On a domination of sums of random variables by sums of conditionally independent ones (Q1323303)

Let \(({\mathcal F}_ n)\) be an increasing sequence of \(\sigma\)-algebras. Two adapted sequences of random variables \((X_ n)\) and \((Y_ n)\) are tangent if, for each \(n = 1,2, \dots,\) the conditional distributions \({\mathcal L} (X_ n \mid {\mathcal F}_ n)\) and \({\mathcal L} (Y_ n \mid {\mathcal F}_ n)\) coincide. \((Y_ n)\) is called the decoupled version of \((X_ n)\) if they are tangent and \((Y_ n)\) is a sequence of conditionally independent random variables. The main result states the existence of an absolute constant \(K\) such that for every \(p\), \(1 \leq p < \infty\), one has \(\| \sum X_ k \|_ p \leq K \| \sum Y_ k \|_ p\), where \((Y_ n)\) is the decoupled version of \((X_ n)\).