Decoupling of Banach-valued multilinear forms in independent symmetric Banach-valued random variables
Let E be a Banach space and \(\Phi\) : \(E\to {\mathbb{R}}_ +\) be symmetric, continuous and convex. Let \(\{U_ i\}\) and \(\{r_ i\}\) be independent sequences of random variables having, respectively, U(0,1) and symmetric Bernoulli distributions, and let \(\{U_ i^{(j)}\}\) and \(\{r_ i^{(j)}\}\) for \(j=1,2,...,d\) be independent copies of these sequences. We prove two-sided inequalities between the quantities \[ E\Phi (\sum _{i\in {\mathbb{Z}}^ d_ +}r_{i_ 1}...r_{i_ d}F_ i(U_{i_ 1},...,U_{i_ d})) \] and their ''decoupled'' versions \[ E\Phi (\sum _{i\in {\mathbb{Z}}^ d_ +}r_{i_ 1}^{(1)}...r_{i_ d}^{(d)}F_ i(U_{i_ 1}^{(1)},...,U_{i_ d}^{(d)})), \] for Bochner integrable \(F_ i: [0,1]^ d\to E\). This generalizes results of Kwapień and of Zinn.
scientific article; zbMATH DE number 3986278