Stochastic Processes and their Applications 32 (1989) 93- 107

North-Holland 93

NECESSARY CONDITIONS FOR NONLINEAR FUNCTIONALS

OF GAUSSIAN PROCESSES TO SATISFY

CENTRAL LIMIT THEOREMS

Daniel CHAMBERS”

Mathematic.s Drpartment, Boston College, Chestnut Hill, MA 02167, USA

Eric SLUD**

Received 28 July 1987

Revised 26 October 1988

Let X=(X,, t CR) be a stationary Gaussian process on (R, .F, P) with time-shift operators

( CJ,, s E R) and let H(X) = L’(f), u(X), P) denote the space of square-integrable functionals of

X. Say that Y t H(X) with EY =O satisfies the Central Limit Theorem (CLT) if

A family of martingales (Z,(t), t 2 0) is exhibited for which 2, (“c) = Z,, and martingale tech-

niques and results are used to provide suficient conditions on X and Y for the CLT. These

conditions are then shown to be necessary for slightly more restrictive central limit behavior of Y.

AMS Subject Ch\jfication.s: hOF05, 60G10, 60644

multiple Wiener-It8 integral 3 stochastic integral with respect to square-integrable

martingale * predictable variance process * martingale Functional Central Limit Theorem * Band-

weighted Central Limit Theorem

1. Introduction

Let X = (X,, t E [w) be a (progressively measurable) stationary Gaussian process on

a probability space (0, 9, P), with EX,, = 0 and E(X,,X,) = I, e”‘a(dx) for a fixed

nonatomic probability measure v on Iw, called the “spectral measure” of X. Take

9= a(X) to be the a-field generated by the variables X,, t E [w, and denote by

{U, E Iw} the family of “time-shift” linear isometries on H(X) = L”(R, 9, P) which

* Research supported by Faculty Research Grant, Boston College.

** Research supported by the Ofhce of Naval Research under contract NOOO14-X6-K-0007

0304.4149/89/S3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

94 D. Chamher.~, E. S/UC/ / Functional central limit theorem

satisfy U, (X,) = X,, , for all s, t E Ft. Call any element Y of H(X) a “square-integrable

functional” of X. We say that Y (with E Y = 0) satisfies the Central Limit Theorem

(CLT) if

1 (V,(T))-“’ U,(Y) ds: ,V(O, 1) as T-co

where V,(T) = E{Jb U,( Y) ds}‘. Many authors have studied the problem of describ-

ing classes of functionals Y (of stationary processes X) which satisfy the CLT. Sun

(1963, 1965), Cuzick (1976), Breuer and Major (1983) and Giraitis and Surgailis

(1985) obtained progressively larger classes of functionals Y of the form G(X,,)

which satisfy the CLT, and the latter two papers included examples with (r con-

tinuous-singular (and a”“’ absolutely continuous with respect to Lebesgue measure

for some m 3 2). Maruyama (1976, 1985, 1986) and Chambers and Slud (1988)

obtained general sufficient conditions for Y = Ir((X,: t E R)) E H(X) to satisfy the

CLT where Y might depend on infinitely many X,. Other authors (Taqqu, 1975,

1979; Rosenblatt, 1979, 1987; Dobrushin and Major, 1979; Fox and Taqqu, 1985)

have obtained many interesting examples and classes of Y for which the suitably

normalized random variables SC; CJ,( Y) ds converge in distribution to non-Gaussian

limits.

This paper is concerned primarily with analytical necessary conditions on (7 and

Y for the CLT to hold. The functionals Y = h(X) are allowed to depend on arbitrarily

many coordinates X,, and as in Maruyama (1976, 1985, 1986) and Chambers and

Slud (1988), conditions on Y will be expressed in terms of the well-known representa-

tion of mean-0 functionals in H(X) as sums CT_, I& (Major, 1981, Section 6;

Chambers and Slud, 1988, Section 1) of multiple complex Wiener-It6 stochastic

integrals, which have the following properties:

(1”) Each integrand ,f;(x,, . . , xk) belongs to the space L’(cr“, sym) of complex-

valued functions which are square-integrable with respect to k-fold product measure

ah on [w“, are invariant under permutations of their arguments and which satisfy

.&(x) =.fA(-x), where “ ” denotes complex conjugation.

(2”) The k-fold stochastic integral operator v%!Z, is a linear isometry from

L”( d, sym) onto the subspace H,(X) of H(X), where H(X) is a real Hilbert space

with inner-product (Y”‘, Y”‘), = E( Y”’ . Y”‘).

(3”) The subspaces Hk = Hk (X) are mutually orthogonal, and H(X) = RO

@F=, H,(X). The subspace rW@@;“_, H,(X) is the closed linear span in H(X) of

the polynomials of degree m or less in the variables X,, r E [w.

(4”) I,( 1) = X,,, and if fA E L’(rr“, sym) and s E Iw, then

u,(r,(fh)) = I,(c “(‘I+ -‘+ ‘i)fA(X, ) . . . ) XL)).

(5’) If r(t)= I,(&_,,,,) and p(t)- I,(-$(xto,,,-Xr r,o,)), for t30, and if T(t)= I’( - t), p(t) = -p( - t) for t < 0, then r( . ) and p( . ) are real independent mean-0

continuous martingales on [0, m). Defining P(t)-f’(t)+ip(t), we have P(-t) =

p(t) a.s.

D. Chamherc, E. S/d / Functional central limit theorem

It follows from these properties that if Y = C,* ~, 1~ (.fi 1, then

U,(Y) ds = f hh.~), h=I

where

A,r(x) =

and

.,(-r,-,:(I,,‘II,(Y)d~)*=~~,(kl) ‘~~;l~i,,(r)l’~(dx,)-..-(d-i,)

The further properties of multiple Wiener-It; integrals which have previously

been used in proving CLT’s via the Method of Moments are formulas (the “diagram

theorem” of Dobrushin, 1979) for expectations of the form

E[(~,,(.f~,))‘l . . (h,,(.h,,)Y”l.

The approach to CLT’s in the present paper is quite different: our results here rely

on stochastic calculus and Martingale (Functional) Central Limit Theory applied

to the functional j,y U,(Y) ds expressed as a stochastic integral with respect to the

square-integrable (complex) martingale (p(t), t > 0). Our starting point is the follow-

ing pair of known properties of multiple Wiener-It8 integrals (cf. Kallianpur (1980,

Theorem 6.7.1), where the same facts are presented in different notations in the case

where (T is absolutely continuous with respect to Lebesgue measure).

(6”) For each m 2 1 and t > 0, let x:,? denote the L2(~“‘, sym) function x, ,,,,f,‘( . ),

and put xb- I and L’(cr”, sym) _= R. Then, given,f, E L7(a”‘, sym), k 2 1, the functions

,fk, and .ff, defined for t > 0 by

A(u, > . . ,Uh ,)=(fh(UI,...,UI,~I,t)+.f~(U,,...,UI,~,,-r))

xx; ,(~,,~.‘,%I)

and

.fL(u,,..., uh ,I = i(.h(u,, . . . , kI, 1) -.h(u,, . . , cl, -t))

xx;Ll(~,,..., %,I

are L’( a’-‘, sym) functions, and for kz2 the random functions Ik_,(lk,) and

Ik_,(j’&) are adapted and predictable with respect to the increasing right-continuous

a-field family

9, = a({P-null sets}, (p(s): 0s s G t)), t 2 0, (1.1)

I

1

I

x

(7”) IL(&) = I,~,(_&,)~‘(dt)+ 1~ ,(.f?,)p(dt) 0 0

where the integrals on the right are L’-sense stochastic integrals with respect to

square-integrable martingales in the sense of Kunita-Watanabe-Meyer.

Property (6”) is immediate from ( lo)-( .5”), with predictability following easily by

induction on k, and (7”)-which is easy to check for special elementary functions

,fL such as those of Major (1981, pp. 24-25)-is proved by the usual techniques of

L’ approximation.

For simplicity of expression, we extend our notation of multiple Wiener-It6

integrals: whenever g,< : [WA +C is a &square-integrable complex valued function

which is symmetric in its arguments, define

r,(gr)-i{I,(g,~(x)+g,,(-x))-iz,(ig,,(x)-ig,,(-x))}.

Taking expected modulus squared of both sides of the last equation and applying

(2”) to the right hand side, one checks easily that again

Now P(t) defined in (5”) can be expressed as I,(x,,~,,,), and formula (7”) becomes

i

i\

(7’) Ik (.L 1 = 1~ ,(.h(. Oxl;‘,(. ))P(df) > LX>

where the ssX is defined separately on (-CO, 0] and on (0, CO) as a complex stochastic

integral.

The primary consequence which we draw from (6”)-(7”) is that the square-

integrable martingale

II

T Zr(r)= E U,(Y) dsl9, JKVI (1.2)

0 II

has an explicit stochastic-integral form for each fixed Y.

Theorem 1. Let Y =I;‘=, IA(_fk)~ H(X), and let {S,} be given 6-v (5”) and (1.1).

Then jbr t E [0, 11, the stochastic process Z,(t) qf (1.2) is up to equiualence an a.s.

continuous 9,martingale with the stochastic-integral representation

I

Z,(f) = I’

G,(s)P(ds) =2 (I

Re( G,(s))T(ds) ~ I

’ Im(G(s))pu(ds) -I 0 0 1

for t 30, where, in the notation qf (7’), the predictable complex random integrand

GT( . ) is dcjined by

L,h,A~, s)xL,(. )) (a a.e.)

and satisfies G,( -s) = G,(s).

In the next section we apply the structural information of Theorem 1 to the

question of asymptotic normality for T +CD of the random variables Z,(a).

2. Statement of results

Necessary and sufficient conditions are known (Rootzen, 1977; Ganssler and

Hausler, 1979) for the Functional Central Limit Theorem to hold for uniformly

square-integrable sequences of continuous martingales. Of course, sufficient condi-

tions for the CLT are available (McLeish, 1974; Helland, 1982), but no good general

necessary conditions for the martingale CLT are known. For this reason, and because

our main interest is in necessary conditions, we introduce a slightly more restrictive

limit-theorem property for (mean-O) functionals Y =I’;=, I,,(.h) E H(X):

Say Y satisfies the Band-weighteti Central Limit Theorem (BCLT) if the family

of %[O, CO) functions of t is uniformly equicontinuous, and if for every positive

integer m, vector (Y E I?“‘, and 0 < t, <. . . < t,,, - CO, the random variable

Y”J= : IL ; cy,~~(X),f~(X) h-l ( I 1 >

either satisfies the CLT or is such that 7- 2 E iJ U,( Ye,‘) ds I/ Vv(T)+O as T-tacl.

0

There now follows easily from McLeish (1974) and Rootzen (1977)

Theorem 2. Suppose that Y = C, I,(,fk) E H(X) and G,( .) are us in Theorem 1.

(a) 1. I lGAs)I’~(d ) s s 1 as T + ~0 then Y satisfies the CLT.

(b) [f Y satisjies the BCLT and for some E > 0, lim sup,,, EZF’(m) COO, then

I IG,(s)l’o(ds) s 1 as T+ cc.

To clarify the relationship between the CLT and BCLT for Y, we next reproduce

the conditions shown by Chambers and Slud (1988, Theorem 1) to be sufficient for

CLT, indicating also how slight is the additional condition needed to imply BCLT.

Theorem 3. Suppose that Y = CT=, Zk(fk) E H(X) ,forfixed m 2 1, and also

(i) u is absolutely continuous with respect to Lebesgue measure on R, and its

density f is such that on compact sets

(min{.f( . ), kf})““’ -,f *n’ un@rmly as M + ~8;

(ii) O<c (k!) -’ lim inf km’ I iwA

I.I&)l’x~,,, t...+,i,l- ,,,o”(dx), I, 11-o+

1 (k!).‘limsup h ~’ I

w” I~(~)l~x~,,,+...+,~~,- ,,,ck(d-~) < 0~; !. I,-01

(iii) for each k 3 m,

lim sup lim sup h -’ M-oc h-O+ I

rm“ lf,(~)12xc,.~,+...+.~,,-~ /t,~/i(x)~ -.v&W) = 0.

Then 0 < lim inf,,, T-’ V,(T) G lim sup,,, T-’ V,(T) <co, and Y satisjies the

CLT. Suppose in addition that Y satisjes the following two conditions:

(ii’) ,for each 0 s s < t, if

O<c (k!)-’ lim sup h-’ I

I.fi(412xc,., + . ..+T.J- ,,,~k(W, k I, -0 + [-‘,f]~~\r-\,,]l~

then also

Then y satisJies the BCLT.

The necessary conditions of Theorem 2(b) for BCLT are still probabilistic rather

than purely analytical. We provide equivalent analytical conditions in the special

case of homogeneous second order stochastic integrals. Here the uniform-integra-

bility hypothesis of Theorem 2(b) is automatically satisfied, according to the

following proposition.

Proposition 4. Jf Y=CT_, I,(fL)rHM(X)c H(X) for some jinite M, then

{ EZ’~(W): T 2 I} is uniformly bounded.

Theorem 5. If Y = Iz(,f2) E H?(X), then a condition which implies the CLT and is

implied by the BCLT is the,following:

1 IX

I I .fr(x, s).f2(4 -s) (e

iCx+t)T_ I)(eitu-t)T_ 1)

V,(T) m.~xfli*l. ldt (s-u)(x+s)

(e i(u+r)T

+.M.% -s)f2(5 s)

_ ,)(,i(.x-*IT_ 1) 2

(u+s)(s-xx)

q(ds) + 0 (2.1)

as an L’(R’, a’) function qf (x, u) as T+w.

Since V,,(T) must necessarily +CO in order that (2.1) hold (without fi vanishing

identically), the expressions (2.1) depend essentially only on the measure q and on

the behavior of f?(u,, u2) in the neighborhood of the diagonals {u: U, = *u?}. For

calculations showing that (2.1) holds under some conditions when u is absolutely

continuous, see Chambers and Slud (1988, Lemma 2.3).

,- (~PMW(~)~ =

0 I

(~Pmo~(~-&v ‘- = 0 I

(nP)d(n-)S(n)bv () !I

0 = [(SP)d!- (SP).Il(S-)~(S)~

I I

uaql .( np)d(n-)/ ’ ,,J p%~u! aql 01 pue.G’a~~! aIq”wpald _ ql!M (nP)d(n)S:‘J LKIOJ aq1 JO ~“.hIalu! 3!lsEq3ols I? sLuJoJsuml n- = n a[q”‘.l”A

-JO-a%Eq3 aq1 1eq1 11ma.l pu\? ‘(S)N E(S-)n Lq s1uauInELIe aiz@au .IOJ (.)fi

auyaa ‘(s)~p (s)~i ‘,)J+(s)wp (S)M ‘:J s! (r)~l(r)w JO md a@u!mm aql aDu!s _ _

1 [(~P)~((~-)~-(~)~)!+(~P)~((~-)8+(~)~)1(~~~ ‘)

SI atI zs

‘(“)P J(s-_)8 - (s)S) ‘I +‘(J>p J(s-)Z + (s)Z) (’ J J =,IWVl , ‘(E’Z’P rua.IoaqL ‘0861 ‘Inddn!lle)l) murua~ s6g1

JO LLLIOJ [maua8 aql 1(8 .sa[%g.mu alqwSalu!-amnbs Ivuo8oql~o OMJ JO ulns aql s!

(sp)rl((s-)8-(s)S) Cl J

!+(sP)J((s-)8+(s)S) ” = I I J

(SP)d(SF ‘- J = (r)Lw I ‘0 e 1 .IOJ Ir?ql MOUy aM ‘JOOld

.(Ylp)d((n)s(S-)s+(L)g(s)s) ‘;l= (S)l/ ‘0~s .Wj‘pWJ (S-)L/ = (S)y c%XjM

(I’E) (SP)d(S)Y ‘- J

+(~Pb$lw ‘- J I = (sp)d(s)8 ‘-

I I z I II 0-C) -q ua$y ‘cc > (sp)n,j( s)Z/ J 3 my1 ynns pus ‘0 e I rof a~qw13.‘pand-‘,s aw (I -)8 puw

(r)8 yioq iwql yms ti uo uo~rnunJuropuw.4 paywn-xa~iuon kuw aq ( . )8 la7 ‘9 eurwaq

‘([I ‘o])$ = K

[I)” JX~+Wf_ ‘1 EJ = (j),dzJ = ‘(d)

c 11

‘([I ‘0,)~:=;1(“.“-Ix;)llia = (1)JY = ‘(J)

Lq uaA!8 ale pm u~opue.~uou ale (?d pue in Jo

slolesuadmo3 laiCan-qooa aql ‘-a.!) rl pua ,I JO sassaDold awv!.uzA alq-empald ayl

‘sluatuamu! (ue!ssnef> iC[lu!o[pue pal%?[aJlo3un asnmaq) luapuadapu! ql!M (cx’ ‘01 uo

sa~t?~u~l.‘??~ luapuadapu! am rl pue J pm ‘( 1.1) JO ‘g uo!lr?~l~y aql 01 1DadsaJ ql!M

(m ‘01 uo a@u!l.mu ue!ssnt?g xa[dwo3 I? s! ( . )d! + ( . ),[ = ( . )d ‘(,g) pue (J) 1cg

‘paxy ale D ammaw [tz.wads ~!uroleuou aql put? x ‘uog3as s!ql lnoqSho_tql T/ ‘J

‘d 01 lDadsa1 ql!M sle.Salu! 3gseq3ols lnoqe suogmap!suo3 Iwaua% ql!M u$aq aM

100 D. Chambers, E. Slud / Functional ceniral limit theorem

so that

M(.s)[(g(s) + g(-s))r(ds) +iMs) -d-.~))p(d.~)l

Now using (r), =(p), = $a([O, s]) and collecting terms, we find

,M(t),‘= 1’ ,g(r),%(d.i)+[ fi(s)g(s)P(ds)+ M(s)g(.s)P(ds) PI I I’ -I

I ’ Ig(s)12g(ds)+

I ’ = [ti(.s)g(s)+M(s)g(-s)]P(ds).

-I --I

Since h(s)= M(s)g(s)+g(-s)A4(s) satisfies h(-s)=h(s) and is equal for ~20 to

d.7) I‘. g(u)P(du) + g(-s) I‘ du)P(du) -5

’ zz k(s)g(-u) + G(-.s)g(u))P(du) -\

our assertion is proved. 0

Corollary 7. If’g( .) is a complex-valued randomfunction such that g(t) is predictable

and g(-t) = S(t) for t 3 0 and such that E 1‘7 Ig(s)12rr(ds) < CO, then the martingale

M(t) = J’_, g(s)P(ds) is reul-valued with predictable variance (M)(t) =

j’, Ig(s)l’@ ) d s an with variance EM’(t) =s’, Elg(.s)\‘m(d.s).

Proof of Theorem 1. Define g(s)- I,,_,(P~,,(. , s)xk!,( .)), and verify that g(-s) =

g(s). Apply (7”), (7’) and Corollary 7 to conclude that

I

IX

WJ~,.))2= ElL,(pk,,(., s)x!~,(.))124ds) -X

=((k-l)!))’ I

X’ IIP~,T(., 4x~!,(.)l/;/~ Ia( 1-

Therefore the orthogonal sum

G(s) = ( VY( T)) 1/z? IL-,(/%.,(., 4xiw))

must exist in L2 sense in N(X) for u-a.e. S, since the corresponding sum of L”-norms

( = expected modulus-squared) is

E(G,(s)I’=( V, (T))-’ 5 ((k-l)!)m’IJp,,,(., s),&!,(.)llf,~ 1

D. Chambers, E. S/d / Funcfional c.en/ml limit theorem 101

the integral of which with respect to a(ds) is

(,,,~))~i,(;,(~~,~))‘=(v,(~))~l~~j~ WY)dr)i=l.

Now apply (7”) to pk.-r/m and sum over k to learn that

I

IX

Z,(m) = G,(s)P(ds) -\-

=I

f u

G,(.~)P(ds) + G(s)P(ds) + -I I, I

--I

G,(s)P(ds), t > 0, -Ix

from which it follows easily that E{Zr(~)l~,} =I’, G,(s)P(ds) a.s. Since the last

stochastic integral is a.s. continuous in t, Theorem 1 is proved. q

Proof of Theorem 2. (a) By Corollary 7 applied to the stochastic integral

Z,(t) = I

,

G(s)P(ds), (Z,)(t) = ’ IGT(S)]‘g(d.s). -I I -I

If (Z,)(a) s 1 as T + a, then the CLT for Y is an easy consequence of the Rebolledo

Martingale CLT (Helland, 1982, Theorem 5.1(a)).

(b) Suppose Y satisfies the BCLT. Then Vy ( T) + 00 as T + 00, so that for 0 < s < t,

(V,(T))-’ I’ b,,Ax)12dd4 \

’ s (2/s”)( V,( T))-’ I,f,(x)]‘v(dx) + 0 as T+KJ.

The assumption of uniform equicontinuity in the BCLT implies by the Arzela-Ascoli

Theorem that for every sequence of numbers T increasing to ~0, there exists a

subsequence of numbers T’ along which

f ’ k-2 k! V,( T’)

IPl;r,(x)(“x:(x)~(dx,) . . q(dxk) + v( 1) as T’-+ cc (3.2)

uniformly for t E [0, an], where u(t) is some nondecreasing uniformly continuous

function on [0, M), and where

{I

T u(a)= lim E u,( Y- I,(_f,)) ds

I’/ V,( T’)s 1.

7--X 0

For the rest of this proof, we will restrict T without further comment to lie in such

a subsequence T’, and we will prove that (Z,)(CO) s 1 as T-CO.

Fix 8 > 0 arbitrarily small. From the previous paragraph, we learn that

E{Zr(r)-Z,(S)]’

= A:, k! V;,(T) j- (x;(x) -x:(x))lp~~.(x)l’cr(dx,) . . cr(dxk)

+v(t)-v(S) as T-,a, uniformly for tE[S,m].

102 D. Chambers, E. Slud / Functional central limit theorem

From the martingale property of Z,, it follows that, as T + 00,

whenever t, > 6. Since

c

I

I

,,I I/,( Y,,‘) ds ( V,(T))“’ = C ~,Zdf,),

J o I ,=I

the BCLT implies for each (Y, t with t, 2 6 that

; a,(Z,(,;)-Z&& ,V(O, R8(q t)) ;=I

as T+co

where the distribution ;V(O, 0) is understood as the distribution degenerate at 0. Then the Cramer-Wold device (Billingsley, 1968, p. 49) implies that the finite-

dimensional distributions of the martingale Z,( . ) - Z,( 6) on [ 6, a] converge weakly

to those of W( u( . )) - W( v( 6)) where W is a standard Wiener process.

In order to conclude for fixed 6 > 0 that

Z,(.)-Z,(S): W(TJ(.))- W(v(6)) in %‘[S,co] as T+oo (3.3)

we need only to verify tightness of the laws of JT(. ) = ZT( .) -Z,(6) on %‘[6, ~1,

which we do as follows. For each /3 > 0 and positive integer M, and each 0 < F’< F,

(Doob inequality), where v-‘(s) = inf{x: u(x) > s}. Now the finite-dimensional dis-

tributional convergence of 17( .) as T -+a, together with the uniform integrability

of {]i,‘r(~)]‘+~‘: 6 s s 4 CO}, implies that as T+ ~0 the last expression converges to

which is bounded above by a constant (not depending on M) multiplied by M mmF”2.

It follows for each p > 0 that M, T,,< cc can be chosen so that, for all Ta T,,,

P 1

sup 5, rt[fi,X]: ~I.O)~L.(I)# Iv-’

lZAr,-z,Ol;-3P} <P.

By a small extension of a well-known criterion (Billingsley 1968, Theorem 8.2),

tightness of the laws of Zr( .)-Z,(S) on %[6, ~1 follows.

D. Chamhtm, E. Slud / Functional central limit theorem 103

Since {Zr( . ) - Z,(S)}, _, is a uniformly square-integrable family of continuous

martingales, it follows from (3.3) and Rootzen (1977, Theorem 6) that

(ZT-ZT(S))(~)~ u(CC)-u(S) as T+oo.

But it is easy to see that (Z,( .) -Z,(~))(W) = (Z,)(a) -(Z,)(6), since the quadratic

variation [Z,]( .) and predictable-variance processes (Z,)( . ) necessarily coincide

by continuity of Z,( . ). Thus we have proved

for each 6>0, (Z,)(m)-(Z,)(S): u(a)-~(6) as T+co.

In addition, for each 6 > 0 we have seen that

lim I

8 IPU( s lP1,r(x)12 T-X _A V,(T)

g(dx) = lim I V,(T)

a(dx) = 1 -u(a). r-s -v

Therefore, since u(O+) = 0, there must exist 6 = 6(T) converging to 0 slowly enough

as T+co so that

and

i7’r) IPI,TbN2 -A(T) VY( 7-1

u(dx) + 1 - ZI(CO). (3.4)

But the martingale

-G(f)- ,, $f+$$ P(dx)

has variance equal to the left-hand side of (3.2), and must therefore converge to 0

as T+ ~0 if f is replaced by 6(T). Moreover, the martingale

has predictable-variance process at t = S(T) equal to

I

ii(T)

I~,,~C#~bWI VY( T).

-ii(T)

We conclude by (3.4) that (Z,)(6( T)) converges to 1 - ~(00) in probability as T+oo,

and then that (ZT)(~) Jk 1, as was to be shown. The observation that (Z,)(a) =

JTr IG,(s)12a(ds) completes the proof. 0

Proof of Theorem 3. The assertion that (i)-(iii) imply the CLT with asymptotically

linear variance was proved as Theorem 1 of Chambers and Slud (1988). The same

theorem will apply to Y a,’ in place of Y, with assumptions (i) and (iii) and the

upper bound in (ii) unchanged, and with the lower bound (ii’) implying

i I ,,I

O<c (k!))’ lim inf h-’ I /l-(1+

~Ai ,;,, ~,x~“(x)~;(x)~~x(,,,+. .t\‘,~ /.,, ,&dx)

whenever the same expression with lim inf replaced by lim sup is positive. Thus

(i)-(iii) and (ii’) for Y imply (i)-(iii) for Ye,‘. Theorem 1 of Chambers and Slud

(1988) implies the CLT for each Y@ for which

7 ?

lim sup E (I

U,( Ye.‘) ds - I/

V,(T)>O. T-X 0

It remains to prove only the assertion of uniform equicontinuity in the BCLT. For

this, it suffices to prove that the nondecreasing, uniformly bounded, and uniformly

continuous functions

(FT(t)= $ (V,(T)k!))’ I, -2 I d

x;(x)IpcT(x)l’~(dx,) . . . ddx,\)

on [0, 001, satisfy, for all s < f,

V,(T)<2 ’ lim sup ((PAN) - (F~(s)) T-

1. y(x)ddx)

r-K (3.5)

where y( . ) is as defined in hypothesis (ii”) of the theorem. Observe first that for

each ks2, by Lemma 2.1 of Chambers and Slud (1988),

lim sup 3 7+.x I

Iw,~ (x;(x)-x;(x))lp~,,-(x)l’o(dx-,) . . d&J

Therefore

v,_(T) lim sup ((PA?) - PAS)) 7

T-u-

(x:(x)-x;(x))lp~,~(x)I’~(dx,) . . . ddx,)

I

s2 y(x)a(dx)

D. C‘hnmhrrc, E. Slud / Func/ional wntral limit theorem 105

by definition of y. Since lim inf,,, ( V,(T)/ T) > 0 under our assumptions, and

since all the functions (oT( .) increase from 0 to a limit less than or equal to 1, it is

easy to deduce uniform equicontinuity from (3.5) by considering the behavior of

(p7 ( .) along finite increasing sequences { tj}ylo for which 5:; ‘I y(x)a(dx) are all small,

where t,, = --Qc: and t,, = ~0. 0

Proof of Proposition 4. As in formula (4.2) and Lemma 3.7 of Chambers and Slud

(1988),

uniformly in T. q

Proof of Theorem 5. Now Y = Iz(,f2), G,(s) = (V,( T))m”‘I,(p2,T( ., s),$,‘( .)), and

Z,(t) =I’_, G,(s)P(ds). Now function G,r (s) takes the special form

of a first-order stochastic integral with nonrandom integrand. Then by Lemma 6,

Jrb, s)P(dx) (3.6)

where the adapted random function JT(. , s) satisfies Jr( -x, s) = .i, (x, s) for 1x1 c s

and is given by

Ill

JT(-T .s) = (V,(T)) ’ I 1 .mx, .~lh(U, -s) (e il\l\)t-_l)(ei(“~‘)r_l)

-IX (x+s)(.s-u)

+.Mx, -sY2(4 .s) (e Illl+<)t _ I)(e”“‘r_ 1)

(u+s)(s-x) P(du),

and where we have used the identity ,f2(-x, s) =,f?(x, -s) in two places.

In the present setting,

2 lp2,,-(x, s)l’cr(dx)a(ds)/ V,,( T) = E * IG,(s))“o(ds) _X

= EZ$(co) = 1.

Since Z,(t) is a continuous .F,-martingale with quadratic variation ( = predictable

variation) process J’, IG,(s)l’cr(ds), it follows from Millar’s (1968, Section 6)

extended Burkholder inequalities that for a universal constant C, for each T

E {I: ,G7 (s),%(ds)} s CEZh,(m).

106 D. Chambers, E. Slud / Functional central limit theorem

By Proposition 4, {j’“,, JG,(s)\‘rr(ds)} . . 7 , IS uniformly square-integrable. Thus the

condition which according to Theorems 2 is sufficient for CLT and necessary for

BCLT becomes

P, 1.2 &(x, s)P(dx)a(ds) - 0. (3.7)

But the left hand side of (3.7) is the stochastic integral

with expectation squared equal by Corollary 7 to

1

I_ 15 ,x 2

E J,-(x, s)a(ds) g(dx) a 1x1

Using the stochastic-integral formula

I’

J,(x, s)c(ds) IX/

(3.8)

and applying Corollary 7, we find

cc 7

E J,(x, s)dds) II

Pz,r(% S)PZ.T(% -.y) +P*,r(x, -~)Pz,,(U, 3) 7

V,(T) a(ds) q(du).

Noting that the last expression in 1.1’ is symmetric in u and x, we find that the

double integral with respect to g(dx)a(du) over R” of the left hand side of (2.1)

is exactly twice the expression (3.8). Thus (3.7) is equivalent to the condition (2.1)

of our theorem. 0

Acknowledgement

The authors are grateful to C.Z. Wei for a helpful conversation related to the

tightness in Theorem 2, and to the referee for suggestions which improved the clarity

of the presentation.

107

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