a functional central limit theorem for weakly dependent...

A Functional Central Limit Theorem for Weakly Dependent Sequences of Random VariablesAuthor(s): Norbert HerrndorfSource: The Annals of Probability, Vol. 12, No. 1 (Feb., 1984), pp. 141-153Published by: Institute of Mathematical StatisticsStable URL: http://www.jstor.org/stable/2243601 .Accessed: 06/02/2011 16:36

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The Annals of Probability 1984, Vol. 12, No. 1, 141-153

A FUNCTIONAL CENTRAL LIMIT THEOREM FOR WEAKLY DEPENDENT SEQUENCES OF RANDOM VARIABLES

By NORBERT HERRNDORF

Mathematisches Institut der Universitat Koin Let (Xn)n>, be a sequence of r.v.'s with E Xn = 0, E(,q=1 X1)2/n a2

> 0, SUPn,mE(Z,+n X,)2/n < oo. We prove the functional c.l.t. for (Xn) under assumptions on an(k) = sup$ I P(A n B) - P(A)P(B) | :A E a(Xi: 1 c i c m), B E a(Xi: m + k < i < n), 1 c m c n - kI and the asymptotic behaviour of 11 Xn II# for some E3 E (2, oo]. For the special cases of strongly mixing sequences (Xn) with a(k) = sup acn(k) = O(k-a) for some a > 1, or ac(k) = O(b k) for some b > 1, we obtain functions f#(n) such that 11 Xn 11,t = o(f6(n)) for some /3 e (2, on] is sufficient for the functional c.l.t., but the c.l.t. may fail to hold if 11 Xn II #= O(f#(n)).

1. Introduction and results. Let (Xn)n, be a sequence of r.v.'s on some probability space (Q, X P), satisfying

(1.1) EXn = 0 EX2 < 00 for n E M.

Put Sn = 1 Xi for n E M. In this paper we will make the following assumptions on the variances:

(1.2) ESn/n -*ne -2m > 0 for some o- > 0

(1.3) supIE(Sm+n - Sm)2/n:m, n E M) < 00.

Consider the space D = D[O, 1] endowed with the Skorokhod topology (see Billingsley, 1968, Section 14) with Borel-a--algebra - and define random functions W Q- D by

(1.4) Wn(t) = S[nt]/(n'12) for t E [0, 1], n E M.

Wn is a measurable map from (Q, X) into (D, ?q). If (Wn) is weakly convergent to a standard Brownian motion W on D, then (Xn) is said to satisfy the invariance principle (i.p.). In this paper we will use the mixing coefficients a0n(k), defined by

an(k) = supI I P(A n B) - P(A)P(B) I:A E a(Xi:1 c i < m), B E o-(Xi: m + k c i c n) , 1 c m c n - k k c n -

an(k) = O for k > n.

The coefficient of strong mixing introduced by Rosenblatt (1956) then can be written as

a(k) = supnEafon(k) for k E N.

Received October 1982. AMS 1980 subject classifications. Primary, 60F05, 60F17. Key words and phrases. Central limit theorem, weak convergence of partial sum processes to

Brownian motion, strongly mixing sequences, Mn-dependent sequences.

141

142 NORBERT HERRNDORF

Let 11 X 11 be defined in the usual way, i.e.

XI11XA = Ell" XV for 3E [1, oo)

jjXjj: = ess sup IXI for = oo.

The following theorem is the main result of this paper.

THEOREM. Let j3 E (2, 0o] and y = 2/f (2/oo = 0). Let (an)nEN be a sequence in [1, oo). If (Xn) satisfies (1.1), (1.2), (1.3) and

(1.5) (SuPin 11 Xi 11 2)(Xian an(i)T' + an, /nl8) nr 0,

then the random functions Wn, n GE A, converge in distribution to the standard Brownian motion W.

The first corollary shows that several known results for strongly mixing sequences are contained in the above theorem.

COROLLARY 1: Let fl E (2, 0o] and y = 2/fl. If (Xn) satisfies (1.1), (1.2) and

(1.6) Den a(i)1'- < oo and lim SUPneN 1I Xn 11A < 00,

then (Wn) converges to W in distribution.

The c.l.t. which follows from Corollary 1 has been proved for the strictly stationary case by Ibragimov (1962), and the general case can be obtained from Theorem 2.1 (A) of Withers (1981). Yoshihara and Oodaira (1972) proved the i.p. for strictly stationary sequences under the assumption (1.6), improving a result of Davydov (1968), but their method does not admit an easy generalization to the non-stationary situation considered in this paper, since the uniform integrability of (Sm+n - Sm)2/n, n, m E [, and of I Xn I ', n E M, if : < 00, is crucial for their argument. Using his theory of mixingales, McLeish (1975) obtained an i.p. for the non-stationary case. The above Corollary 1 improves McLeish's result, by weakening and simplifying the mixing condition and drop- ping the assumption 3.8 (c) of his paper.

The following corollary shows that the moment condition in (1.6) can be considerably relaxed, if one imposes stronger mixing conditions and adds the assumption (1.3).

COROLLARY 2. Let fl E (2, 0o] and y = 2/fl. Assume that (Xn) satisfies (1.1), (1.2), (1.3) and one of the following conditions

(1.7) a(k) = O(k-a) and

l1 Xn li = o(n(l1-)/2-(l1-/2)/(l+a)(l+a)) for some a > 1/(1-y)

a(k) = O(bk-) for some b > 1 and (1.8) 11 Xn 11 i = o(n (1-y)2/(log n)l-',/2)

then (Wn) converges to W in distribution.

FUNCTIONAL CENTRAL LIMIT THEOREM 143

The moment conditions in the above corollary cannot be weakened. In Section 3 we will show by examples that the c.l.t. may fail to hold, if the moment condition in (1.7) or (1.8) is weakened, by replacing "o" by "0".

In the following corollary the above theorem is applied to a class of processes which need not satisfy a (k) -> 0. Define for n GE 8:

Mn= inflm GE N U IO1:(Xi: 1 c i c k) and (Xi: k + m < i c n)

are independent for every k E1, * ... n - m - 18.

COROLLARY 3. Let j3 E (2, oo] and y = 2/f3. Assume that (Xn) satisifies (1.1), (1.2), (1.3) and

(1.9) SUPijn || Xi L3 = o(n 8

Then (Wn) converges to W in distribution.

In general (1.9) cannot be weakened. The processes in Example 1 and 2 of Section 3 satisfy SUPi-n II 11 = Q(n(1-y)/2/Mn1-/2), but the c.l.t. does not hold in these examples.

If m = supnelnMn < 00, then (Xn) is called r-dependent. For such sequences we obtain:

COROLLARY 4. Let 3 E8 (2, oo] and y = 2/f3. Assume that (Xn) is r-dependent and satisfies (1.1), (1.2), (1.3)

(1.10) Xn 110 = o(n (1--)/2)

Then (Wn) converges to W in distribution.

The condition (1.10) cannot be weakened even in the independent case. If (X,7)ncN are independent r.v.'s with P{Xn = +n1/2j = 1/(2n), then as a consequence of the classical Lindeberg Theorem the c.l.t. is not valid, and one has 11 Xn 11 = n~'-812. In Example 3 of Section 3 we will show that it is impossible to find a-(k) > 0 and imn E8 N U 101 with rn -> 00 such that every process (Xn) which fulfills (1.1), (1.2), (1.3), (1.10) and a (k) c a-(k), Mn < thn satisfies the c.l.t. Hence the assumption of r-dependence in Corollary 4 cannot be replaced by any weaker assumption of strong mixing and rn-dependence without strengthening the moment condition.

Under the assumptions of our Theorem one can find Pn E8 N such that Pn - o(n1/2)- and an(Pn - k) >nEN 0 for every k E N. Therefore it follows from arguments of Billingsley (1968) that the assertion of our Theorem can be strengthened to "(Wn) is R-mixing to W", i.e. P(Wn E B I A) +nEN W(B) for every W-continuity set B E E and every A E vY with P(A) > 0. This extension is useful for the study of weak convergence, if the indices are r.v.'s (see Durrett and Resnick, 1977). The proof of our results is given in Section 2. To prove the c.l.t. we use Bernstein's blocking argument, which is discussed in Lemma 3.1 of Withers (1981). We do not use Theorem 2.1 of Withers, since his moment assumptions are too strong for our purpose. The tightness of (Wn) is also


established by a blocking argument. An important tool is Lemma 2.2, which extends an inequality proved by Ottaviani for independent r.v.'s.

The following notations are frequently used: [x] =max~n E Z: n < xj for x E R. an bn means an/bn -nEN 1.

For a set A C Q, I(A) is the indicator function of A, i.e. I(A)(w) = 1 for w E A and I(A) (w) = 0 for w ? A. For a collection 9 of r.v.'s a( S ) denotes the a-- algebra generated by these r.v.'s.

2. Proofs. The following lemma follows from formula (2.2) of Davydov (1968).

2.1 LEMMA. Let (Xn)nEm be a sequence of r.v.'s with EXn = 0. Let : E (2, oo] and y = 2/A. Then for all i, j E {1, * * * n with i = j the following inequality holds:

I EXiXj I < 12aen(I i _jl-Y i Xi II# 11Xi II#

The proof of the following lemma is omitted, since it is very similar to the proof of Lemma 1.1.6 of Iosifescu, Theodorescu (1969).

2.2 LEMMA. Let n, ... *,nn be r.v.'s. Put

a = supf I P(AfnB)-P(A)P(B) 1: A E u(ni, f**lk)9

B E- U(7nk+l** 7nA 1 C k c n- 1}.

Then for every c > 0:

Pl zIl i I > -t + na

Pfmaxlcrcn I Zi fi I > e} {

minl--r--n-lPt l r+1 n~i I C 2

PROOF OF THE THEOREM. Let (Xn)nEN be a sequence of r.v.'s. Assume that (1.1), (1.2), (1.3) are satisfied. Let A E8 (2, oo], y = 2/A3. Let. (an)nE be a sequence in [1, oo) such that (1.5) is fulfilled. We define a sequence (bn)neN in [1, 0o) by

b~n-/nl-- = Xiuan an(i)'-y + an-7/nl-'.

Then the following conditions are satisfied:

(2.3) (a) b2-7/n e 0nEr 0

(c) sup1 jX jn(i)2-- < b2-Y/nl--

(c) SUPi--n #lil = o(nl-'Y/b~n-)


Now we will construct sequences of positive integers p(n), q(n) with the following properties:

(2.4) (a) q(n) , p(n)

(b) p(n) 40 n

(c) Xi>q(n) an(i)1 SUpisn || X1 P - 0

(d) p(n 2,y supi---n 1Xi 1 2__ >0

(e) a n(q(n)) -- 0 p(n)n

(2.3) (a) and (c) imply that there exists a sequence r(n) of positive integers with r(n) -- oo, r(n)b(n) = o(n) and SUpisn 11 Xi IIP = o(nl-,/(r(n)2-"b2-,)). Choose q(n) - 2 infti E H: i > bn) and p(n) = r(n)q(n). Then (2.4) (a)-(d) follow immediately. Since k -- an(k) is non-increasing, one obtains

( ( ) Otn(q(n))) C ()l-t Ei':-bn an(i)1 t < 0.

Now we proceed as in Lemma 3.1 of Withers (1981). With p = p(n), q = q(n) and k = [n/(p + q)] we define

(i = Z JXi~j(p + q) + 1 c i c

;(p + q) + pj O j i k- 1

tj = Z jXi~j(p + q) + p + 1 c i c (j + l)(p + q)} O c j c k- 1

gk = Z jXi:k(p + q) + 1 c i c n}

Sn j=0S-? Sn J= i?i

To prove the c.l.t. it suffices to show that for n - oo

(2.5) (a) n-'ESn2 - 0

(b) n-1 X0oi<j-k-1 I EtijI --+ 0

(c) k - oo

(d) | E exp(iuS') - =1J E exp(iutj) | n 0

uniformly over u GE R.

(e) n-1 ,j=- E(QyII I (I I > ccn'12j) nEN 0 for every c > 0.


To (a): Using 2.1 and (1.3), one obtains

ES"'2 < Zi=o E~j2 + 2 Xl1isn >i+p<j-n I EXiXj I < C(kq + p + q) + 24n i>p an(i)-'Supi'n jj Xi II

where C is the constant of (1.3). Now (2.5) (a) follows from (2.4) (a), (b), (c). To (b): By a similar application of 2.1 follows

XO-i<j-k-1 I Etjtj I c 24n 2i>,q an(i) SUPicn 1l Xi 11 2

Thus (2.4) (c) implies (2.5) (b). (c) follows immediately from the definition of k and (2.4)(a), (b). Arguing as

in the proof of Theorem 18.4.1 of Ibragimov and Linnik (1971), one obtains (d) from (2.4)(e).

To (e): 1st case: A = 0o, y = 0. Then (2.4)(d) implies

SUPo0jck-1 11 j 11 x = o(n1/ ).

Hence the sum in (2.5)(e) equals 0 for all sufficiently large n. 2nd case: A E8 (2, oo), y E8 (0, 1). Then we estimate the l.h.s. of (e) by

nlk(ean1/2)2s 8UPo0y*k-lE I il 1 n-lk(I an</2)2- 1p SUP IIX 11 10

(Co)2 0( p2-ttlsc 11 Xi 11 2)11/2

which goes to 0 according to (2.4)(d). All points of (2.5) are proved. Hence we have proved the c.l.t. under the

assumptions of our Theorem. Using (1.2) we obtain

(2.6) WK(t) -n.N W(t) in distribution for every t E [0, 1].

Let 0 c tj < ... < tk< 1 be given. We wish to show:

(2.7) (Wn(t )9 .. * Wn(tk)) >-)nEF4 (W(t )~ .. * W(tk)) in distribution.

It follows from Prohorov's classical characterization of tightness that (2.6) implies the tightness of (Wn(t1), .. , Wn(tk))nEM. Let Q I _ (R]ti, . tkA) be the weak limit distribution of some subsequence of (Wn(t1), ... , Wn(tk)). According to (2.6) the marginal distributions Qir-1 are normal with variance ti. Take rn = (2 + an)/n, where (an) is the sequence in the assumption of the Theorem. Then rn -.+ 0 and an([nrn] - 1) -O 0. Using (1.3) one obtains E(Wn(ti + rn) - Wn(ti))2 0nE . Therefore Q(7rtl, rt, - lrtl, ... * rt -rtk, )- is the weak limit distribution of some subsequence of (Wn(t), WK(t2) - Wn(t1 + rn), ..., W (tk) - Wn(tk1 + rn)). Now an([nrn] - 1) -- 0 implies that lrt,, 7rt2 -lrt, 9 ..., rtk -rtk are independent under Q. Hence Q is the distribution of (W(t1), ... , W(tk)). This argument proves (2.7).

Finally, we have to prove the tightness of the sequence (Wn). According to Billingsley (1968) Theorem 15.5, it suffices to show for every C > 0:

(2.8) lim'olim S3PnEMPjW(Wn9 6) > e} = 0.

Let c > 0 be given. For 6 > 0 and n E M holds

Pfw(Wn, 6) > el < ha-L/o Pjmax[nab]<r-[n(a+l)(5] I Sr - S[nab] I > 1/3 cfn 1/2[


For the next step of the proof let a 8 10 *.., [1/6]) be fixed. Let p = p(n), q = q(n) be sequences of positive integers satisfying (2.4). Put m = m(n) =

maxli 8 H: (p + q)i < [n(a + 1)6] - [nab]j. For j 8 10, ..., m - lj define

'Pj = X 1X: [nab] + j(p + q) + 1 < i < [nab] + j(p + q) + pj

4j = Z JX1:[nab] + j(p + q) + p + 1 < i < [nab] + (j + l)(p + q)}. Then we have

Pjmax[nab]<r<[n(a+1)h] I Sr - S[nab] I > 1/3 can' /I

< (m + 1) maxo-b-n-(p+q)Pfmaxl rcp+q I Sb+r - Sb I > 1/9&eonl/2 I

+ Plmaxo<r-m-1 I ZX=o (j I > 1/9Cconf/}2

+ Plmaxocr-m-1 I Zj=O VI; I > 1/9Cefnl/21 = I + II + III. To estimate I we distinguish two cases.

1st case:3 = oo, y = 0. Then we have in I

ess sup | Sb+r - Sb I C (p + q)supicn 1 Xi 1_ - o(n1/ )

according to (2.4) (a), (d). Hence I equals 0 for all sufficiently large n. 2nd case: 3E (2, oo) y E (O. 1). Let b E1-0, ... , n-(p + q)}. Then we have

Pfmaxjcrcp+q I Sb+r - Sb I > 1/9eornl/2 I ' PI Z1X=+'q I Xb+i I > 1/9cofl1/2}

C (co/9)-"n-1/2(p + q) #SUpicn 11 Xi 11s

Using the definition of m and (2.4) (a), (d), we obtain

(m + 1)n 3/2 (p + q)'3supi<n 11 Xi 11 b(p2-n7 S1Pi-n||Xi 112) -nEH 0.

Hence I goes to 0 for n -> oo.

Now we will estimate II and III. Using 2.1 and (1.3), we obtain:

maxJc(o,.. .,m-1)E(EjEJ tj)2/(of2n)

<ZOsj-rnm-j Ei'j2/(of2n) + 2 Xl-icn Ii+p5j-n I EXiXj l/(o2n) < Cmq/( n) + 242 Zip an(i) 'Supicn || Xi - >nCN 0.

Similarly:

maxjc0o,. .,m-jE(XjEj (pj)2/(o2n)

< ZO:5j-m-iEeP2/(jfln) + 2 Xlliln Ei+q<jcn EXiXj /(of2n)

< Cmp/(o n) + 24 Zi>q an(i)'0supicn || Xi || A

Using (2.4) and the definition of m, we obtain that the last expression converges to C-26 for n -> oo, where C is the constant of (1.3). We can choose 60(c) > 0 such that

(F8/ra e 2C-2eo(e) < 1/2w

From now on we will assume 6 < 60(c). Then we obtain by an application of


Tschebyscheffs inequality

minocrm-2PI I X'i=r+l (Pi I < 1/18 cn'l12I > 1/2

for all sufficiently large n. Now we apply Lemma 2.2 and obtain:

PlmaxO<r-m-i I ZX=o (j I > 1/9 cofnl/'}

< 2PI I ZX=' (Pj I > 1/18 con'12I + 2man(q + 1).

(2.4) implies man(q + 1) --n, 0. Since

1j zm-1 g4/(o 1/2) 11 2 >neM 0

11 (S[n(a+1)6] - S[nab] - Ej=o1 (Pj + 4Ijj))/(o n/2 ) 112 0

we obtain:

lim supneNPImaxo<r-mr-1 I Z=o Pj I > 1/9 ccnf}2

c 2 lim supn(=PI I SSn(a+l)- Sinab] I > 1/1?co-nl/2 I

Another application of Tschebyscheffs inequality and Lemma 2.2 yields

lim supneFPjmaxor-m-1 I ZX=O g1j I > 1/9 canf/}2

c 2 lim SupINEPj I |jm=% 4'. I > 1/18e0n' I} + 2 lim supne-maon(p + 1) = 0.

Summing up our results for a = 0.**, [1/6], we obtain:

lim sup~~~P Iw( Wn, 3) > e} < 2 z'a6) lim sup NPI I S[n(a+1)6] - S[nab] | e

a-n"' 19J

< 2( + 1)N(O, ){x E lxi > IjI.

Here we have used the weak convergence of Wn((a + 1)6) - Wn(ab) to N(O, 3), the normal distribution with mean 0 and variance 3. Now the proof of (2.8) is complete, and this was the last step of the proof of our Theorem.

PROOF OF COROLLARY 1. Since finitely many Xn may be truncated, without affecting the asymptotic behaviour of (Wn), we may assume w.l.g. supnEE 11 Xn II <oo. Then (1.6) implies (1.5) for every sequence (an) with an -+ ?? and a2-7/n1-

- 0. Using Lemma 2.1 one obtains

E(Sm+n - Sm)2 < n supien || Xi 11 2 + 24n i= a(i)lsupie || Xi 11 2

Hence (1.3) follows from the assumptions of Corollary 1.

PROOF OF COROLLARY 2. W.l.g. we may assume 11 Xn 11ii < oo for all n E M.


Then the moment conditions in (1.7), respectively (1.8), imply

supi.,n || Xi |, = o(n=((1-)/2)(a/(a+1))-(1/2)(1/(a+

respectively

supicn 11 Xi 11- = o(n '1-y)2/(log n)1-/2).

If (1.7) is fulfilled, then apply the Theorem with an = nl/(a+l) If (1.8) is fulfilled, then take an = max(1, log n/log b).

Corollary 3 follows by an application of the Theorem with an = mn + 1. To prove Corollary one may assume w.l.g. 11 Xn 113 < o? for all n 8 M. Then one has Supicn 11 Xi 113 = o(n (-y)/2) and the assertion follows as a special case of Corollary 3.

3. Examples. In Lemma 3.1 the common structure of the examples in this Section is discussed.

3.1 LEMMA. Let g: Hf '- H have the following properties (i) g is non-decreasing (ii) g(n) -- oo for n --+ oo (iii) g(2n) c Mg(n) for all n E8- H with some constant M. Denote g-'(n) = minji E M: g(i) 2 n}, n E M, G(n) = E;'=1 g(i), n 8i H U {O, G-1(n) = minmi E M: G(i) 2 n}, n E8 M. Let (Yj)jEN be an independent sequence of r.v.'s with

G_____ 1/ g(i) g(i)

I g(i) 2 G(i) ' G(i)

for i E M. Consider the process (Xn)nl , defined by Xn = Yi for G(i - 1) < n < G(i). Then (X,) has the following properties:

(3.2) EXn = 0, EX2 < 00 for n E N (3.3) ES2/n >n.r= 1 (3.4) E(Sm+n -Sm)2 < n for n, m E H (3.5) a(k) c sup g(i)/G(i):g(i) 2 k + 1} for k 8i N (3.6) mIn g(G-1(n)) - 1 for n 8 H (3.7) For j 8 (2, oo], y = 2/3 holds

II XnIIp = (G(G-1(n))('1-)/2/(g(G-1(n))1-'/2 for n E H

(3.8) (Xn) does not satisfy the c.l.t.

PROOF. (3.2) is obvious. To (3.3): (i), (ii), (iii) imply g(n)/G(n) -+ 0. Hence

g(G-1(n)) g(G-1(n)) g(G-1(n)) n - G(G-1(n)-1) G(G-1(n)) - g(G'1(n)) nEE .


Let n E M, i = G-'(n). Then G(i - 1) < n c G(i) and one has:

ES' = Z>'- g(v)2EY' + (n - G(i - 1))2EY

= G(i - 1) + (n - G(i - 1))2/g(i)

ES2/n - 1 I = I (G(i - 1) - n)/n + (n - G(i - 1))2/(ng(i))

Cg(i)/n--nE=M ?.

To (3.4): Let m, n E H and G(i - 1) < m + 1 c G(i), G(i + t) < m + n < G(i + t + 1) for some i E M, t E H U 101. Then one has:

E(Sm+n - Sm)2

= (G(i) - m)2EY? + XZi+j g(v)2EY' + (m + n - G(i + t))2EY2+t+1

= (G(i) - m)2/g(i) + Zi+i+j g(v) + (m + n - G(i + t))2/g(i + t + 1)

c G(i) - m + Zt+1+, g(v) + m + n - G(i + t) = n.

If G(i - 1) < m + 1 c m + n s G(i), the assertion follows similarly. To (3.5): (X1, *..., Xn) and (Xn+k, Xn+k+l, * * ) are independent, unless there

exists i E8- H with G(i - 1) < n, n + k c G(i). In the latter case it follows from Lemma 8 of Bradley (1981) that

supf I P(A n B) - P(A)P(B) 1: A E u(X1, ** , Xn), B 8 U(Xn+k, Xn+k+l, ...

< supf I P(A n B) - P(A)P(B) 1: A, B E o(Yi)}.

Since P I o-(Yi) has an atom with measure 1 - g(i)/G(i), it is easy to check that I P(A n B) - P(A)P(B) I ' g(i)/G(i) for A, B E o(Yi). Hence a(k) < sup{g(i)/G(i):g(i) 2 k + 11.

To (3.6): mn ' max{g(i) - 1:G(i - 1) < n$ = g(G-1(n)) - 1. (3.7) follows easily from the definition of Xn. To (3.8): Consider Zn = G)(n-l)+l X = g(n) Yn. Then (Zn )nle) is an independent

sequence with EZn = 0, EZn = g(n), E(E1 Z1)2 = G(n). It suffices to show that D=1 Zj/G(n)112 is not asymptotically normal. Since max1ci-ng(i)/G(n) - g(n)/G(n) -> 0, the Lindeberg condition is necessary for asymptotic normality. Hence it suffices to find e > 0 such that

G(n)-1 Z,7=1 E(Z2Ij I Zi I > eG(n)12j) 41 0,

i.e. G(n)-1 Z {g(i):1 c i c n, G(i)1/2 2 eG(n)1/2} 4 0.

The assumption (iii) implies G(2n) c 2M G(n) for n E [. Put e = (2M)-1/2. Then one obtains

G(2n)-1 h ig(i):c h i s 2n, G(i) (2M)mG(2n)

2- G(2n)-1 Z, {g(i):n c i c 2n} >- 1/2,

which proves our claim.


EXAMPLE 1. For every a > 1 there exists a process ( such that (1.1), (1.2), (1.3) are fulfilled, and

a(k) = O(k-a), mn = 0(nl/(a+l))

11 Xn ig = 0(n(lY-)/2-(lY-/2)/(a+l)) for all 3 E (2, oo], y = 2/f

but (Xn) does not satisfy the c.l.t.

PROOF. Define g(n) = [nl/a]. Then (i), (ii), (iii) of Lemma 3.1 hold, and one has the following asymptotic relations:

gAn) - n '/a , 9-l (n) - na

G(n) - a n(a+l)/a G-1(n) - a + aa

na/(a+l)

g(n) a+1 a

___P _ _ _ _ _ _ a +1k 1 k-a [upG(i):

~)k+1 a 9-1(k + 1) a

(a + 1) 1/a1)/(a+l) g(G-1(n)) - (

1 n/(a+l) a

G(G-1(n)) - n.

Using these relations, one obtains the assertion from Lemma 3.1.

EXAMPLE 2. There exists a process (Xn)ney, such that (1.1), (1.2), (1.3) are fulfilled, and

a(k) = O(e-k), mn = O(log n) II Xn 11 I = 0(n(l1-y)2/(log n)l-'/2) for all G 8 (2, oo], y = 2/f


PROOF. Define g(n) = 1 + [log n]. Then (i), (ii), (iii) of Lemma 3.1 hold, and one has the following asymptotic relations:

g(n) - log n, g-'(n) ~ e

G(n) - n log n, G-1(n) - n/log n

g(n)/G(n) - n-1

supjg(i)/G(i):g(i) 2 k + 11 - 1/g-1(k + 1) e -k

g(G-1(n)) - log n, G(G-1(n)) - n.

The assertion now follows from 3.1.


EXAMPLE 3. Let a-(k) > O . k G , and hn G H with mn o o for n oo, be given. Then there exists a process (Xn)ni; such that (1.1), (1.2), (1.3) are fulfilled, and

a(k) c a-(k) for all k E , Mn ' hn for all n E H

II Xn 11 _ = o(n(1-7)/2) for all 3 8 (2, oo], y = 2/3


PROOF. W.l.g. we assume ca(k + 1) c a-(k) and 7hn ' fhn+l for all k, n 8 H. Define g(l) = 1 and inductively: g(n + 1) = g(n) + 1, if

g(n) + 1 C 2 g([(n + 1)/2)

and g(n) + 1 c rhn+1 + 1 and g(n) + 1 c a-(g(n))n, and g(n + 1) = g(n) otherwise. Then g(n) is non-decreasing and g(n) -- oo. It is easy to prove by induction: g(n) C 2 g([n/2]), g(n) C fin + 1 for all n GE , and g(n) c a(k)n for all n GE H and k < g(n). Let (Xn)nE- be the process of Lemma 3.1. Then (1.1), (1.2), (1.3) are fulfilled, and

a(k) c supjg(i)/i:g(i) 2 k + 1} - ca(k) for k Ek H

Mn ' g(G-1(n)) - 1 c g(n) - 1 < tn for n E H

Xni 1 = (G(G-l(n) - 1) + g(G-(n)))(1-f)12/(g(G-1(n)))1-e/2 < (n + o(n)) (-y)/2/(g(G-1(n)))1-'/2 = o(n-(-y)/2

for # E (2, oo], y = 2/f3.

In the last line we have used the relation g(G-1(n)) = o(n), which has been established in the proof of (3.3). Finally we obtain from Lemma 3.1 that (Xn) does not satisfy the c.l.t.

REFERENCES

[1] BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York. [2] BRADLEY, R. C. (1981). Central limit theorems under weak dependence. J. Multivariate Anal.

11 1-16. [3] DAVYDOV, Y. A. (1968). Convergence of distributions generated by stationary stochastic pro-

cesses. Theor. Probab. Appl. 13 691-696. [4] DURRETT, R. T. and RESNICK, S. I. (1977). Weak convergence with random indices. Stochastic

Process. Appl. 5 213-220. [5] IBRAGIMOV, I. A. (1962). Some limit theorems for stationary processes. Theor. Probab. Appl. 7

349-382. 16] IBRAGIMOV, I. A. and LINNIK, YU. V. (1971). Independent and Stationary Sequences of Random

Variables. Wolters-Noordhoff, Groningen. [7] IOSIFESCU, M. and THEODORESCU, R. (1969). Random Processes and Learning. Springer, Berlin. [8] MCLEISH, D. L. (1975). Invariance principles for dependent variables. Z. Wahrsch. verw. Gebiete

32 165-178. [9] OODAIRA, H. and YOSHIHARA, K. I. (1972). Functional central limit theorems for strictly

stationary processes satisfying the strong mixing condition. Kddai Math. Sem. Rep. 24 259-269.


[10] ROSENBLATT, M. (1956). A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. USA 42 43-47.

[11] WITHERS, C. S. (1981). Central limit theorems for dependent variables I. Z. Wahrsch. verw. Gebiete 57 509-534.

MATHEMATISCHES INSTITUT DER UNIVERSITAT KOLN WEYERTHAL 86-90 5000 KOLN 41 FEDERAL REPUBLIC OF GERMANY

a functional central limit theorem for weakly dependent...

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