Series of independent mean zero random variables in rearrangement invariant spaces with the Kruglov property

This paper compares sequences of independent mean zero random variables in a rearrangement invariant space $X$ on $[0,1]$ with sequences of disjoint copies of individual terms in the corresponding rearrangement invariant space $Z_X^2$ on $[0,\infty)$. Principal results of the paper show that these sequences are equivalent in $X$ and $Z_X^2$ respectively if and only if $X$ possesses the (so-called) Kruglov property. We also apply our technique to complement well-known results concerning isomorphism between rearrangement invariant spaces on $[0,1]$ and $[0,\infty)$.