Abstract
This paper is devoted to some applications of probability theory in infinite dimensional spaces to problems of analysis related to rearrangements of summands in normed spaces. In particular, the problem on linearity of the set of sums of a conditionally convergent series in a normed space, as well as the problem on permutational a.s. convergence of functional series (both scalar and vector) are typical examples of such problems. The last problem is connected with the well-known long standing Kolmogorov problem on existence of a rearrangement of an orthogonal system, converting the latter into a system of convergence.
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Chobanyan, S. (1994). Convergence A.S. of Rearranged Random Series in Banach Space and Associated Inequalities. In: Hoffmann-Jørgensen, J., Kuelbs, J., Marcus, M.B. (eds) Probability in Banach Spaces, 9. Progress in Probability, vol 35. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0253-0_1
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