THE EFFECT OF RANDOM SCALE CHANGES ON LIMITS OF INFINITESIMAL SYSTEMS

Suppose S={{Xnj,   j=1,2,…,kn}} 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 (γ,σ2,M). If {Yj,   j=1,2,…} are independent indentically distributed random variables independent of S, then the system S′={{YjXnj,j=1,2,…,kn}} is obtained by randomizing the scale parameters in S according to the distribution of Y1. 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. If such sums converge to a distribution determined by (γ′,(σ′)2,Λ), then the exact relationship between (γ,σ2,M) and (γ′,(σ′)2,Λ) is established. Also investigated is when limit distributions from S and S′ are of the same type, and conditions insuring products of random variables belong to the domain of attraction of a stable law.


INTRODUCTION AND SUMMARY.
The classical linear model for the relationship between empirical data Y and theoretical or "true" data X is to assume Y X + e where e (the error) is a random variable independent of X with E(e) 0. In some cases however the error tends to depend upon X.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 population 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.Specifically, if S {{Xnj j=l,2,...,kn }} is an infinitesimal system of random variables whose centered sums converge in distribution to some (infinitely divisible) random variable X, and if {Yj, j=l,2 is a sequence of independent iden- 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 converge, say to Z.In case S' does converge, we wish to determine the exact relationship between X and Z.
Here we take an index of convergence S of the system S and under + hypothesis E IYII S < for some 8 > O, we obtain a necessary the weak and sufficient condition for convergence of the system S'.When S' con- verges we then obtain the exact relationship between the limit distribu- tions 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 .In this case we find S=.In the classical central limit theorem (= 2) we find that if EY 2 < , 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 < 2 we find that if E IY 1 + < and X is attracted to a stable of index , 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.

PRELI>] NARIES AN INDEX OF CONVERGENCE
Let us recall several important facts from [3].
If S [{Xnj j 1,2, ...,kn is an infinitesimal system of random variables, the functions M are defined by where F X is the distribution function of Xnj n,j We shall say that the system S is convergent to X if there exists a k of real numbers [C n= 1,2,...} such that E nIX -C sequence converges n j= n,j n in distribution to X as n + .In this case X is infinitely divisible with characteristic function whose logarithm is of the form 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 We shall hence forth use % to denote %j=l and Fnj to denote F X nj DEFINITION.Let S be an infinitesimal system.The index S for the system S i__s defined by S infiX.> O" lira (x n) + (0,),J dM (t) O} n with S 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 M for the Levy As a similar result in [4] it was shown that then S converges to X~(,2,M) we have for a e Cont(M) Ix I<: Ix < with S if the above set is empty.An easy way of calculating S is then given by noting that I im in f n+ Olxi< ,}xlYdMn,,[x)= for any e > 0 if Y < >S" We next recall a variational sum result proved in [4].
THEOREM I. Suppose S is an infinitesimal system converging t__o X (,2,M), and suppose that g is a bounded function which satisfies g(x) 0(Ixl ) a__s x 0 for some > S and which is either continuous o__r is of bounded variation over (-,0) and over (0,) with discont(g) Q discont(M) @.Then we have lim Z E(g(Xnj)) lim n+ n+< (-,) g(x)dM n (x) f(_ ,") g(x)(x).R]D4ARK I. Some comments about the index S are in order.If the system S is convergent to X~(cr,e2,M), then M <_ S <--and either of the .M see [4] where two equalities is possible.For an example where S it is shown that if [XI,X2, ...] 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/Bn, j 1,2, ...,n}} M if < 2 In [3] then S =aJ.It is clear that in this case 0 if 2 an example is given showing S can be anything.Namely for <_ a conver- gent system S is found such that S =%" To show that S 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 % > M there exists a system S converging to (0,0,M) with S =%" The index S has proved to be the appropriate index for studying variational sums of in- finitesimal 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).
is that the following three conditions all hold A) Mn(X + M(x) for all x e Cont(M) Jlim sup [ x/(l+x2)dM(x) for any r > 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 yields the needed tool for obtain- ing conditions insuring S' is convergent.The precise formulation, and the exact relationship between the limit laws of S and of S' is given in the following theorem.Also to be noted is that the convergence properties of S' are the same as that of S (i.e., S S ') and the behavior of the limiting Levy functions are the same (M= ).In particular since an infinitely divisible distribution is continuous if and only if the corres- ponding Levy spectral function is unbounded, it follows that the limit distributions of S and S' are simultaneous continuous or not continuous.THEOREM 3. Suppose that S [[Xnj j 1,2, ...,kn}] is convergent t_9_o 2 d yj XN (, ,M) i.e., EXnj -c >X.Let J= 1,2,...} be a sequence of n independent identically distributed random variables with non-degenerate S+ common distribution function Fy and with E IY < for some > O.
NDTE: I.If S < 2 then (3.1) is automatically satisfied with CO= 0 by (2.3) and necessarily q2 0 (otherwise S >-2), so 30 2 says () 0 in this case.Also when S 2 and (3.1) holds we need only assume E(Y2) < to obtain the conclusion.20 2 It should be noted that the mere meaningfulness of formula may not be sufficient for the conclusion of the theorem.Examples of this are given in [3].
Here XA(X I if x e A and 0 if x 4 A. Let x e A and 0 if x 4 A. Let and let (x Then using (3.2) in (3.3) we have for x < 0 while for x > 0 we have [l-Fy (x/t) ]dM n (t) + S(0,)Fy (x/t)dMn (t) g (x/t)dMn(t) z{n (-, O) g(x/t)dFnj (t) S(0,) -Ig(x/t)dM n(t).
3) In this part of the proof we shall determine how to calculate integrals of the form S(.,) f(x)dA n(x) and S(-oo,) f(x)dA(x) where A (a,b] Xi (u) dry (u)dM n (t) An(b)-An(a) S(_,)S(_,) (a,b] S S X (tu)dFy (u)dM n (t) so that S f(x>dA (x) S S f(ux)dFy(u)dMn(x) 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 by A and M by M.
However concerning the second term above we see that lira lim sup Ix ul dFy(U)dMn(x) 0.
Thus to show S S' we must only show lim lira sup Ix lu dFy(U)dMn(x) 0 if 5 > S and lira inf S Ix15 S lulgdFy(U)dMn(X) for any e > 0 if 5 < S" Suppose then that 5 < S and choose e I so that f Ix lSdFy(X) > 0.
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.
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 we have Z(S x S UdFy (u) dFnj (x)) I< I<,} lul<:/ : (e)= UdFy (u)dFnj (x) UdFy (u)dFnj (x) {S udFy (u)dFnj (x) }.
Thus to complete the proof of Claim 3 we must show llm lim sup I f2  (2)) 0 so that (3.16) holds.In a similar manner, concerning (3.17  (so that lim lira sup n () (') lim lira inf 2()) then necessarily n +0 n 0 n (3.1) holds.By part 7) we know that we may assume S 2dN lim sup x (x) < so that all of the previous equations leading 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, there exists norming constants [B n= 1,2,...} and centering constants [An, n= 1,2,...} such that E n I X /B -A converges in law to a distri- j= j n 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. Le__t [XI,YI,X2,Y2,...} be independent random variables x having distribution function F X and Y havin$ distribution function n n Fy I) l__f EIYI 2 < , th.en X (2) .impliesYX e (2) an__d the sne norming constants wor k.Conversely i__f YX e D2) .t.he.nX D(2) an__d the sne norming constants work.
PROOF.The direct statement in I) and 2) will follow from Theorem 3 once we show that (3.1) holds.Let S= {[Xj/Bn, j= 1,2,...,n}} then as proven in [4], X e O() implies S =" For < 2 we have (3.1)holding with CO=0 by (2.4).For =2 we know (see Feller (1971), [6] pg. 314) that necessarily B-Yl <B y dFX (y) + C0 as n + Let us suppose now that YX e (2) and EIYI 2 < .We shall use the general limit theorem to show E n. i X /B -A converges to a normal distri- j= j n n bution.With M given by (2.1) and A given by (3.3)we shall show n n A (x) + 0 for x0 implies M (x) + 0 for x 0. Indeed let t be such that O, thus the limiting Levy function is 0.
I =C0 and Z n.j=Ixj/Bn-An converges to a normal 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-2 ducts.He shows that if EY < and X e () then X+Y e () and the 2 same norming constants work.If X +Y e () and E(Y) < m, then a slight modification of his methods yields X e D().ii) In Theorem 4 we can actually calculate the scale change in- volved in the distribution of the limit laws of S and S' when <  iii) It can be shown that for < 2, YX e D() and EIYI + < implies that EIXI c' < and EIXI c+6= for all 8, hence S " I suspect that in fact X e D() however, I have been unable to establish this for #2.---we have A (x) + + A(x) where A(x) n IS 0 is given by 2 of Theorem 3.This is a Levy spectral function when S < 2. Thus we see that some random scale changes introduce a stable component into the limit law.
PROOF.This follows immediately from Theorem 4 by utilizing the necessary and sufficient conditions given in Feller (1971) for a dis- tribution 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 EIYI S < and we obtained a new system S' which was convergent.Moreover the limit distributions of S and S' were of the same type.In problems where X represents a "true" or nl theoretical measurement of some occurrence and Y the scale change in 1 the measuring device used to measure the occurrence, we obtain as an observation the product X .Y.
It is of interest to determine when nl I limit distribution calculated from the empirical data [[XnjYj, j 1,2,...,kn}} is of the same type as that from [[Xnj j 1,2,...,kn }" For normed sums in the domain of attraction of a stable law, Theorem 4 answers the question and Remark 2ii) allows us to calculate the scale change.In general the following theorem tells us that in non-stable limits the empirical data may yield a different type distribution than the theoretical data [[Xnj J 1,2,...,kn }" THEOREM 5.In order that a limit distribution b__e preserved i__n type S+5 under all random scale mixtures with E IYI < it is necessary an__d sufficient that the limit distributio type be either purely stable or a mixture of stable and normal.
PROOF.The sufficiency follows from 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.Then with S and S' as defined in Theorem 3 we know Z~(', (')2,A) is of the same type as X~(,2,M), thus (,)2 2 2 a and A(x)=M(x/a) for some constant a.As in 2) of the proof of Theorem 3 we know that I f IM(x/t)IdFy(t) A(x) IM(x/t) IdFy (t) x<0 x>O (3.25)If M(x) 0, then both X and Z are normally distributed.If M(x) 0, then we must show M is given by (3.22) of Theorem 4. Using the fact that A(x) =M(x/a) in (3.25) U) iu -+ [e -i }dM(x) l+x We shall simply write X (,2,M) to express the fact that X has such a d d b characteristic function.The symbol S represents l:n[f +S a 0 -c a -c and b 0}.All integrals are taken in the Lebesgue-Stieltjes sense.

Ix 2 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- S UdFy (u)dF (x)) 2 <-S x S u dFy(U)dFnj(x)

2 Theorem 3 (
Y) is a slowly varying function of t yields(3.1)  in this case also.Applying Theorem 3 we obtain the convergence of S' [[XjYj/Bn, j 1,2,..o,n}}.If = 2 then Mm0 so by 20 of Theorem 3, A= 0 and S' converges to a normal distribution.If < 2, then by 30 of t)d(t -) if I-Fy (x/t)d t -) -C2 f(0, ) '(_ oo, o) C I -C1 S Fy(X/t)d([tl-c)-C 2 f {Fy(X/t)-l)d(t -c) if ., X Y is in the domain of attraction of a stable law with character- n n istic exponent , and the same norming constants work.
finite.In view of part 7) of the proof of Theorem 3 have CO= C I Now by Fatou's lemma and by(3.21) 23) by the change of variables z= xt Combining our re- sult with his we find that if X,Y,e are independent random variables, 2.
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.