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.