Independent functions in rearrangement invariant spaces and the Kruglov property

Let $X$ be a separable or maximal rearrangement invariant space on $[0,1]$. It is shown that the inequality
$$\begin{equation*} \biggl\|\,\sum_{k=1}^\infty f_k\biggr\|_{X} \le C\biggl\|\biggl(\,\sum_{k=1}^\infty f_k^2\biggl)^{1/2}\biggr\|_X \end{equation*}$$
holds for an arbitrary sequence of independent functions $\{f_k\}_{k=1}^\infty\subset X$, $\displaystyle\int_0^1f_k(t)\,dt=0$, $k=1,2,\dots$, if and only if $X$ has the Kruglov property. As a consequence, it is proved that the same property is necessary and sufficient for a version of Maurey's well-known inequality for vector-valued Rademacher series with independent coefficients to hold in $X$.
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