Inequalities for Walsh like random variables
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 inequal...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1990-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171290000527 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |