INEQUALITIES FOR WALSH LIKE RANDOM VARIABLES

Let (X) be a sequence of mean zero independent random variables. n k Wk j{=l Xij II ' il 0 and K > 0 and let C(p,m) 16(5/ p2"m-l ()" for < p < m. We show that for f e (Y the p- logp m following inequalities hold:


INTRODUCTION.
Let (X) be a sequence of independent mean zero random variables.Let W k be n n)l products of length k of the (Xn)n) i.e.W k {Xi2Xi2... Xikll < i 2 < < ik}, let Yk jJj and let [Yk be the linear span of functions in Yk" The object of this note is to show that for functions in [Yk the p'th mean is of the same order as the second moment.
As such this generalizes classical inequalities such as Khlnchln's inequality in Zygmund [I] as well as more recent inequalities on Walsh functions such as those of H. Rosenthal [2] and A. Bonaml [3].Precisely we prove the following: THEOREM 2.4.
Let (Xn) n be a sequence of independent mean zero random variables on a probability space (fl, 1).Suppose there exist > 0 and K > 0 such that IXnl K for all n.For f e [Y we have n D. HAJELA <C(q,n:>llll , o,: <p < 2 + Iif112 C(q n) Ilfllp " " (1.2) where C(p,n) 16 (5 pL_21) n-1 p, ()n logp We assume of course that the (Xn)n> belong to some LP(fl).Recall that for p < (R), LP(fl) is the space of all measurable functions f such that f]f()]Pdp ( and the norm of f is lfl Ip (flz()lPd.)x/p-We assume the reader is familiar with martingales and refer to Garsia [2] for unexplained notation.

PROOF OF THE INEQUALITIES.
We require three preliminary facts in order to prove theorem 2.4.We denote by E(X), the expectation of a random variable X. (Johnson, Schechtman, and Zinn [5]) Let (Xn)n> be a sequence of independent mean zero variables and let (ak)nk=l n.Then for p > 2 l/p) Recall that for a martingale f (fn)n>l its difference sequence is d n fn fn-1 and its square function is S(f) (y.d2n )1/2 n The last fact that we need is: THEOREM 2.3 [4].For a martingale f (fn), we have p2 I1( d2)/211p lfn IIp p k. for < p < (R).
We may now prove Theorem 2.4 quite easily.THEORE 2.4.Let (Xn)n> be a sequence of independent mean zero random variables on a probability space (R,).
PROOF.The proof is by induction on m.We will first consider the case p >  .fnXn where fn [Ym and fn only depends on the random variables n )I Xj, (J < n.It is clear then that f is a sum of a martingale difference sequence.Applylng Theorem 2. THEOREM 2.1.[I].Let r (t) be the Rademacher functions on [0 I]. n Then f I akrk(t)Idt I (.lak12)I/2 for any complex numbers (ak)k=_C.