Convergence of Bernoulli series and the set of sums of a conditionally convergent functional series

We survey a. s. convergence criteria for series $\sum a_k\varepsilon_k$ where $(\varepsilon_k)$ is a sequence of independent Bernoulli random variables, and $a1,a2,\dots$ are elements of a Banach space $X$. These criteria are applied to investigate the set $\mathfrak S_{(a_k)}$ of sums of a conditionally convergent series $\sum a_k$. The following problem is posed: does the a. s. convergence of $\sum a_k\varepsilon_k$ imply that $\mathfrak S_{(a_k)}$ is a shifted closed subspace of $X$. The answer is affirmative, if $X$ is of cotype $q$, $q<4$, and possesses the local unconditional structure.