Khinchin's inequality and the asymptotic behavior of the sums \(\sum \epsilon_ nx_ n\) in Banach lattices (Q923467)

Let \((X,\| \cdot \|)\) be a Banach lattice and let \(\{r_ i\}_{i\geq 1}\) be a Rademacher sequence of i.i.d. rv's. The Maurey- Khintchine inequality \[ C^{-1}\| (\sum^{n}_{i=1}| x_ i|^ 2)^{1/2}\| \leq E \| \sum^{n}_{i=1}r_ ix_ i\| \leq C\| (\sum^{n}_{i=1}| x_ i|^ 2)^{1/2}\| \] for the sequence \(\{r_ ix_ i\}_{i\geq 1}\) is extended for a sequence of i.i.d. X-valued rv's and for a sequence of X-valued martingale differences. Moreover, an exponential probability inequality for the rv \(\| \sum^{n}_{i=1}r_ ix_ i\|\) is also proved.