International Journal of Mathematics and Mathematical Sciences
Volume 13 (1990), Issue 2, Pages 353-356

Inequalities for Walsh like random variables

D. Hajela

Bell Communications Research, 2P-390, 445 South Street, Morristown, New Jersey 07960, USA

Received 28 March 1988

Copyright © 1990 D. Hajela. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let (Xn)n1 be a sequence of mean zero independent random variables. Let Wk={j=1kXij|1i1<i2<ik}, Yk=jkWj and let [Yk] be the linear span of Yk. Assume δ|Xn|K for some δ>0 and K>0 and let C(p,m)=16(52p2p1)m1plogp(Kδ)m for 1<p<. We show that for f[Ym] the following inequalities hold:f2fpC(p,m)f2      for2<p<f2C(q,m)fpC(q,m)f2for1<p<2,1p+1q=1and f2C(4,m)2f1C(4,m)2f2. These generalize various well known inequalities on Walsh functions.