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GEOMETRIC EQUIVALENCES TO THE ASYMPTOTICNORMALITY CONDITION
by
Galen R. Shorack
TECHNICAL REPORT No. 336
November 1998
Department of Statistics
Box 354322
University of Washington
Seattle, Washington 98195 USA
GEOMETRIC EQUIVALENCESTO THE ASYMPTOTIC NORMALITY CONDITION
G. R. SHORACK
Department of Statsiiics, University of Washington, Seattle, WA 98195, USA.
Abstract: The asymptotic normality of the sample mean of iid rv 's is equivalent to the wellknown conditions of Levy or Feller. More recently, additional equivalences have been developedin terms of the quantile function (qf). And other useful probabilistic equivalences could be cited.The emphasis here is on equivalences, and many other useful and informative equivalenceswill be developed. Many depend only on simple comparisons of areas, and those are theones that will be developed herein. [Because the asymptotic normality above is equivalent toappropriately phrased consistency of the sample second moment, many additional equivalencescan be developed in the simpler context of the weak law of large numbers. But this will be doneelsewhere.] Roughly, one can learn all about asymptotic normality by studying the conditionsin simpler settings. Here, we do the geometric part.
O. Introduction
, ... , ... be iid with distribution function (df) F(·) and quantile functionK (.) == F-1(-). The classical central limit theorem (CLT) states that when this distributionhas finite variance, then == y'n[.¥n - 1tJ/(1 -7d N(O, 1). When (12 is finite, it holds that
(0.1) t [K;(ct) V K 2 (1 ct)] / (12 -7 0 as t -7 0, for each fixed c> O.
But what if (12 is infinite? Let a> 0 be tiny. We agree that dom(a, a) denotes [0,1 a), (a, 1]or (a, 1 - a) according as X 2:: 0, X ~ 0 or general X, and that I<a,a (.) denotes K Winsorizedoutside dom(a, a). [For example, when X takes on both positive and negative values, thenI<a,a(-) equals K+(a), K(t), K(l - a) according as t ~ a, a < t < 1 - a, 1 - a ~ t.] Let ~denote a Uniform(O, 1) rv. Let X == K(~), so that X has df P, and let X(a) == I<a,a(O. ThenX(a) has mean (called the (a, a)- Winsorized mean) given by ji(a) == f01 I<a,a(t) dt and variance(called the (a, a)- Winsorized variance) given by
(0.2) o-2(a) = f~ f~ [rAs-rs] dI<a,a(r) dI<a,a(s) = f domre.a) f dom(a,a) [rAs-rs] dK(r) dK(s) ,
that increases to (12 as a \,. 0, whether (12 is finite or infinite. And this expression makes noreference to any mean (such as It or ji(a)). This is very useful! It turns out that
(0.3) z; == y'n[Xn ji(1/n)] /o-(l/n) -7d N(O, 1)
where -¥n == I:~ "Yk(l/n)/n) if and only if
as t -7 0, for each fixed c > O.
And condition (0.4) holds if and only if o-2(t) doesn't grow too fast; that is, if it is a slowlyvarying function of t as t --1' 0 (in the sense of the definition below). Thus it is appropriate toexamine slow variation in the infinite variance case. Shoud we examine it in the df domain orthe qf domain? We'll do both, in the next section. And since the problem is so 'central', we'lldo it very carefully.
This problem could be made harder, by examining it in the context of the CLT and. in thecontext of the general theory of slowly varying functions. But in fact, the equivalence of all theconditions in the next section follows in an elementary fashion from simple pictures and a dashof Cauchy-Schwarz (hence the title of this paper). In fact, the CLT will not even be mentionedin the theorems of the next section. It is also simpler if one ignores the general theory of slowlyvarying functions and uses only the fact that F and K are monotone, so properties of slowlyvarying functions won't be mentioned in the next section either.
But one to connect probability theory by showingpnlliv;~lpllt. conditions in next section is 0.3) (0
refined source for both is Feller(l966, p. 303). Properties of quantile functions (or, qf's)were developed in S. Csorgo, Haeusler and Mason(1989, 1988) (in an approach to CLT's usingweighted constructions of empirical processes). The Theorem 1.2 versions of conditions (0.4),(1.4), (1.6) (1. figured heavily in their work. Importantly, they brought out clearly thevalue of the Gnedenko and Kolmogorov condition (0.7) below. Of course, variations appearin various texts and articles on probability theory. The net result is that many conditionsequivalent to the ANC are now known, both in the df domain and in the qf domain. One resultof this paper is to emphasize and expand on this literature in a treatment specifically designedto provide many equivalent versions of the ANC.
Geometry: In this article we empahsize equivalences that can be established based onlyon simple geometric comparisons of areas), and many are possible. Since the ANC above isessentially a condition about lXI, Theorem l.l(A)-(D) deals with qf conditions re IXI. Thesecan be equated (using only pictorial comparisons) to all the classic conditions about IXI in thedf domain; see Theorem 1.1(E). But the qf versions of the ANC referred to above are not interms of lXI, and so Theorem 1.2(A)-(D) gives these other important qf phrasings of the ANC.
Variance consistency: Feller(1966, p.233) established a condition equivalent to the consistency of the sample mean of non negative observations. When this condition is rephrasedin terms of the consistency of the sample mean X~ of the squared values (with all Xl > 0,always) it becomes the well known equivalence (1.29) again. Maller(1979) also obtained theresult of Feller's problem, and discussed it for X 2 in Maller(1980). See also Raikov(1938).[See O'Brien(1980), S. Csorgo and Mason(1989), and Gotze and Gine(l995) for equivalencesphrased in terms of the negligibility of the maximal summand.] Thus the ANC can be studiedin the context of the simpler WLLN problem, and this is therefore a preferable context in whichto study and develop equivalences. Then, the most convenient equivalence is available for theeasiest possible proof, while all equivalences are available for applications and understanding.The full list of informative equivalences obtained is large.
Related work: The reader is also referred to Shorack(l998b) where the consistency of samplemoments is studied in the possibly infinite moment case for non iid rv's (conditions reduce to(0.4) in the iid case). While in this simpler context, many other equivalences are developed(probabilistic conditions, not geometric). And the reader is referred to Shorack(1998c) wherequite general necessary and sufficient conditions for normality are given (the non iid case iscovered with necessary conditions and with sufficient conditions, and these both reduce to(0.4) in the iid case). Both papers put emphasis on the fact that many equivalences arepossible. Having them available offers insight and makes other proofs simpler. point ofview and notation of Shorack{1998b,c) are specifically geared to that of the current paper, andso papers seem natural to mention next theorem is a teaser {given in
a J's row mdependent., F.
In
that L = log(I/t) and lex) = log(x) are slowly varying. They are the prototypes.Note also that when 0-
2 is finite, the Winsorized variance function ij2(t) is always slowly varying,since -t 0-
2/0-2 = 1 for each c > O.
Infinite variance facts. Whenever the variance in infinite, the Winsorized variance ij2 (a)completely dominates the square j12(a) of the Winsorized mean. Let ka,a' denote I< Winsorizedoutside (a, I-al
) . Gnedenko-Kolmogorov(I954, p.I83) showed (0.7) below, while (0.8) and (0.9)are then trivial. For every non degenerate qf I< having EI<2(.;) 00, we have:
(0.7) (J~-al II< (t)1 dt)2/ J~-al I<2(t) dt = 0 as (a Val) -t O.
(0.8) {alI<+(a)I+ aliI< (1 - al)/}2 / Ek~,al(';) -t 0 as (a Val) -t O.
(0.9) Var[ka,al(O]/Ek~,a'(';) -t 1 as (a Val) -t O.
1. Slowly Varying Partial Variance
Many functions can be learned from simple pictures. We concentrateon just such facts in section. Condition (1.29) has as the necessary and sufficientcondition of choice for the CLT (though (1.25) will receive the major focus in the programoutlined in the previous section); and we will soon be prepared with many equivalent waysto demonstrate either. [Necessary and sufficient conditions for consistent estimation of thevariance parameters V(l/n) and O' 2 (1/ n) (see below) are equivalent to those on the currentlist.]Notation Let Y denote an arbitrary rv , and let X IYI. Let F and I< denote the df and qfof IXI. For 0 < t < 1, let Xt == I«1- t). Let yt denote Y Winsorized outside Xt]. Define:
(1.1) vet) == ](2(1 - t), Vet) == f[t,I] v(s) ds, O'2(t ) == Var[yt].
(1.2) q(t) == 1«(1 - t), met) == f[t,l]q(s) ds, {t(t) == Eyt.
(1.3) U(x) == f[o,x] y2 dF(y), i\if(x) == f[o,x] ydF(y), Vet) == EY? Vet) + t v(t).
Theorem 1.1 (Partial variance, with symmetric Winsorizing)(A): The following are equivalent (as t -+ 0).
(1.4) 0'2(.) ERa = 12.
(1.5)V(t) == Vet) + tv(t) has \/(.) E R o =[.
(1.6) ret) t v(t)/V(t) -+ o.(1.7) VERa = £ .
(1.8) t [v(et) - v(t)J1V(t) -+ 0 for all 0 < e < 1.
(1.9) d(t) == Vi [q(ct) q(t)]/ JV(t) -+ 0 for all 0 < e < 1.
(1.10) t v(et)/V(t) -+ 0 for all 0 < e < 1.
(1.11) [meet) - m(t)]/ vlt Vet) -+ 0 for all 0 < e < 1.
(B): The following may be added to the above list of equivalences (as n -+ (0).always assume that an \. 0 satisfies sup <
(1. -+1 anyone specific an 0 and O<c<l.
any one SPE~CltlC < .
(C): The following are also equivalences (in which V(an) has been replaced by o-2(an) in thedenominator of the previous seven conditions). [This is very convenient.]
(1.20) [V(can) - V o-2(an) -+ 0 for anyone specific an ':,., 0 and all O<c<l.
(1.21) anv(an)jo-2(an) -+ 0 for anyone specific an \. O.
(1.22) [V(can) V(an)]jo-2(an) -+ 1 for anyone specific an \. 0 and all O<c<1.
(1.23) an [v(can)-v(an)]jo-2(an) -+ 0 for anyone specific an \. 0 and all O<c<1.
(1.24) va;; [q(can) -q(an)]j o-(an) -+ 0 for anyone specific an \. 0 and all O<c<1.
(1.25) anv(can) j o-2(an) -+ 0 for anyone specific an \. 0 and all O<c<1.
(1.26) [m(can) - m(an)]j [~o-(an)] -+ 0 for anyone specific an \. 0 and all O<c<1.
(1.27) r(t) == tv(t)jo-2(t) -+ O.
(D): The most useful choices are an ef n, or an == ljn with c == f, or an == fnjn and f n \. o.
(E): The following are equivalent (as x -+ (0) to the previous conditions.
(1.28) U E Uo
(1.29) R(x) x2 P(X > x)jU(x) -+ 0
(1.30) x[M(cx) lvf(x)]/U(x) -+ 0
(that is, U is slowly varying at (0).
(equivalently,R(x n ) -+ 0 with limxn+dxn < (0).
for anyone (equivalently, all) fixed c> 1.
(1.31) U(x) defines a function in Uo .
Theorem 1.2 (Partial variance, Winsorizing equal fractions) Consider an arbitraryrv X with df F and qf J<. Let ka,aO denote J<O Winsorized outside dom(a, a), and nowredefine
(1.33) q(t) [J<+ (1 - t) + Ie; (t)] ,
tneorem we
V(t) == f[i,l]
d d -() E[k.'3 an f.L t:
(a)
(c)
(b)
we show that T~2 0 is equivalent to a list £4 that contains the (1.14) (based on (1.32))of £2. Thus 0 is on both £3 and £4. Thus all conditions on the lists £1, £2, £3and £4 are In Shorack(1998b) we add to the combined list the condition (0.5)that S;J(j2(1jn) 1, and also (0.6). In Shorack(1998c) we add to the list the condition(0.3) that z; == yIn[X(an) jl(an)]j(j(an) -7d N(O,l) (for an = and other appropriatean)' Knowing these last two facts, it is indeed very interesting to have a large list of suchequivalences. 0
Proof. Clearly, all of (1.4 )-( 1.27) (in the context of both Theorems 1.1 and 1.2) hold when(J2 < 00. So from here on we assume that (J2 00.
(A): Now (1.5) implies (1.6), since for each d » 1 Figure 1.1(a) demonstrates that
t < tv(dt)+t[v(t) v(dt)]V(t) - V(dt) + (d - 1) t v(dt)
< _1_ d t v (d t) + ~ t [v (t) - v (d t)]d-1 V(dt)
< 1 . d V(t)-V(dt)- d - 1 + d 1 V(d t)
(d) -7 1j (d 1) + 0 .
Since this holds for any d > 1, it gives (1.6).Next, (1.6) implies (1.7) since for each fixed < c < 1 Figure 1.1(b) demonstrates that
(e) [V(ct) - V(t)]jV(ct) ::; [(1 - c)jc][ctv(ct)jV(et)] -7 O.
Suppose (1.7), that V E R o. This implies [V(ctj2) V(t)]jV(t) -7 0, and Figure Ll(c)then demonstrates that
(f) (ctj2) [v(ct) - v(t)]jV(t) ::; [V(ctj2) - V(t)]jV(t) -70;
and so (1.8) holds.Supposing (1.8) about vO, we will establish (1.9) about d(·). Now
(g) x [d(t) If
-=}
---'--==::-'-'- = D(t) -t 0 .
We obtain (1.11) implies (1.6) (start on the right hand side below) via
{tv } 1/2
(1 - c) = (1 - e)--=====(i)
(k)
Then (1.7) implies (1.11) via Cauchy-Schwarz, in
l : q(s) ds {(I-e)t q2(S\)dS}1/2 {V }1/2(j) = d < '< -to
JtV(t) - tV - .
We next show that (1.4) is equivalent to the simpler (1.5). When a2 ( . ) E no, we use theGnedenko and Kolmogorov results (0.7)-(0.9) to write
V(et~ V(t) = [a2(ct) - a2(t ) + p,2(ct) . P,2(t)] X a2 x ~2(t)V(t) a2(et ) a2(ct) (t) V(t)
[a2(ct ) - a2(t ) ] a2(ct) \. \
= a2(ct ) + 0(1) + 0(1) x a2(t ) X [1 + 0(1)] = 0(1);
which implies V(.) E no. When V E no, the same Gnedenko-Kolmogorov results (0.7)-(0.9)give
a2 (ct) - a2 (t )a2 (t )
V(ct) - V(t) p,2(t) - p,2(ct)- + '--'-----;;,-;-'-,-;-'--'-V(t)[1 + 0(1)] a2(ct )
(I) . --'--'::----'-'-[1 + 0(1)] +0(1) + 0(1) 0(1) ;
which implies a2( . ) E no.
(B): We next show that (1.14) implies (1.6). Suppose that (1.14) holds for even one an '\. 0having an/an+1 < 00. We are given that T'n anv(an)/V(an) -t O. This implies (1.6) via
t v(t) an an+1 v(an+1) an V(an+d(m) sup -- < -- -- T'n+1 ---'--:--'-:.-'-
Un+l:5t:5Un V(t) an+1 \!(an+d V(an) an+1 V(an)
since
(n)V
1 2': V
We now show that (1.21) implies (1.27), the converse being trivial. We could do so byreplacing V(·) by 0'2 in the denominator as we read (m) and (n); but in fact, (0.8) allowsus to replace 0'2(-) by instead while repeating and (n). Note that equation (n) (usingV(.)) now introduces the two new terms ) and an+lv(an+r)/V(an+r), both ofwhich also converges to 0 (by (1.21), with a (an) in its denominator). Thus (1.27) is on ourlist. Having established (1.27), we can trivially claim that (1.21) holds with an replaced bycan in all three locations. This allows us to reread (e) (with the denominator V(an) replacedby O'2(an ) throughout), to see that (1.21) and (1.27) imply (1.22). Rereading (f)-(h) (with thesame denominator replacement) shows that (1.22) implies (1.23), which implies (1.24), whichimplies (1.20). We now close the circle on (1.20)-( 1.24) by noting that (1.20) implies (1.13),again using the Gnedenko and Kolmogorov result (0.9). We can add (1.25) by the same trivialargument as before. Rereading (i)-(j) (with the new denominator) then allows us to add (1.26).
The proof of Theorem 1.2 is nearly identical. All but lines (h) and (j) are identical; line(i) is identical because m is still the integral of q. But the other two are not identical becausewe no longer have q2 = v. But we do have v2 = [q+F + [q-F where q = q+ + q'>, and that isenough. Just factor the two pieces separately in (h) and (j), and apply the trivial inequalities(a + b)2 ~ 2(a2+ b2 ) and a V b ~ a + b ~ 2(a V b). 0
Proof. (E): We will prove this part separately. We first show that (1.29) implies (1.28).Suppose that (1.29) holds, so that R(x) x2p(X > x)jU(x) ---+ 0 as x ---+ 00. If c < 1, then
(a) [U(x) - U(c.:r)]jU(c:r) J(cx,xj y2 dF(y)jU(cx) ~ c- 2[(cx)2p(X > cx)jU(cx)] ---+ 0;
and for c > 1 it is analogous that [U(cx) U(x)]jU(x) ---+ O. Thus, U is slowly varying, as in(1.28). Suppose U is slowly varying,as in (1.28). Then for all x 2 (some x,) :
(b) ---'--..,.--'-- < E.
(d)
E U(2x)< 4" U(x) ~ £[(1 +
E U (4x) U(2x ) r
< 42 U(2x) U(x) <q(l + , ......
So for x 2 XI' we add these to
> < <E'
_~ --'- < ~ _U_(4_x_)- 42U(2x)
£:::; (1 +
£< (1 + 2£){1 + --'-----,------"-}
Add these to get R(x) --1- 0 as x --1- 00, as in (e), establishing (1.29).We obtain (1.28) implies (1.30) via Cauchy-Schwarz, in
(i)y2dF(y) [U(ex) - U(x)]
-"---'-=-'::-;--,-----'-'-=- < :::..::::.---:::-:::--;--::---'- < T - ( , --1- 0 .- U z )
(j)
Now (1.28) implies (1.31) follows from
U(ex) - U(x) U(ex) - U(x) , (e2+ 1)x2P(X > x)- < T"( ) --1- O.U(x) - U(x) U x '
since (1.28) now brings (1.29) with it. Then (1.31) implies (1.29) since
[U(ex) - U(x)]jU(x) = [J~x y2 dF(y) + (ex)2p(X > ex) x2P(X > e)]jU(x)
> ([x2[P(X > x) - P(X > ex)] + e2x2P(X > ex) - x2P(X > x)}jU(x)
(k) = [(e2 1)je2]j {I + 1/[(ex)2p(X > ex)jU(ex)]} ,
and the extreme left term going to 0 forces (ex)2p(X > ex)jU(ex) --1- O.The second condition in (1.29) suffices since
(1) x2
P(..X > x) < [-1'- X;+1] x; P(X. .> xn ) < 0(1) R(,) 0sup U"(x) - 1m x
n2 UT(X~) _ X Xn --1- .
xnS: XS:Xn+l "
That (1.29) and (1.6) are equivalent is the subject of the next section. 0
2. Specific Tail Relationships
We list two the re the CLT and variance estimationand second re the WLLN. They compare the qf with partial moments.
Theorem 2.1 (The (z , t)-tail equivalence; CLT) Consider any X ~ 0, with qf K, Then
(2.1) lim sup tIC'(l-t-+O
[(1'(1 - s) ds = lim sup x"P(X > x)j f[o.x] yr dF(y) ,X-t-OO '
for each r > O. The same is true for the lim inf , and for the lim (if it exists).
Proof. By change of variable
(2.2) f[o,F(xlJ [(I' (s) ds f[o,x] yr dF(y) for all x.
Define t = 1 F(x). Then
(a) r(t) tI<"(l.~ t) = R(x) == xl'P(~ > x)f[0,1 K (s) ds f[o,x] y dF(y)
if t 1<"(1 t) = xl'P(X > x).
But examination of the figure shows that (from an x point of view)
when and only when
(2.4) x is any point in the domain of some flat spot of F, other than the left endpoint.
Also, R is /" continuously across any such flat spot; and R(·) approaches a limit at both endsof each flat spot. So far we have considered all values of z , but we have omitted values of t notin the range of the F(x)'s. But equality fails in (2.3) (from a t point of view) when and onlywhen
(2.5) t is any point in the domain of some flat spot of K, other than the left endpoint.
Also, r is continuously /" across any such flat spot as t /" ; and r(·) approaches a limit at bothends of each a flat spot. Finally, note that all right and left limits achieved by R(·) across flatspots are achieved as and right limits of . and vice versa. suffices for the claim.o
Definition 2.1 [Order-r qf's) A K is order-r >
-+ O.
> 1(.
Proof. As the figure shows, the values (or limit of the values) oftIK(l-t}l'" and x" P(X >are equal at the associated pairs of end points re any flat spot of F or any flat spot of K; and
are monotone across flat spots. At all non flat spot pairs of points, theyare equal. That at all of the key values (i.e., local extremes) the quantities in question areequal. 0
References
[1] Csorgo, S., Haeusler, E. and Mason, D. (1988). The asymptotic distribution of trimmedsums. Ann. Probability 16, 672-699.
[2] Csorgo, S., Haeusler, E. and Mason, D. (1989). A probabilistic approach to the asymptoticdistribution of sums of independent identically distributed random variables. Adv. inAppl. Math. 9, 259-333.
[3] Csorgo.S. and Mason, D. (1989). Bootstrapping empirical functions. Ann. Statist. 171447-1471.
[4] Feller, W. (1966). An Introduction to Probability Theory and It's Applications. Vol. 2.John Wiley and Sons, NY.
[.5] Gine, E. and Gotze, F. (1995). On the domain of attraction to the normal law and convergence of selfnormalized sums. Universitiit of Bielefeld Mathematics Technical Report9.5-99.
[6] Gine, E., Gotze, F. and Mason, D. (1998). When is the student t-statistic asymptoticallystandard normal? Ann. Probability 25 1.514-1531.
[7] Gnedenko, B. and Kolmogorov, A. (19.54). Limit Distributions for Sums of IndependentRandom Variables. Addison-Wesley, Cambridge, Mass.
[8] Maller, R. (1979). Relative stability, characteristic functions and stochastic compactness.J. Austral. Math. Soc. Ser. A 28,499-509.
[9] Maller, R. (1980). On one-sided boundedness of normal partial sums. Bull. Austral.Math. Soc. 21, 37:3-391.
O'Brien, G. (1980). A limit theorem for sample maxima and heavy branches in Galtonwatson trees. J. Appl. Probab. 17 539-54.5.
