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THE DISTRIBUTION OF

VECTOR-VALUED RADEMACHER SERIES

S.J. Dilworth and S.J. Montgomery-Smith

Abstract. Let X =

nxn be a Rademacher series with vector-valued coefficients.

We obtain an approximate formula for the distribution of the random variable ||X||in terms of its mean and a certain quantity derived from the K-functional of inter-polation theory. Several applications of the formula are given.

1. Results

In [6] the second-named author calculated the distribution of a scalar Rademacherseries

nan. The principal result of the present paper extends the results of [6]

to the case of a Rademacher series

nxn with coefficients (xn) belonging to anarbitrary Banach space E. Its proof relies on a deviation inequality for Rademacherseries obtained by Talagrand [9]. A somewhat curious feature of the proof is thatit appears to exploit in a non-trivial way (see Lemma 2) the platitude that everyseparable Banach space is isometric to a closed subspace of . The principal re-sult is applied to yield a precise form of the Kahane-Khintchine inequalities and tocompute certain Orlicz norms for Rademacher series.

First we recall some notation and terminology from interpolation theory (see e.g.[1]). Let (E1, ||.||1) and (E2, ||.||2) be two Banach spaces which are continuouslyembedded into some larger topological vector space. For t > 0, the K-functionalK(x, t; E1, E2) is the norm on E1 + E2 defined by

K(x, t; E1, E2) = inf{||x1||1 + t||x2||2 : x = x1 + x2, xi Ei}.

For a sequence (an) 2, we shall denote the K-functional K((an), t; 1, 2) byK1,2((an), t) for short. For 1 p < , a sequence (xn) in a Banach space (E, ||.||) issaid to be weakly-p if the scalar sequence (x

(xn)) belongs to p for every x E.The collection of all weakly-p sequences is a Banach space, denoted

wp (E), with the

norm given by lwp ((xn)) = sup||x||1 ||(x(xn))||p (where ||(an)||p = ( |an|p)1/p).

If (xn) w2 (E), we make the following definition:

Kw1,2((xn), t) = sup||x||1

K1,2((x(xn)), t).

Observe that Kw1,2((xn), t) is a continuous increasing function of t. In fact, it is a

Lipschitz function with Lipschitz constant at most w2 ((xn)).

1991 Mathematics Subject Classification. Primary 46B20; Secondary 60B11, 60G50.

Key words and phrases. Rademacher series, K-functional, Banach space.The second author was supported in part by NSF DMS-9001796

Typeset by AMS-TEX

1

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2 S.J. DILWORTH AND S.J. MONTGOMERY-SMITH

Next we set up some function space notation. Let (, , P) be a probabilityspace. A Rademacher (or Bernoulli) sequence (n) is a sequence of independentidentically distributed random variables such that P(n = 1) = P(n = 1) = 12 .For a random variable Y defined on , its decreasing rearrangement, Y, is thefunction on [0, 1] defined by Y(t) = inf

{s > 0 : P(

|Y

|> s)

t}

. For 0 < p 0, we have

(1) P(||X|| > 2E||X|| + 6Kw1,2((xn), t)) 4et2/8,

and, for some absolute constant c, we have

(2) P

||X|| > 1

2E||X|| + cKw1,2((xn), t)

cet2/c.

The proof of the Main Theorem will be deferred until the end of the paper inorder to proceed at once with the applications.

Corollary 1. Let X =

nxn be an almost surely convergent Rademacher seriesin a Banach space. Then, for 0 < t 110 , we have(3) S(t)

E

||X

||+ Kw1

,2((xn),log(1/t)),

where S denotes the real random variable ||X||. The implied constant is absolute.Proof. (1) and (2) give rise to the inequalities S(4et

2/8) 2E||X||+6Kw1,2((xn), t)and S(cet

2/c) 12E||X|| + cKw1,2((xn), t), respectively, whence (3) follows for allsufficiently small t by an appropriate change of variable. To see that the lowerestimate implicit in (3) is valid in the whole range 0 < t < 110 , we recall from [2]that E||X||2 9E2||X||. Hence, by the Paley-Zygmund inequality (see e.g. [4,p.8]),for 0 < < 1, we have

P(||X|| > E||X||) (1 )2E2X

EX2

19

(1 )2,

whence P(||X|| > (1 310

)E||X||) 110 , which easily implies (3). In [4] Kahane proved that if P(||X|| > t) = , where X is a Rademacher

series in a Banach space, then P(||X|| > 2t) 42. By iteration this impliesP(||X|| > st) 14(4)s for s = 2n. According to our next corollary the exponent sin the latter result may be improved to be a certain multiple of s2.

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THE DISTRIBUTION OF VECTOR-VALUED RADEMACHER SERIES 3

Corollary 2. Let X =

nxn be an almost surely convergent Rademacher seriesin a Banach space. Then, for t > 0 and s 1, we have

P(

||X

||> st)

1

c1

P(

||X

||> t)

c1s2

for some absolute constant c1.

Proof. By choosing c1 < c, where c is the constant which appears in (2), theresult becomes trivial whenever P(||X|| > t) c. Hence we may assume thatP(||X|| > t) < c. Choose > 0 such that P(||X|| > t) = ce2/c. Then (2) givest 12E||X|| + cKw1,2((xn), ). Thus,

st s2E||X|| + scKw1,2((xn), )

2E||X|| + Kw1,2((xn),cs)

provided s max(4, 1/c). Now (1) gives

P(||X|| > st) 4e(cs)2/8

= 4

1

c(ce

2/c)

c3s2/8

= 4

1

c(P(||X|| > t)

c3s2/8,

which gives the result.

Our next corollary, which is the vector-valued version of a recent result ofHitczenko [3], is a rather precise form of the Kahane-Khintchine inequalities.

Corollary 3. Let X = nxn be a Rademacher series in a Banach space. Then,for 1 p < , we have

(E||X||p)1/p E||X|| + Kw1,2((xn),

p).

The implied constant is absolute.

Proof. We may assume that p 2. It follows from a result of Borell [2] that(E||X||2p)1/2p 3(E||X||p)1/p. Since 12 ||Y||p ||Y||2p, ||Y||2p for every ran-dom variable Y (as is easily verified), it follows (letting S denote the randomvariable ||X||) that 12 ||S||p ||S||2p,

3||S||p. So it suffices to prove that

||S||p, ES+ Kw1,2((xn),

p) to obtain the desired conclusion. By Corollary 1,we have

||S

||p,

ES+ sup0

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To evaluate the expression in brackets we shall make use once more (see Corollary2) of the elementary inequality K1,2((an), s) max(1,s/t)K1,2((an), t). Thus,

sup0

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2. Proof of main result

The principal ingredient in the proof of the Main Theorem is the following de-viation inequality of Talagrand [9].

Theorem A. Let X =N

n=1 nxn be a finite Rademacher series in a Banachspace and let M be a median of ||X||. Then, for t > 0, we have

P

|||

Nn=1

nxn|| M| > t

4et2/82 ,

where = w2 ((xn)Nn=1).

Lemma 1. Let X =N

n=1 nxn be a finite Rademacher series in a Banach spaceE. Then, for t > 0, we have

P(||X|| > 2E||X|| + 3K((xn)Nn=1, t; w1 (E), w2 (E))) 4et2/8.

Proof. It follows from Theorem A that for all y1, . . . , yN in E, we have

(4) P(||

nyn|| > 2E||

nyn|| + tw2 ((yn))) 4et2/8.

On the other hand, since max || nyn|| = w1 ((yn)), we have the trivial estimate(5) P(||

nyn|| > w1 ((yn))) = 0.

Let xn = x(1)n + x

(2)n for 1 n N, let X(1) =

nx

(1)n , and let X(2) =

nx

(2)n .

Then

w1 ((x(1)n )) + t

w2 ((x

(2)n )) + 2E||X(2)|| w1 ((x(1)n )) + tw2 ((x(2)n )) + 2E||X(1)|| + 2E||X||

3w1 ((x(1)n )) + tw2 ((x(2)n )) + 2E||X|| 2E||X|| + 3(w1 ((x(1)n )) + tw2 ((x(2)n ))).

Let Q denote 2E||X|| + 3(w1 ((x(1)n )) + tw2 ((x(2)n ))). Then, by (4) and (5) and bythe above inequality, we have

P(||X|| > Q) P(||X(1)|| + ||X(2)|| > w1 ((x(1)n )) + tw2 ((x(2)n )) + 2E||X(2)||) P(||X(1)|| > w1 ((x(1)n ))) + P(||X(2)|| > 2E||X(2)|| + tw2 ((x(2)n )))< 0 + 4et

2/8

The desired conclusion now follows from the definition of the K-functional.

Lemma 2. Let x1, . . . , xN be elements of the Banach space . Then

K((xn)Nn=1, t;

w1 (),

w2 ()) 2Kw1,2((xn)Nn=1, t).

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Proof. For 1 n N, let xn = (xn,j)j=1 . A simple convexity argument gives

||(xn)||wp() = sup

1

j

N

n=1|xn,j|p

(1/p).

It follows that the mapping which associates an element (yn)n=1 wp () with

the element in (p) whose jth coordinate equals (yn,j)n=1 is an isometry. HenceK((xn), t;

w1 ,

w2 ) = K(((xn)), t; (1), (2)). Let (yn)

n=1 (2) and let

> 0. For each n there exists a splitting yn = z(1)n + z

(2)n such that

||(z(1)n,j)j=1||1 + t||(z(2)n,j)j=1||2 K1,2((yn,j)j=1, t) + .

It follows that

||(z(1)n )||(1) + t||(z(2)n )||(2) = sup1n dK1,2((x(xn)), t))

inf||x||1

P(

nx(xn) > dK1,2((x(xn)), t))

det2/d.

The Paley-Zygmund inequality now gives

P

||X|| > 1

2E||X|| + d

6Kw1,2((xn), t)

min

P

||X|| > 3

4E||X||

, P

||X|| > d

2Kw1,2((xn), t)

min

1

144, det

2/d.

.

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References

1. J. Bergh and J. Lofstrom, Interpolation Spaces. An Introduction, Grundlehren der mathema-

tischen Wissenschaften, Vol. 223, Springer-Verlag, Berlin- Heidelberg-New York, 1976.2. C. Borell, On the integrability of Banach valued Walsh polynomials, Seminaire de Probabilites

XIII Lecture Notes in Mathematics, Vol. 721, Springer-Verlag, Berlin-Heidelberg- New York,

1979, pp. 13.3. P. Hitczenko, Domination inequality for martingale transforms of a Rademacher sequence,

preprint (1992).4. J.P. Kahane, Some Random Series of Functions, Cambridge Studies in Advanced Mathemat-

ics Vol. 5, Cambridge University Press, Cambridge, 1985.5. S. Kwapien, A theorem on the Rademacher series with vector valued coefficients , Probability

in Banach spaces, Oberwolfach 1975, Lecture Notes in Mathematics, Vol. 526, Springer-Verlag,Berlin-Heidelberg-New York, 1976, pp. 157158.

6. S.J. Montgomery-Smith, The distribution of Rademacher sums, Proc. Amer. Math. Soc. 109(1990), 517522.

7. G. Pisier, De nouvelles caracterisations des ensembles de Sidon, Mathematical Analysis and

Applications, Adv. Math. Supp. Stud. 7B (1981), 686725.8. V.A. Rodin and E.M. Semyonov, Rademacher series in symmetric spaces, Analyse Math. 1

(1975), 207222.9. M. Talagrand, An isoperimetric theorem on the cube and the Khintchine-Kahane inequalities,

Proc. Amer. Math. Soc. 104 (1988), 905909.

Department of Mathematics, University of South Carolina, Columbia, South Car-

olina 29208, U.S.A.

Department of Mathematics, University of Missouri, Columbia, Missouri 65211,

U.S.A.

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