On uniqueness of distribution of a random variable whose independent copies span a subspace in $L_p$
Tom 230 / 2015
Studia Mathematica 230 (2015), 41-57 MSC: 46E30, 46B20, 46B09. DOI: 10.4064/sm8089-1-2016 Opublikowany online: 21 January 2016
Let $1\leq p \lt 2$ and let $L_p=L_p[0,1]$ be the classical $L_p$-space of all (classes of) $p$-integrable functions on $[0,1]$. It is known that a sequence of independent copies of a mean zero random variable $f\in L_p$ spans in $L_p$ a subspace isomorphic to some Orlicz sequence space $l_M$. We give precise connections between $M$ and $f$ and establish conditions under which the distribution of a random variable $f\in L_p$ whose independent copies span $l_M$ in $L_p$ is essentially unique.