Data: wtorek, 25.01.2022, godz. 11:00-12:00
Prelegent: dr Maciej Rzeczut (IM PAN, Warszawa)
Abstrakt: Suppose \(X_1,X_2,\ldots\) and \(Y_1,Y_2,\ldots\) are symmetric and iid random variables. The well known fact that \(\left|\sum_i a_i X_i\right|_{L^p}\) is for a given \(0<p<\infty\) equivalent to a specific quasi-Orlicz norm of the sequence \(a_1,a_2,\ldots\) is a consequence of inequalities of Rosenthal for \(p>2\) and Johnson and Schechtman for \(p<2\). The same question for \(\left|\sum_{i,j} a_{i,j} X_iY_j \right|_{L^p}\) for \(p>2\) is answered easily by iterating the Rosenthal inequality for \(p>2\). For \(p<2\) the answer is much more involved. We provide a description of matrices \(a_{i,j}\) in the unit ball of span of \(X_iY_j\) in \(L^p\) as a modular space dependent on specific quasi-Orlicz norms of rows and columns of the matrix. This leads to an explicit closed formula in the case of \(X_i,Y_j\) with \(\mathbb{P}(X>t)\simeq \mathbb{P}(Y>t)\simeq t^{-r}\) for \(0<r<2\).
Miejsce: A1-33/34 (sala RW)