On uniqueness of distribution of a random variable whose independent copies span a subspace in L_p

TL;DR

This paper investigates the conditions under which the distribution of a mean zero random variable in L_p is uniquely determined by the subspace spanned by its independent copies, linking it to Orlicz sequence spaces.

Contribution

It establishes precise conditions connecting the distribution of a random variable and the Orlicz space spanned by its independent copies in L_p.

Findings

Conditions for the essential uniqueness of the distribution of the random variable.

Connections between the Orlicz function M and the original random variable.

Characterization of subspaces spanned by independent copies in L_p.

Abstract

Let 1\leq p<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 from L_p spans in L_p a subspace isomorphic to some Orlicz sequence space l_M. We present precise connections between M and f and establish conditions under which the distribution of a random variable f whose independent copies span l_M in L_p is essentially unique.