Inequalities for Walsh like random variables

Abstract

Let (Xn)n≥1 be a sequence of mean zero independent random variables. Let Wk={∏j=1kXij|1≤i1<i2…<ik}, Yk=⋃j≤kWj 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(52p2p−1)m−1plogp(Kδ)m for 1<p<∞. We show that for f∈[Ym] the following inequalities hold:‖f‖2≤‖f‖p≤C(p,m)‖f‖2                       for   2<p<∞‖f‖2≤C(q,m)‖f‖p≤C(q,m)‖f‖2     for   1<p<2,   1p+1q=1and ‖f‖2≤C(4,m)2‖f‖1≤C(4,m)2‖f‖2. These generalize various well known inequalities on Walsh functions.