On the limit theorems for random variables with values in...

Z. Wahrscheinlichkeitstheorie verw. Gebiete 41, 289-304 (1978)

Zeitschrift far

Wahrschein l ichkei t s theor ie und verwandte Gebiete

�9 by Springer-Verlag 1978

On the Limit Theorems for Random Variables with Values in the Spaces Lp (2_=p<

Gilles Pisier 1 and Joel Zinn 2 .

1 Centre de Math6matiques de l'Ecole Polytechnique, Plateau de Palaiseau F-91128 Palaiseau Cedex- France "Laboratoire de Recherche Associ6 au C.N.R.S." 2 Department of Mathematics and Statistics, University of Massachusets, Amherst - USA

Summary. We prove that whenever B is an infinite dimensional Banach space, there exists a B-valued random variable X failing the Central Limit Theorem (in short the CLT) and such that IE IlXl[2 = oo but yet satisfying the Law of the Iterated Logari thm (in short the LIL). We obtain a new sufficient condition for the LIL in Hilbert space and we characterize the random variables with values in lp or Lp with 2 < p < oo which satisfy the CLT. As an application we show that in Ip (2 < p < oo) the stochastic boundedness of the weighed partial sums does not imply the CLT.

1. Introduction

Let X, X t ,X2 .... be an independent identically distributed (in short i.i.d.) sequence of scalar random variables. In [19], Strassen proved the converse to

the classical Har tman-Wintner LIL, namely: if sup IX I + . . . + X n / ~ l < c~ a.s. then necessarily IEIX[2<oo and also (but this is trivial) I E X = 0 . On the other hand, it is well known that the classical conditions I E X = 0 and IEIXI2< o0 are also necessary and sufficient for the CLT. These results have trivial extensions for random variables with values in a finite dimensional space.

In Section 3, we investigate the class of Banach spaces for which Strassen's converse to the LIL is valid, i.e. for which every r.v. X satisfying the LIL in the space must also verify IEIrXI[2<oo. It turns out that only finite dimensional Banach spaces have this property. To prove this, we first obtain a new LIL in Hilbert space and then apply a theorem of Dvoretzky [5].

Similarly, we show that the implication LIL ~ CLT also characterizes finite dimensional spaces. The first example of a r.v. (with values in Co) satisfying the LIL and failing the CLT was given in [13]. We give an explicit example of a r.v. X with values in a Hilbert space H satisfying the LIL but such that IE IIXI[ 2 = o9 and X fails the CLT. This is in contrast with the (obvious) fact that if a r.v. X

* Research partially supported by NSF Grant MCS 75-07605 A01

0044-3719/78/0041/0289/$03.20

290 G. Pisier and J. Zinn

satisfies the CLT on H then necessarily IgllXll2<oo. More generally, it was proved in [-1] (cf. also [10]) that such a converse to the CLT holds in a space B if and only if B is of "cotype 2". Our Theorem 4.2 may be considered as the analogue for the LIL of this last result.

In Section 5, we show that a mean zero r.v. X with values in l e or Lp, for 2 c}--,0 when c ~ o o .

Using this characterization we exhibit an i.i.d, sequence X, X1,X2, ... of r.v.'s

with values in lp (2 < p < m) such that sup IE [IX1 + - ' - + x~/1/~ll < ~ (therefore X is pregaussian) but X fails the CLT.

2. Conventions and Preliminary Facts

All the Banach space valued random variables (in short r.v.'s) which appear in this paper are assumed to induce a Radon measure on the Banach space under consideration; equivalently, we may assume that they essentially have a separable range. All the Gaussian r.v.'s will be assumed symmetric.

Throughout the paper, X, X 1, X 2 . . . . will be an i.i.d, sequence of r.v.'s with values in some Banach space B. We will use freely the notation S n = X I + . . . + X,

and a, = l / 2 n L L n (with the convent ion:LLx = 1 if x < ee). If K is a subset of B, we write d(x,K) for the distance of x to K, i.e. inf{l lx-kl i ; k~K}. If {x,} is a sequence of elements of B, we denote C{xn} the (closed) set of all the cluster points of the sequence {xn; neN}.

We now pass to a description of the LIL in the Banach space setting: Assume that sup I[S,/a, II < co a.s. Then V ~eB*, sup I~(SJa~)l < oo a.s. Therefore, by Strassen's result [,19], we know that IE I~(X)12< oo. Moreover,

by the classical LIL of [8], we have:

V ~eB* (IE I~(X)12) x/z = lim ~(S,)/a, n o oo

< 11s lim IIS,/a,][ a.s. n- -~ ao

In particular, we have sup{IEl~(X)[Z; 11411_-<U<oo. Let J x c B * be the set of those ~ in B* such that IEI~(X)[2< 1. We denote K x the closed, bounded and convex subset of B which is the polar of Jx; i.e.

K x = { x e B ; g { e J x l ( { , x ) ] < l } .

In the literature, K x is usually referred to as the unit ball of the reproducing kernel Hilbert space associated to X.

We first recall a result of Kuelbs [-14] (actually, Kuelbs assumes that IE IlX[I2< o% but this hypothesis is used only to ensure that K x is compact and it is easy to check that it may be omitted; for details, see [,17] th6or6me 1.1).

On the Limit Theorems for Random Variables with Values in the Spaces L v (2 <p < oo) 291

Theorem 2.1 ([143 Th. 3.1). Let X, X1,X2, . . . be as above. Assume that the set {Sn/a, lneN } is relatively compact almost surely. Then the set K x described above is compact and verifies:

i) d(S,/an,Kx)~O a.s.

ii) C {S,/a,} = K x a.s.

This theorem naturally leads to the following

Definition 2.1. If X satisfies the assumptions (hence the conclusions) of the above theorem, we will say that X satisfies the law of the iterated logarithm (in short the LIL).

Remark 2.1. The relative compactness is essential in the above theorem: for an example where {S,/an} is a.s. bounded but a.s. not relatively compact although K x is compact, see [173 Example 7.3.

For a later use, we mention the following obvious

Observation. A sequence {x,} in B forms a relatively compact subset of B if (and only if) there exists a sequence 6N ~ 0 and a sequence K u of compact subsets of B such that:

V N ~ N lim d(x,,KN)<6N. n ~ o o

3. The LIL in Hilbert Spaces

It is known (see [14]) that if X is a r.v. with values in a Hilbert space the classical conditions I E X = 0 , IE I/xjI2< c~ are sufficient for X to satisfy the LIL. However, the next theorem and example show that these conditions are not necessary.

Theorem 3.1. Let X, XI, X 2 . . . . be a sequence of i.i.d, r.v.'s on a probability space ( f2 ,d ,P) with values in a real separable Hilbert space H (with inner product denoted (., .)) . Then the following three conditions imply that X satisfies the LIL on H:

IIXrl 2 IELL]rX[~-~< oo; (3.1)

I E X = 0 , (3.2)

IE(fX1, X2)) 2 < Go. (3.3)

Remarks 3.1. i) The preceding statement is obviously true also for a complex Hilbert space H.

ii) A routine computation shows that condition (3.1) is equivalent to: sup IlX,/aa]F < ov a.s.; hence this condition is necessary. Moreover, (3.1) clearly implies IE IIXl] < oo so that the meaning of (3.2) is unambiguous. Of course, (3.2) is also necessary. We do not know (but we doubt it) whether (3.3) is necessary or not.

292 G. Pisier and J, Zinn

iii) Note that if H is unidimensional then (3.3) reduces to the classical condition: (IE 1212) 2 = IE IX1 ' X2I 2 < ~ .

Remark 3.2. It is known (cf. [3]) that a Banach space valued r.v. X with mean zero satisfies the LIL if (and only if) X 1 - X 2 also does; (this also follows from Remark 4.1iii) below). Moreover, if X satisfies (3.1), (3.2), (3.3) then the sym- metrized r.v. X I - X 2 also does. Therefore, it is enough to prove the above theorem assuming that X is symmetric. In this case, we reduce the proof to a study of the behaviour of certain Bernoulli (or Rademacher) series with coefficients in H: let {e,} be a Bernoulli sequence [i.e. a sequence of i.i.d.r.v.'s taking the values + 1 and - 1 with equal probability �89 defined on an auxiliary probability space ((2', ~4', P'); if we assume that X is symmetric, the distribution of the sequence {X,} on (f2, P) is the same as that of {e, X,} on (f2, P) | (f2', P'). The proof of the theorem will require two lemmas:

Lemma 3.1. For any infinite matrix of real numbers (cqj), we have:

lira <i<j<n a<' 2K ~i l<_i<j<=n 2 ~ m n 2

n ~ oo a n

where K is an absolute constant.

Notations. We will write

M,= ~ ei~j~ij and M * = sup IMkl. l ~ i < j < n l<--k<n

The preceding lemma follows from a result of Bonami [2], which we now reformulate:

Sublemma. There is an absolute constant K such that for any integer n:

V c > 0 ] P { M , > c ( ~ ~2)1/2}<-Ke -cm. l <=i<j<n

Proof of the Sublemma. By homogeneity, we assume as we may that ~ c~ 2 = 1. We then have by a result of A. B onami (cf. [2] Theorem 6): 1= 2

is an even integer, and therefore

(IEIM,lP)I/P<=2p for all p>2.

Clearly M, is a martingale, therefore we have by Doob's inequality ([4] p. 317):

V ; > l (IEM*Pll/p<pPl(IEIM,IV)I/P,

hence

Vp>2 (IEM*V)I/V<4p.

On the Limit Theorems for Random Variables with Values in the Spaces Lp (2__0, Vp>2 IP {M, > c } <

C The choice of p = 4ee gives lg {M* > c} < e- ~/4 ~ at least when c > 8 e; if c < 8 e then

IP' {M* > c} _< 1 _< e 2 e ~/'~ ~.

In conclusion, we have

Vc>O IP'{M*>c}Ge2e -~/4~

and the sublemma follows with K=4e.

Proof of Lemma 3.I. We follow a standard argument: it is clearly enough to prove that

i f 2 2 > l i m 1=<i<1=<. ~ c~{;/n2 then ~ I P ' { ~ ~ ; 2k_Sup > 2 2 K <oo; n~oo <n__<2 k a n J

hence, it suffices to prove that

~ P'{M*k> 22Ka{k_l} < oo ; k

by the sublemma, since224k>62 ~ ~2 for k large enough for some 6>1, it l 1. k

Lemma 3.2. Assume that IF. ( X1, Xz)Z< oo ; then the sequence

2 z . - <y~< ( x i , x j } 2 n 2 - - n l=i<j=n

converges almost surely to the constant IE ( XI, X 2 ) 2 when n ~ oo.

Proof. Let sO, (n>2) be the ~-algebra generated by the sequence Z,, Z,+ I . . . . . We claim that Vn>2, Z,=IE~"Z2. Indeed, by the exchangeability of the sequence (X,), we may observe that if 1 < i < j < n:

�9 ~~ x2) 2) = m~~ x j) 2);

averaging these equalities, we obtain

IE~"((X1, X2) 2) = IE ~" Z, = Z,.

By the reverse martingale convergence theorem (cf. [-4]) it follows that Z, converges a.s. to a limit Zoo. Clearly Zoo is independent of {X~,...,Xu} for any fixed integer N; by the zero-one law Zoo must be a constant which can only be equal to 1EZ 2.


Remark. The preceding argument is inspired from a classical proof of the strong law of large numbers (cf. [4] p. 341).

Proof of Theorem 3.1. We will apply Remark 3.2, therefore we do assume that X is symmetric. First we claim that

lira <2 K {IE((XI,X2))2} 1/2. t l~oo

A routine calculation shows that the condition 3.1 implies

~lP{llXlI2>c~a2} < oe for some constant e;

therefore sup []X~12<oo a.s.; it then follows from a result of Feller ([6] n ~

Theorem2) that: (if IEIIX][2<o% this follows from the strong law of large numbers)

I IX1112 Af_ . . . _[_ iiX~ll 2 ~ o 2

a n

Therefore we may write

lim IIx~ +"+x"ll2 2

n ~ 0o G

a.s. when n--, oe.

- l i m ~ <X~,Xj>/a2~ n ~ o o l <=i,j<=n

= 2 lira (3.4) n ~ o e l < i < j < n

Now fix co in O; by Lemma 3.1, we have co'-almost surely:

lira [ Y', ei(co')ej(co')<Xi(co), X~(co))l/a2,<l/2g lira (Z,(co)) 1/2. n ~ o o l < i < j < n n ~ o o

Moreover, by Lemma 3.2: limZ,,(co)=lE<X1,X2) 2 co-a.s. Applying Re-

mark 3.2, we have

lim ~ <Xi, Xj)/a 2 = lira ~ z~ zj (X~, X~)/a~ n ~ o o l < i < j < n n ~ o o l < i < j < n

< 1/2 K (m <xl,

and combining with (3.4), we establish the above claim. This yields only the a.s. boundedness of {SJG }. We now give an approxi-

mation argument to complete the proof: Let d be the a-algebra generated by the variable X. We know that d is

countably generated (remember we assume that the distribution of X is a Radon measure); so we can find an increasing sequence (tiN) of finite sub-a-algebras of d , such that U ~'N generates ~r

Now let XN=IEd~(X), so that X N is a step function for each integer N; moreover it follows from (3.1) and the martingale convergence theorem that X N tends to X a.s. when N ~ ~ .

On the Limit Theorems for Random Variables with Values in the Spaces Lp (2 < p < co) 295

Let N N . . . a (X1)N~N, be sequence of i.i.d, copies of the martingale X N ( )N~N, and denote X1,X 2 . . . . the a.s. limits of these martingales. It is easily

checked that the sequences

( (X N, X2))NeN , ((X1, X~))N~N, and N N ( ( X l , X 2 > ) N e N

are martingales (with respect to three different sequences of a-algebras) and that each of them converges a.s. to ( X 1, X2); since (X1, X2) is square integrable by assumption (3.3), it follows that each of these three martingales converges in quadratic mean to (X~ ,X2) . Hence:

(~N = (]E ( X 1 - X N, X 2 - x2N)e) 1/2 --+ 0

when N ~ oo. By the above claim, we have (writing S, u for XJ + -.. +X,).N.

lira I[&- SN./a.II < 2]/2 K a N (3.5)

and

lira rlSU,/a,]d =<2]f2K( lE/xx', I,XN\2~1/22/~ for each N. (3.6) n~oo

Being a step function, X N takes its values in a finite dimensional subspace EN; let K N be the ball in E N of radius less than 4K (IE ( X t u, xN)2) */2.

By (3.6), SU,/a, e K N a.s. for n large enough, hence by (3.5), we have

lim d (S,/a,, K N) <= 2 ]/2 U 6 N . n~ oo

By the observation in Section 2 we conclude that X satisfies the LIL.

Remark 3.3. Throughout the above proof, we could work with a sequence of standard Ganssian random variables instead of the sequence {e,} (and refer either to the result of [,-18] or to [22] Corollary 2, instead of [-2] in Lemma3.1) but our presentation seems more natural.

As we have already mentioned, if a Banach space valued r.v. X satisfies the LIL then sup [[S,/a,J I < oo a.s. therefore sup IrXja, J I < oe a.s. and it follows easily (using Borel-Cantelli's lemma) that: g p < 2 lErlXlrP< oo. On the other hand, if the range of X is finite dimensional, then Strassen's result [19] implies that IE IlXfj2< c~. The next example shows that this is no longer true in the infinite dimensional case.

Example 3.I. There is a random variable X with values in the space 12 (of square summable sequences) satisfying the LIL and yet such that IEI/X]r 2= c~.

Proof We denote (e~)~ the canonical basis in l 2. Let N(co) be an integer valued 1

r.v. such that IP{N=~} c~aLc~LLc~ for large ~. Let e be a Bernouilli r.v.

independent of N and set

~<N(m)


Clearly X is symmetric and IEIIXII2=IEN=c~ Moreover

+ ) (since ~ 1 = Go . \ ~=lctLc~LLc~

IE I] X II 2/LL ]1X II = ~ L L - ~ IP{N = ~} ~

Also,

c~ Lo:(LLo:) 2 ~0C.

IE<XI, X2) 2 = IE(N t A N2) 2 =~, ~2 IP{N, A N 2 =~}

~y ~IP{N, A N2>__~} ~F~(~{N>~}) 2

eLeLLr

but

IP(N > ~) ~ - -

1 therefore the convergence of the series ~o:(LoO2(LLo02 I E ( X 1 , X 2 ) 2 ,~ oo.

By Theorem3.1, X satisfies the LIL on 12. q.e.d.

ensures that

4. Some Consequences

We first recall some notations and a result of [17]: If (X+) is an i.i.d.sequence of copies of a r.v. X we define the (not necessarily finite) quantities CL(X) and IL(X) respectively as

suplEHS,/l/nl[ and IEsup LlS,/a,,ll n=>l n>l

where S, = X 1 +- . . + X, and a, = ~ . Given a probability space (g2, P) and a Banach space B, we denote CL~(Q, P; B) (resp. ILoo(~2, P; B)) the linear space of all B-valued r.v.'s X on (f2, P) which satisfy CL(X)<oo (resp. IL(X)<oo). Equipped with the norm X ~ CL(X) (resp. IL(X)) this space becomes a Banach space. Let us denote 5Po(~2,P;B) the space of B-valued mean zero step functions on (f2, P); clearly (by the finite dimensional limit theorems) 5P0(~2, P;B) is a subset of both CL~(Q,P;B) and IL+(~2, P;B). Note that either condition CL(X)< oo or IL(X)< ~ implies: IEX=0.

We will use below the following result from [17]:

Theorem 4.1. A B-valued r.v. on (f2, P) satisfies the CLT (resp. the LIL) / f and only if it belongs to the closure of 5eo(~2, P ;B ) in CL~o(f2,P;B) (resp. in IL~o(~,P;B)). 7his closure is denoted CL(~2, P; B) (resp. IL(f2, P; B)).

Remark 4.1. i) Let ~o(f2,P;B) be the linear space of all the mean zero r.v. X on (f2,P) which take their values in some finite dimensional subspace of B and satisfy IE JIX ]l 2< oo;it is clear that we may replace 5~ P; B) by ~,~o(~, P; B) in the preceding statement.

On the Limit Theorems for Random Variables with Values in the Spaces L, (2<p < oo) 297

ii) The main use of the preceding theorem is to avoid some routine con- structions by invoking the closed graph theorem.

iii) It also gives some immediate corollaries such as: if X satisfies the CLT or the LIL then any conditional expectation of X also does.

Our next result is based on

Dvoretzky's Theorem ([-5, 7]). For any infinite dimensional Banach space B, any e > 0 and any integer n, there is an n-dimensional subspace B. of B and an isomorphism T. from B. onto the usual n-dimensional euclidian space such that f] T.lr" II Z.-i II < l + e .

We can deduce from it a characterization of finite dimensional spaces:

Theorem 4.2. I f (and only if) the dimension of a Banach space B is infinite, there is a B-valued r.v. X satisfying the LIL but failing the CLT and such that 1EI[X[I 2 =OQ).

Proof We will prove below that there are B-valued r.v. X' and X" each of them satisfying the LIL but such that CL(X')=IEf]X"I[2=oo. Of course we may assume that X' and X" are independent. Let X = X ' + X " ; since IEX'=IEX"=O necessarily, we have also CL(X)= IE [IX II 2= oo and X has the desired properties.

Assume on the contrary that there exists no such variable X'. Then, for any given (/2, P), we have (by Theorem 4.1) IL(D, P; B) c CLoo((2, P; B). Therefore by the closed graph theorem there is a constant 2 such that:

VZ~9~o(f2,P;B) CL(Z) < 2IL(Z). (4.1)

Applying Dvoretzky's theorem, we immediately infer that (4.1) remains true when B is replaced by any Hilbert space, in particular by 12; therefore we must have IL(f2,P;lz)cCL(~2,P;lz) but this (according to Theorem4.1) contradicts the example in Section 3. This final contradiction shows the existence of X'; the existence of X" is proved by a completely analogous argument, q.e.d.

Remark 4.2. The preceding proof shows (and is based on) the following fact: for a Banach space B, the property that all B-valued r.v. which satisfy the LIL also satisfy the CLT (resp. IEI[X]p2 < oo) is a super-property in the sense of [11].

5. The CLT in the Spaces l~, and Lp for 2 < p <

Let X be a r.v. with values in a real Banach space B. We say that X satisfies the

central limit theorem (in short the CLT) if S,,/]//n converges in distribution when n ~ ~ . We say that X is pregaussian 1 if there exists a Gaussian Radon probability measure 7x such that

V~eB* ~ ( X ) ~ =S r ~ ~(d~).

1 In [9], the term pregaussian has a more restrictive meaning


Our original motivation for this section was to produce an example for a r.v. X which fails the CLT but is pregaussian and satisfies

sup IEI]S,/]~ H < oo.

Apparently no such example was known; the examples of [207 fail the CLT precisely because IENS,/]/nll--,oo. We will give our examples in the spaces lp for 2 <p < oo; consequently, they exist also in the space C[0, 1] since any separable Banach space is isomorphic to a subspace of CE0, 1]. We should mention that in Hilbert space and more generally in any space "of cotype 2" (in particular in lp for p < 2) no such example can exist because any pregaussian r.v. automatically satisfies the CLT (cf. [10] Theorem 5),

It is often not necessary to check the "pregaussianness" because of the following

Proposition 5.1. The following two properties of a Banach space B are equivalent:

i) B does not contain an isomorphic copy of Co.

ii) For any B-valued r.v. X the condition sup IE [IS,/]fnl[ < oe ensures that X is pregaussian.

Proof. i i )~ i) First observe that the space Co fails ii): X = (G/V~Logn),>=2 (where (e,) is a Bernoulli-sequence) is a well-known counterexample. This shows that ii) ~ i).

i )~ i i ) Assume that X verifies sup IELIS,/t/nl] < o e; let (XN)N~s be a martingale of B-valued step functions such that IE [] X - X N[I ~ 0 when N ---, 0% and let (X~'MN, " (X, )u~N, ... be sequence of independent copies of (XN)N~W (Xz)N~N, a We write S~ for X~ +.- . + X, N. Then, clearly

IE ]IS.~]J < IEllS.II. (5.1)

By examining the covariances, we observe that there independent B-valued Gaussian r.v.'s (G~) such that:

V~ffB* IF, ~, q i =IE(~,xN) 2.

By the CLT in the finite dimensional case, we have:

exists a sequence of

therefore by (5.1):

supIEN ~ G ~suplglLsdl~nll <oo. (5.2)

On the Limit Theorems for Random Variables with Values in the Spaces L v (2 <p < os) 299

By the result of [15], if B does not contain Co, then (5.2) implies that the series N

G~ converges a.s. to a Gaussian r.v. G which must satisfy 1

V~B* ~ ( 6 ) 2 =~E~(X)~;

in particular, this proves that X is pregaussian, q.e.d.

In what follows, we will say that {S,/]/n} is stochastically bounded if

sup 1P{[4S,/l/nd[ >e} ~ 0 when c ~oo. n

As shown by Jain ([10] Section 3) this is equivalent, for each p<2, to:

sup 1EJISjl/nlf < oo.

Of course if the CLT holds then afortiori {S,/]/n} is stochastically bounded. We will need the following formulation of Jain's result:

Lemma 5.1 (1-10]). There is a constant L such that an arbitrary Banach space valued r.v. X must satisfy:

{sup c 2 IP{ I[X]l •C}} 1/2 ~_~g sup IE I1S,,/]/n[I ; c > 0 n

(of course, this is meaningful only if the righthand side is finite).

Notation. In the sequel we will denote simply A(X) the quantity (sup c 2 IP{ [IXll > c}) 1/2.

c > 0

The preceding lemma and Theorem 4.1 imply the following result (which was independently observed by A. de Araujo and E. Gine using a different argument):

Proposition 5.2. A necessary condition for a r.v. to satisfy the CLT is that c2lP{lIXll >c} -.-~0 when c ~oo .

Proof. Assume that X satisfies the CLT; then by Theorem4.1: for each e>0, there is a mean zero step function !/ such that C L ( X - Y ) < e . Since ]lYIJ is bounded, cZIp(llYll>c)~0 when c ~ o o . Also, by Lemma5.1: A ( X - Y ) < L e . Hence, we can write:

lim c 2 IP{I[X[I >c} < lim c 2 IP{]IX- Y/J >c/2} + lira c 2 IP{JJ Y/J >c/2}

<(2Le)2+0,

and since ~ > 0 is arbitrary, we conclude as announced that

lira c2 IP{I[X[[ >c} =0. c ~ o o

Notation. In the sequel, we denote (e~)~N the canonical basis of IR ~. For 0 < p < o o , the norm in Ip will be denoted simply It.lip, so that we have by


definition:

v( ,~ ) s~ N IlY,.~=e~ll~--(y,l.~=l~) ~/~.

We now state the main result of this section:

Theorem 5.1. Let X = ~, X~e~ be a mean zero r.v. with values in lp, for 2 c} --,0 where c -~oo,

supc2IP{HX[Ip>C} < oo. c>.O

(5.3)

(5.4)

(5.5)

Then Xsatisfies the CLT if and only if (5.3) and (5.4) hold; moreover {S,/1/n } is stochastically bounded if and only if (5.3) and (5.5) hold.

Proof. Condition (5.3) is equivalent to "X is pregaussian" (this is simple and known, cf. [21]). The necessity of the other conditions follows from Propo- sition 5.2, Lemma 5.1 and the remarks preceding it. We pass on to the sufficiency. By a standard symmetrization argument, it is enough to treat the symmetric case, so let X be a symmetric r.v. with values in lp. We will write simply a~ for (IEIX~[2) 1/2. We claim that there are constants K 1, K z independent of X such that

CL(X) <= I~1 A(X) + K2(~ ~)I/P.

To prove this claim, we will use the following

Lemma 5.2. Let 1/1, V 2 . . . . , V~ be independent symmetric r.v.'s with values in /q(0<q< oo); denote Vi ~ the a-th coordinate of Vii; then i f 0 < r < o% we have."

where At , q is a constant depending only on r and q.

This fact is an easy and well known consequence of Hin6in's inequality for the Rademacher functions and we omit its proof. To shorten the formulae appearing below we introduce the following notation:

If 0 < q < m, and if x = ~ x~e~ is an element of IR ~, we will write simply Ixl q s e n

the element of IR ~ which admits the development ~ Ix=] q e=.

We now proceed to prove the above claim: A first application of (5.6) gives us with these notations:

On the Limit Theorems for Random Variables with Values in the Spaces Lv (2__<p < oo) 301

IF, ~lXi p<=Al, p]E /n 2\1/2

Now let (Y~) be an independent copy of the sequence (Xi). We denote g the conditional expectation with respect to the a-algebra spanned by the sequence {Xi}. We may write Vc~eN

(i~1 'X~[2)1/2 ~ (i~1 'X'[2)1/2-~ (i~l 'Y/~'2)1/2 -~ (i_-~1 '~ic~]2) 1/2

. Ii/Zx

for the last line we have used the elementary inequality:

va, b>0

This gives immediately:

IX~I2 p<g ~ IXil2- Y~ p/2 +l /n(2a~)l /P '

hence, by (5.7):

n (AI,p)-I]EIISnIIp~IE ~lXil 2_ 2 el2q_]~(2Gp)l/p" (5.8) ~ii p/2

Applying (5.6) again (this time to the variables ~=lX~12-1y~l 2 with q=p/2), we obtain:

2 2 \x/zllx/2 ~g ~1 ]Xi'2-]~ii'2[[1/2~A1/2'p/2]g (~ (]Xi]2-]gi') ) p / 2 p/2

hence

c 1 1 / " \x/2111/2 1Y/14)1/2 1/2] ~-~A1/2'p/2[IE t~l [Xi'4) p/2 AVIE (~1 p/2

n d- 1/4 <=2A~/a'p/2IE ~ [Xi[ p/4" (5.9)

Now let s = p/4 if p/4 < 1 and s = 1 if p/4 > 1. In both cases we have s > 1/2 since p > 2; moreover:

i[~l iXi,4lp/4~(~i \l/s


therefore:

~g ~lx~141f~ <~lg(~l llSihl~S)i/~s, (5.10)

To complete the proof of the above claim we will use the following known fact:

Lemma 5.3. Let Z, Z 1, Z2, . . .Z , . . . be an i.i.d, sequence of r.v's. For each q>2 there is a constant Bq (depending only on q) such that:

VnsN IF. Z q <B~l/n(supc21P{Z>c}) 1/2. c > 0

For lack of a suitable reference, we give the

Proof. Assume that Vc>0, IP{Z>c}<l/c 2. Let us write F(c) for

IP( t /" ~l/q ) t~ Zq) >ct . Wehave:

f ( c ) < _ P Zql{z~<__c}>c q +P {Zi>e } i

<c-qYlEZ~l~z~<_~+ IP{Z~>c} 1 i=1

< n c - q ] E ( Z q l{z<__~}) + n c - 2.

But

F(c)<nc-2(1 +q/q-2), and hence:

]E Z~ F(c)dc< ~ F(c)dc+ ~ F(c)dc 0 0 Vn

o o

__<]/n+n(l +q/q--2) ~_c -2 dc gn

<tfn(2+q/q-2)

IEZq l{z<=r <-i qtq- l P{Z >t} dt ~ i qtq- 3 d t = q ~ C q-2 0 0 - -

therefore

and the lemma follows by homogeneity with Ba < 2 + q/q- 2.

This lemma, combined with (5.8), (5.9) and (5.10) yields (since 4s>2)"

(A l,p) -1 IE ILS.H.<~2AI/2,p/2 B4s ~n A(X) + ] ~ ( ~ a~) lip.

Therefore, we conclude as claimed earlier that:

CL(X) G(2A 1/2,p/2 B 4~ A l.p) A (X) + A 1,p(~ a~) 1/p. (5.11)

On one hand, this inequality clearly shows that the conditions (5.3) and (5.5) imply the stochastic boundedness of {Sn/]/~}. On the other hand, consider, for

On the Limit Theorems for Random Variables with Values in the Spaces Lp (2<p < oo) 303

each integer N, the variable X(N)= ~ X ~ % It is easy to check that condition r162

(5.4) implies that A(X(N))-~ 0 when N ~ c~; if we assume also that ~ a~ < o% then (5.1l) ensures that CL(X(N))--,O when N ~ o o ; b y Theorem 4.1 we conclude that X satisfies the CLT and this finishes the proof.

From Theorem 5.1, we deduce

Example 5.1. For each p with 2 <p < oo there is a r.v. X with values in lp which is

pregaussian and for which the sequence J-~(X~+...+X,) is stochastically

bounded but which fails the CLT.

Proof We keep the same notations as in Theorem 5.1. Let N(o)) be an integer valued r.v. such that

IP {N >= n} ~ 1/n "~/~. (5.12)

Let e be a r.v. independent of N such that I P { e = l } = I P { e = - l } = l / 2 . We consider the r.v. X = e ~ e~ so that I]Xl[p=N 1/p.

N2<7.<Na+N

By (5.12), we know that c2]p { H X[I p > c} remains bounded but does not tend to zero when c--+ ~ .

On the other hand, an easy computation shows that

1 1

cr =(ip{N2 <ct<=N2 + N})a/2 eO(o: 4 2p);

p 1

therefore ~o-~<oo since ~c~ 4 2 < o r if p>2. By Theorem5.1, X has all the announced properties.

Remark 5.1. Let (~,P) be a probability space and let (M,#) be an arbitrary measure space. Let X be a mean zero r.v. on ((2, P) with values in Lp(M, #); we may identify X with a r.v. J?(co, a) on (Q, P ) x (M, #). It is easy to check that (if 1 = p < oo) X is pregaussian if and only if

(m J2(o, c~)V) p/2 ~(dc~) < c~ (5.33 M

(where the expectation sign is meant with respect to (s Moreover, Theo- rem 5.1 remains valid, with (5.3') instead of (5.3), with almost the same proof.

Remark 5,2. Actually, the properties of Ip that were used in the proof of Theorem 5.1 are shared by a wide class of Banach lattices. Indeed the proof works with lp replaced by any Banach lattice L which is (in the terminology of [12]) of type >p for some p > 2 and of type __<q for some q<oo.

In this general setting we obtain: a r.v. X with values in such a lattice L satisfies the CLT if and only if X is pregaussian and c21P{]IXH >c} -~0 when

c ~ o o . Similarly {S,/~n} is stochastically bounded iff X is pregaussian and sup c 2 IP{rlXN >e} < oo. c > 0

We have chosen our notations to make it easier for the reader to check this statement using the approach of [12] (see also [16]). This class of lattices


includes the spaces of the form

Lvo(L;l(... (Lw)) with min{po,pl . . . . ,Pk} > 2 and max{po,. . . ,pk} < oe.

We do not know if the preceding statement is true when L is a Banach space of type 2 in the sense of [91.

References

1. Aldous, D.: A characterization of Hilbert space using the central limit theorem. [To appear] 2. Bonami, A.: Etude des coefficients de Fourier des fonctions de L~(G). Ann. Inst. Fourier

(Grenoble) 20, 335~402 (1970) 3. Crawford, J.: Ph. D Dissertation. University of Wisconsin at Madison (1976) 4. Doob, J.L.: Stochastic Processes. New York: Wiley !953 5. Dvoretzky, A.: Some results on convex bodies and Banach spaces. Proc. Sympos. Lin. Spaces.

Jerusalem (1961) 123-160 6. Feller, W. : A limit theorem for random variables with infinite moments. Amer. J. Math. 68, 257-

262 (1946) 7. Figiel, T.: A short proof of Dvoretzky's theorem on almost spherical sections of convex bodies.

Compositio Math. 33, 297-302 (1976) 8. Hartman, S, Wintner, A.: On the law of the iterated logarithm. Amer. J. Math. 43, 169-176

(1941) 9. Hoffmann-Jorgensen, J., Pisier, G.: The strong law of large numbers and the central limit

theorem in Banach spaces. Ann. Probability 4, 587-599 (1976) 10. Jain, N.C.: Central limit theorem and related questions in Banach spaces. To appear in the

Proceedings of the Amer. Math. Soc. Probab. Sympos. Urbana, Illinois (1976) 11. James, R.C.: Some self dual properties of Banach spaces. Ann. Math. Studies 69, 159-175 (1972) 12. Krivine, J.L.: Th~or6mes de factorisation dans les espaces r6ticul6s. S6minaire Maurey-Schwartz,

exp. 21-22 (1973 1974) 13. Kuelbs, J.: A counterexample for Banach space valued random variables Ann. Probability 4,

684-689 (1976) 14. Kuelbs, J.: The LIL and related strong convergence theorems for Banach space valued random

variables. Ecole d'Et6 de Calcul des Probabilit6s de St. Flour 1975. Lect. Notes in Math. No 539. Berlin-Heidelberg-New York: Springer 1975

15. Kwapiefi, S.: On Banach spaces containing co, A supplement to the paper by J. Hoffmann- Jorgensen "Sums of independent Banach space valued random variables". Studia Math. 52, 187- 188 (1974)

16. Maurey, B.: Sur les espaces de Banach munis de structures locales inconditionnelles. S6minaire Maurey-Schwartz exp. 24-25 (1973-1974)

17. Pisier, G.: Le th~or~me de la limite centrale et la loi du logarithme it6r6 dans les espaces de Banach.S6minaire Maurey-Schwartz, exp. 3-4 and Annexe 1 (1975-1976)

18. Schreiber, M.: Fermeture en probabilit6 des chaos de Wiener. C.R. Acad. Sci. Paris, S6r. AI 265, 859 (1967)

19. Strassen, V.: A converse to the law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie verw. Gebiete 3, 265-268 (1966)

20. Strassen, V., Dudley, R.M.: The central limit theorem and e-entropy. Lecture Notes in Math. 89, 224-231. Berlin-Heidelberg-New York: Springer 1969

21. Vakhania, N.V.: Sur une propri6t~ des r6partions normales de probabilit~s dans les espaces 1 v ( l < p < c ~ ) et H. C.R. Acad. Sci. Paris S6r A 260, 1334-1336 (1965)

22. Varberg, D.: Convergence of quadratic forms in independent random variables. Ann. Math. Statist. 37, 567-576 (1966)

R e f u le 1 Mars 1977

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