Geodesic
Convergence of Bernoulli series and the set of sums of a conditionally convergent functional series
Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 2, pp. 420-429 
S. A. Čobanjan. Convergence of Bernoulli series and the set of sums of a conditionally convergent functional series. Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 2, pp. 420-429. http://geodesic.mathdoc.fr/item/TVP_1983_28_2_a18/
@article{TVP_1983_28_2_a18,
     author = {S. A. \v{C}obanjan},
     title = {Convergence of {Bernoulli} series and the set of sums of a~conditionally convergent functional series},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {420--429},
     year = {1983},
     volume = {28},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1983_28_2_a18/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.