Internat. J. Math. & Math. Sci.
VOL. 13 N0. 2 (1990) 353-356
353
INEQUALITIES FOR WALSH
LIKE RANDOM VARIABLES
D. HAJELA
Bell Communications Research, 2P-390
445 South Street
Morristown, New Jersey, 07960, U.S.A.
(Received March 28, 1988)
ABSTRACT.
Let (X)
be a sequence of mean zero independent random variables.
n
k
j{=l Xij II ’ il < i2"’" <ik }’ Yk JUnk Wj and let [Yk be the
Yk" Assume
IXnl K for some > 0 and K > 0 and let
16(5/ p-p2"m-l
We show that for f e [Y
C(p,m)
for
<p <
logp ()"
Let
linear span of
Wk
(
m.
the
m
following inequalities hold:
These
inequalities on
Wlsh
KEY WORDS AND PHRASES.
well
various
Walsh Functions, Martingales, Square Function.
Primary 60E15, Secondary 60A99.
INTRODUCTION.
Let (X) n)l be a sequence of independent mean zero random variables.
n
products of length k of the
W
k
let
known
functions.
1980 AMS SUBJECT CLASSIFICATION CODE.
1.
generalize
Yk
jJj
(Xn)n)
{Xi2Xi2...
and let
[Yk
Let Wk be
i.e.
Xikll
<
i
<
2
< ik},
be the linear span of functions in
note is to show that for functions in
[Yk
The object of this
Yk"
the p’th mean is of the same order as the
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:
second moment.
(Xn)
THEOREM 2.4.
Let
be
n
variables on a probability space (fl,
that
IXnl
K for all n.
a
sequence
of
independent
1). Suppose there
For f e [Y we have
n
exist
mean
>
zero
random
>
0 such
0 and K
D. HAJELA
354
Iif112
IIllz
where C(p,n)
C(q n)
<
16
Ilfllp
C(4,n)2
(5
pL_21)
<C(q,n:>llll
Ilflll
p,
n-1
logp
We assume of course that the
p
<
LP(fl)
(R),
f]f()]Pdp
(
C(4,n)2
,
o,:
<p <
+
""
2
11112
(1.2)
()n
(Xn)n>
belong to some
LP(fl).
Recall that for
is the space of all measurable functions f such that
lfl Ip (flz()lPd.) x/p-
and the norm of f is
is familiar with martingales and refer to Garsia
We assume the reader
[2] for unexplained notation.
PROOF OF THE INEQUALITIES.
2.
We require three preliminary facts in order to prove theorem 2.4.
We denote by
E(X), the expectation of a random variable X.
THEOREM 2.1. [I]. Let r (t) be the Rademacher functions on [0 I].
Then
(. lak12)I/2
I akrk(t)Idt I
/2
f
n
(ak)k=
for any complex numbers
ken
0
THEORE 2.2.
(Johnson, Schechtman, and Zinn [5]) Let
(ak)nk=l n.
independent mean zero variables and let
(Xn)n>
Then for p
>
_C.
be a sequence of
2
l/p)
Recall that for a martingale f
fn-1
and its square function is S(f)
(fn)n>l
d2n )1/2
(y.
n
THEOREM 2.3 [4].
lfn IIp
p
For a martingale f
p2 I1(
k.
d2)/211p for
its difference sequence is d n
(fn),
<
p
<
fn
The last fact that we need is:
we have
(R).
We may now prove Theorem 2.4 quite easily.
Let
be a sequence of independent mean zero random variables
THEORE 2.4.
on a probability space
(Xn)n>
(R,).
Suppose there exist 5
>
0 and K
I1112 ’ I111p C<p m)llll 2 o,: 2 < <
I1112 ’ C(q,m)l I11p c(q.,m)l I112 or < p < 2
p
and lip
+ llq
>
0 such that
(2.)
(2.2)
I.
I1112 ’ C(4,m)
C(4,m)
(2.3)
INEQUALITIES FOR WALSH LIKE RANDOM VARIABLES
2
logpP ()m.
(5 p--L
)m-l-
where C(p,m)= 16
The proof is by induction on m.
PROOF.
Suppose m
[YI].
and f e
355
We will first consider the case p
I auX
Then f
some a e C, k
k
for
kn
Theorem 2.2 we have,
kn
>
n.
2.
By
kn
((112.11) / (11.11) /)
(Snce
ndependenL)
EXk=0 and Lhe
16 pK
p) 1/ )
logp max (( k I1 21/2, k Il
max
logp
ae
.
16p___K
logp
kn
[
> 6(
follows for m
.
I.
n )I
Xj,
(J
sequence.
<
n.
[Ym
fnXn where fn
It
(2.4)
[a12)I/2 and
so
by
(2.4)
the
result
kn
kn
We assume the result is valid for f e
write f as f
lakl2)l/2
that
then
clear
is
and
[Ym ].
Let f e
[Ym+l ].
Note that we may
fn
only depends on the random variables
f
is
sum
a
of
a
martingale
difference
Applylng Theorem 2.3 we have
-
5 p
2
2
2
,
p-1
2)I/2
II
p
n
n>l
IXn[
==(=)f=[d=[Ip
Y.
f
(since
0
K)
(by Theorem 2.1)
2
,7
0
c(p,.)
p
n
k)l
2
f
0
II n>l =n (=)= II 2dt
2
(by nducton)
2 I/2
0
nl
2
0
2
p
52
K C(p,m)
(f f
G 0
2
52
K C(p,m)
(f [
nl
n
[ rn(t)fn ()2dtdu)l/2
n)l
f2()d)I/2
(since
(by Fublnl s
eorem)
demacher’s
the
orthogonal)
are
D. HAJELA
356
K
5
<
en
p
<
.
Ilftl
(since f is a martlngle)
2 we employ the classical trick of der.
l-O
0
+--p for
0
1/2 (since q
lfll ’ lfl12
0
0
+T for o 1/3,
.h
2/3
4
1/3
P
+ q
1.
(by HSlder’s inequality)
1/2
see (2.3) note that
defined by
[Ym ]"
t f
/2llfl1112
t q
/3
>
2).
" o.o=., p,ov.g (2.2).
Finally to
so again by HOXder’s inequality,
2/3
1/3 (for f : [Y ])
is automatic, proving (2.3).
REFERENCES
1.
2.
ZYGMUND, A., Trigonometric Series, Cambridge University Press, 1959.
ROSENTHAL, H., Biased Coin Convolution Operators, The Alteld Book, University
of Illinois at Urbana, 1976.
3.
4.
5.
BONAMI, A., Etude des Coefficients de Fourier des Fonctions de Lp, Ann. Inst.
Fourier 20 (1970), 335-402.
GARSIA, A., Martingale Inequalities, Seminar Notes on Recent Progress, Math.
Lecture Note Series, W.A. Benjamin Inc., 1973.
JOHNSON, W., SCHECHTMAN, G. and ZINN, J., Best Constants in Moment Inequalities
For Linear Combinations of Independent and Exchangeable Randon Variables,
Ann. of Probability 13 (1985), 234-253.