## Sums of independent random variables in rearrangement invariant function spaces.(English)Zbl 0674.60051

The aim of the paper is to formalize one aspect of the general principle that independent random variables behave like disjoint functions. The authors’ abstract runs as follows:
Let X be a quasinormed rearrangement invariant function space on (0,1) which contains $$L_ q(0,1)$$ for some finite q. There is an extension of X to a quasinormed rearrangement invariant function space Y on (0,$$\infty)$$ so that for any sequence $$(f_ i)^{\infty}_{i=1}$$ of symmetric random variables on (0,1), the quasinorm of $$\sum f_ i$$ in X is equivalent to the quasinorm of $$\sum \hat f_ i$$ in Y, where (\^f$${}_ i)^{\infty}_{i=1}$$ is a sequence of disjoint functions on (0,$$\infty)$$ such that for each i, $$\hat f_ i$$ has the same decreasing rearrangement as $$f_ i$$. When specialized to the case $$X=L_ q(0,1)$$, this result gives new information on the quantitative local structure of $$L_ q$$.
Reviewer: A.Gut

### MSC:

 60G50 Sums of independent random variables; random walks 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Full Text: