13)
and upon adding and subtracting
Z(S
x
I< I<,}
S lul<:/UdFy (u) dFnj (x))
we have
: (e)=
2
u2dFy (u)dMn (x)
(3" u
(3.14)
iI<
2
ii<<
I <e2
ux
i<e
nj
+ gl (n, e)
EFFECT OF RANDOM SCALE CHANGES ON LIMITS
where lira lira sup
PROOF.
u I<:/
(S
x
Ix i<E
gl(n’E) =0.
Using the equality a
2
357
2
b
2
(a-b) (a +b) we have
)dFnj
S
UdFy (u)dF
(x))
2
UdFy (u)dFnj (x)
UdFy (u)dFnj (x)
{S
udFy (u)dFnj (x) }.
For simplicity we denote
UdFy (u) dF n j (x).
Then the right hand side of (3.15) becomes equal to
2fnj(E)
S
x
S
UdFy(U)dFnj(X)
+
f2nj().
P, L. BROCKETT
358
Thus to complete the proof of Claim 3 we must show
f2nj ()
llm lim sup I
9-0
n+
(3.16)
0
and
lira lira sup[E f
nj
n 9e 0
Concerning
(e)
S l<e2 S lu I<i/UdFy(U)dFnj (x)]
(3.17)
x
(3.16), note that by the Schwarz inequality
f2nj () <- S
and letting a(e
2)
x
2
S
lira sup
n+m
hence
lim lira sup Z
so that (3.16) holds.
fnj ()
0.
x
f2
x
Ixl<
2
u
2dMn (x)
() < lim
2
dFy(U)dFnj (x)
we have a(e
S
u
2
C
O
2dFy (u)a (2))
as e
0 and
0
In a similar manner, concerning (3.17) we have
S Ixl<e S lul <I/UdFy (u)dFnj (x)
x
2
lu dFy (u)dFnj (x)) [E IYI
< E (Y)
[S
oluldFy(U)}[
2
x dF
ixi<e2
.(x)}.
n3
S
2
Ix IdFnj (x)]
EFFECT OF RANDOM SCALE CHANGES ON LIMITS
2
x dM
lu IdFy(U)
(x))
359
from which (3.17) follows
easily.
Now using Claim 3 in (3.14) and upon adding and subtracting
2
2
x dM (x), we obtain from (3.14),
udFY(u)>
n
2
S
(lul<le
G
n
S ixl<,
(c)
+
x
g2(n’E)
UdFy(u))2
(u)-
I,,1</I,1
2
where
lim lira sup
:+0
n+oo
g2(n’E)
(3.:8)
O.
Then, if we use the inequalities
{S
ix j<(::2
2
x dM (x)
n
iI<2
in
{S lu i<1/(: u2 dFy (u)
u
(3.18), we obtain the following
ii<2
two sided bound for d
2
(E):
n
n
P. L. BROCKETT
360
(3.19)
Using the fact that
Ilim+0
Elim
sup
of the extremities in (3.19) are
lim
equal when (3.1) holds, we obtain
C
0
Var(Y) +
by
O
(,)2
<_
lim lim inf
e+O
<_
where C
(E(y))22
lim lim sup
e +0
n+
Var(Y) + (E(Y))
0
2(e)n
2
n (e) <- CO Var(Y) + (E(y))22
(3.1), so that (0’) 2 exists and is given
is the constant given in
C
n
2 2
(y
9) In this part of the proof we show that if S’ is convergent
(so that lim lira sup
+0
(3.1) holds.
lim sup
n +
n
n2 ()
(’)
2
lim lira inf
0 n
2())
n
then necessarily
By part 7) we know that we may assume
S Ixl<e 2dN (x) <
x
n
so that all of the previous equations leading
up to (3.19) still remain valid.
Let us then subtract the quantity
361
EFFECT OF RANDOM SCALE CHANGES ON LIMITS
Ix I<i/e
UdFy (u))2{
x
x
i<2
n
(x)
Y.
xdF
nj
2
(x))2
+ g2 (n, )
throughout the inequality (3.19) to obtain
{S
x
ix i<2
2
<
n
Z
<
Now,
2dMn (x)
(S ixl<i/gXdFy(X))2{S
(g)
(S
xdF
2
{S
lim
o|lim
lim
[S Ixl<i/e u2 dFy(U)-(S
x
2
sUPn
inf
nj
i<i/e
x
xl<g
2
UdFy (u)) 2
(3.20)
2dMn (x)
(x))2 g2(n’e)
2dMn (x)}{E(y2)
(S lul<I/eUHFy (u))2}"
of the inside of (3.20) equals
()2
(E(Y))
2 2
while on the outside of (3.20) the above limits yield
Var(Y) lim
+0
holds with
I
lim
lira
sup S
infn+
2
xl 2dMn (x).
Ixl<E
Var(Y)C0= (,)2_ (E(y))2 2.
We thus conclude (3.i)
This completes the proof of the
theorem.
Let us now turn to the frnework of the more classical central
limit theorem.
If
[XI,X2,...}
is a sequence of independent identically
distributed random variables with common distribution function F, we
shall write X
I e ()
to denote the fact that F
(or
XI)
is in the domain
of attraction of a stable law with characteristic exponent
.
That is,
362
P. L. BROCKETT
there exists norming constants
[An, n= 1,2,...}
[B
1,2,...} and
n=
centering constants
n
such that E
X /B -A converges in law to a distrin
j= I j n
<
bution which is stable with exponent
2.
In this case the relation between the scale mixed system S’ and
the original system S takes on a particularly appealing form.
Our con-
dition for convergence only depends upon the moments of Y and the index
THEOREM 4.
xn
Le__t
[XI,YI,X2,Y2,...}
having distribution function F
X
be independent random
and Y
variables
havin$ distribution function
n
Fy
I) l__f
EIYI 2 < ,
th.en
Conversely i__f YX e D2)
norming constants work.
the sne norming constants
2) l_f E
IYI +8 <
(2) .implies YX e (2) an__d the sne
X
.t.he.n
X
D(2) an__d
work.
> 0,
for some
and the sme norming constants
then X e D() implies YX e
.wprk.
PROOF. The direct statement in I) and 2) will follow from Theorem
3 once we show that (3.1) holds.
as proven in [4], X e O() implies
holding with
CO=0 by
(2.4).
Let S=
S ="
{[Xj/Bn,
<
For
j=
1,2,...,n}} then
2 we have (3.1)
For =2 we know (see Feller (1971), [6]
pg. 314) that necessarily
B-
y
Yl <B
dFX (y) + C0
as
n
+
(3.21)
363
EFFECT OF RANDOM SCALE CHANGES ON LIMITS
2
n
y
y dM (y) and that
Using the fact that
(y)
n
e
[y
[<
n
f
S
y2dFx(Y)
case also.
S’
2dFx S Ix
[<Bn
is a slowly varying function of t yields (3.1) in this
Applying Theorem 3 we obtain the convergence of
[[XjYj/B n,
1,2,..o,n}}.
j
If
=2
then Mm0 so by
A= 0 and S’ converges to a normal distribution.
Theorem 3 (’)
2
M (x)
-C2x
A(x)
20
of Theorem 3
C
’(_
I
of Theorem 3,
30
< 2, then by
of
0 and with
Cllx[-C
we have by
If
20
-
x
<
0
if
x
>
0
(3.22)
I-Fy (x/t)d
oo, o)
if
t
-) -C2
f(0,
)
Fy(X/t)d(t -)
-C1 S (- ,o) Fy(X/t)d([tl-c)-C 2 f (o,) {Fy(X/t)-l)d(t -c)
if
x<O
if
x>O,
or equivalently, upon integrating by parts
A(x)
(-,o)
(o,)
(- ,o)
(o,)
(3.23)
Upon simplifying (3.23) we obtain
A(x)
:I
al
]x[--
-a2x
if
x
<
0
(3.24)
if
x
> O,
364
P.L. BROCKETT
i.e., X Y
n n
is in the domain of attraction of a stable law with character-
,
istic exponent
and the same norming constants work.
Let us suppose now that YX e (2) and
general limit theorem to show E
With M
bution.
IYI
> t] >
O--S
O.
IYI
.
We shall use the
to a normal distri-
j=
n
given by (3.3)we shall show
x 0.
implies M (x) + 0 for
n
Indeed let t be such that
Then
dA (x)=
Ixl>t
2
n
f
Since P[
<
n. iX j /Bn -An converges
given by (2.1) and A
x0
A (x) + 0 for
n
P[
n
EIYI 2
Sf
luxl>t 2
dFy(U)dMn(X)
SIxI>tP[IYI
2dFy(u)dMn(X) >-
> t] > O,
>
t]dMn(X)"
O, thus the limiting Levy
dM (x)
function
is 0.
Let us complete the proof by showing
E
0
lim inf
exists and is finite.
x
n->oo
Ix[<e
n
(x)-n
XdFx(x ))
In view of part 7) of the proof of Theorem 3 we
know we must have
CO
lim lim sup
e ->" 0
n->oo
x
Ixl<e
2dMn (x)
<
EFFECT OF RANDOM SCALE CHANGES ON LIMITS
365
Utilizing the inequality (3.20) of Theorem 3 we see that we must only
show for
C
we have
C
lim
x
lim
e 0
S
lira inf
n +oo
On the other hand for 7
(I-)E2).
n
Ixl<e
n+
<_
Then C
2
>_
+
(x)
>
n
(x)
> 0 choose
lim sup
n -->-
x
ix i<e/5
0 is arbitrary C
I
x u dM
{ux
+
x
lux I<
2dMn (x)dFy (u)
5 so small that
x
u
u
2dM.n (x)
(I-D)E(y2)c 0 <_ C 2
Since 7
x
lim inf
n
u2 S
(I- G) E (y2) I im sup
n
n+
0
Now by Fatou’s lemma and by (3.21)
CO= C I
2
S Ixl<e
im lim inf
I
=C0
<_
Ix
I
C E
1
(x)dFyU
(y2)
u2dFy(U) >
f
n
n
(x)dFy(U) >
Thus
E(y2)cI <_ E(y2)c0
and Z
n.j=Ixj/Bn-An converges
to a normal
distribution.
REMARK 2.
i) The first part of Theorem 4 should be compared to a
result of H. Tucker (1968) who considered sums of random variables in the
domain of attraction of the stable distributions instead of their pro-
ducts.
He shows that if EY
2
same norming constants work.
<
and X e () then X+Y e () and the
If X +Y e () and E(Y)
2
< m,
then a
366
P.L. BROCKETT
slight modification of his methods yields X e D().
Combining our re-
sult with his we find that if X,Y,e are independent random variables,
Ey2 <
and EE
2
constants work.
,
<
then X e D()YX+ e D(o)and the same norming
For =2, X e )12)=> YX+E e D(2) with the same
norming constants.
ii) In Theorem 4 we can actually calculate the scale change involved in the distribution of the limit laws of S and S’ when
<
2.
Namely, in going from (3.23) to (3.24) we have
x<O
x>O
f (o,) tdFy(t) f (-,o) tlCdFN(t)
a2=C2 S (o,) tdFy(t) I f (-,o) ItldFy(t).
where
+C
al=C 1
2
+C
This follows immediately
from (3.23) by the change of variables z= xt
iii) It can be shown that for
implies that
EIXI c’
<
and
and
< 2, YX
EIXI c+6=
-1
EIYI + <
hence
S " I
e D() and
for all 8,
suspect that in fact X e D() however, I have been unable to establish
this for
#2.
iv) Suppose that
lira
+0
lira sup
lira inf
Ix[
+
ixl<
dM
n(x) =C O
-
EFFECT OF RANDOM SCALE CHANGES ON LIMITS
C
then for i
-Fy(X) +Fy(-X)
is given by 2
S<
2.
0
of Theorem 3.
I
we have A (x)
n
+
C0C 1
IS
367
+ A(x) where A(x)
This is a Levy spectral function when
Thus we see that some random scale changes introduce a stable
component into the limit law.
We can also use Theorem 4 to derive some interesting statements
about slowly varying functions which would be difficult to prove by
other means.
The precise formulations are given in the following
corollary.
COROLLARY i
0
varying function of t if
slowly
20
function
2
o
x2dFx(X)
and only ifS lu l<txmumdFx(X)dFy(U)
Suppose EY
<
Then
o__f
t.
S x2dFx(X)
function of t and also
P[X > x]
P[
>_ xl
IXl
b)
x
+
+<
an__d EIYI
_
varies regularly with exponent 2-
-
p’
P[X
p[Ixl
for some 8
<-x]
> 0,
x]
)
x
+
q
then
S(_ , Slu l<tx2u2dFy (u)dFx
varies
is a
o
Suppose
a)
is a slowly
o
(x)
x
regu!arly with exponent 2- a__s
a function of t and also
a__s
a
P. L. BROCKETT
368
PItY >_
(o )
S (-,)
x]dFx(t
ItYl >-- x] dFx (t)
P[
S <- ). PItY <_ x(t)
S P[ ItYI >- x] dFx (t)
-x]dF
for some
p’
PROOF.
q
>__
0
.w.ith p’ +q’ >
q’
0.
This follows immediately from Theorem 4 by utilizing the
necessary and sufficient conditions given in Feller (1971) for a distribution to belong to the domain of attraction of a stable law.
In the previous theorem we observed an interesting phenomenon.
Namely we took an infinitesimal system S and subjected it to an arbitrary
random scale change with
which was convergent.
EIYI
S
<
and we obtained a new system S’
Moreover the limit distributions of S and S’
were of the same type.
In problems where X
nl
represents a "true" or
theoretical measurement of some occurrence and Y
1
the scale change in
the measuring device used to measure the occurrence, we obtain as an
observation the product X .Y.
nl
I
It is of interest to determine when
limit distribution calculated from the empirical data
[[XnjYj, j 1,2,...,kn}}
[[Xnj j 1,2,...,kn }"
is of the same type as that from
For normed sums in the domain of attraction
of a stable law, Theorem 4 answers the question and Remark 2ii) allows
EFFECT OF RANDOM SCALE CHANGES ON LIMITS
us to calculate the scale change.
369
In general the following theorem tells
us that in non-stable limits the empirical data may yield a different
J 1,2,...,kn }"
[[Xnj
type distribution than the theoretical data
In order that a limit distribution b__e preserved i__n type
THEOREM 5.
under all random scale mixtures with E
IYI S+5 <
it is
necessary an__d
sufficient that the limit distributio type be either purely stable or a
mixture of stable and normal.
The sufficiency follows from
PROOF.
20
and
30
of Theorem 3 of
Theorem 3 as we calculated in Theorem 4, and in fact the scale change
involved is given in Remark 2ii).
Suppose now that the limit distribution is preserved in type when
subject to random scale change.
3 we know Z~ (’,
(,)2
a
2 2
and
(’)2,A)
Then with S and
S’ as defined
is of the same type as X~
A(x)=M(x/a)
for some constant a.
(,2,M),
in Theorem
thus
As in 2) of the proof
of Theorem 3 we know that
A(x)
If
IM(x/t)IdFy(t)
x<0
(3.25)
IM(x/t) IdFy (t)
If M(x)
x>O
0, then both X and Z are normally distributed.
then we must show M is given by (3.22) of Theorem 4.
that A(x)
h(t)
=M(x/a)
If M(x)
0,
Using the fact
in (3.25) and letting
j(x/t)[
M(x/a)
we see that
h
is a function of
t
(3.26)
P.L. BROCKETT
370
a.eo
alone, the equality holding
[dFy]. Since Y could be chosen to be
absolutely continuous we have (3.26) holding on a dense set of points t.
For simplicity in calculation we shall consider the case x
a
>
0 and denote
similarly.
N(x)=M(x)/M(1).
O,
t
O,
The other cases may be considered
Rewriting the equation (3.26) yields
N(x/a)h(t)
N(x/t)
and hence for x= a
N (xy/a
N (x/a)
h(a/y)
N(y).
Thus
N(y2)
N(y)h(a/y)
(N(y))2
and by induction for any k
N(y
N(yk-l)h(a/y) Nk(y).
U(x)=-nlN(e x) I,
the above equation becomes
k)
Letting
I/k U(knx)
i/k U(ky)=U(y).
U(x)
U(n x)
or
By Lemma 3, page 277 of Feller (1971), this implies
x 0 or equivalently N(x)
x
O.
That -2
<
0
<
0 follows from M being
EFFECT OF RANDOM SCALE CHANGES ON LIMITS
a Levy Spectral function.
Thus
M(x)=M(I
for x
>
0.
371
Similarly we can
X
show
M(x)=M(-I)
for x
<
0.
We see that
=
<
is of the same type whether multiplied by Y
since the limit distribution
0 or Y
>
0.
The result then
follows easily from the calculation of A(x) in both cases.
This completes
the proof of the theorem.
REMARK.
The problem considered here could have been solved by de-
fining the class C
C
S
[Y" lim
of random variables by
S
lira
sup[
[l-Fy(xt) +Fy(-Xt)}dMn(x)
and proving the main theorem for members of this class.
must be made of the complex interaction of M
l-Fy(t)
as t
+
.
Some such account
(t) as (t,n) + (0,) and
Since moment conditions are perhaps a more natural
approach, we choose to introduce the index
borderline case
n
0}}
l-Fy(y)~y
"s
S
instead.
we choose to assume E
To avoid the
IYI s
+
<
.
ACKNOWLEDGMENT
This paper is based in part upon results obtained in the author’s
doctoral dissertation completed in 1975 at the University of California,
Irvine, under the direction of Professor Howard G. Tucker.
I would like
to thank Professor Tucker for posing the problem solved here, and con-
stantly expressing an interest in it.
versations with him.
I benefited considerably from con-
P. L. BROCKETT
372
REFERENCES
I.
Berman, S.M.
2.
Blumenthal, R. M. and Getoor, R.K. Sample Functions of Stochastic
Processes with Stationary Independent Increments, J. Math. Mech.
IO (1961) 493-516, MR 23, #A689.
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