Stochastics Seminar

Speaker: Olav Kallenberg

Title: On \(L^p\)-symmetric sequences and processes

Abstract: By the Cramér--Wold theorem, the distribution of a random sequence \(\xi=(\xi_1,\xi_2,\dots)\) is determined by the distributions of all finite linear combinations \(u\xi=u_1\xi_1+\cdots+u_n\xi_n\). By a theorem of Rudin and Hardin, it is enough to know the \(L^p$-norms $\|1+u\xi\|_p\) for some fixed \(p>0\) that is not an even integer, assuming of course that they are finite. This yields some interesting characterizations involving only the \(p\)-th absolute moments \(E|u\xi|^p\). In this talk, we consider \(L^p\)-versions of all the basic probabilistic symmetries, along with their continuous-time counterparts.