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Subseries and signed series

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This work is supported by Ministry of Economy, Industry and Competitiveness of the Spanish Government through grants MINECO/FEDER MTM2016-77278-P and MTM2017-86795-C3-1-P and by Generalitat de Catalunya, grant number 2017 SGR 1617.
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  • For any positive decreasing to zero sequence $a_n$ such that $\sum a_n$ diverges we consider the related series $\sum k_na_n$ and $\sum j_na_n.$ Here, $k_n$ and $j_n$ are real sequences such that $k_n∈\{0,1\}$ and $j_n∈\{-1,1\}.$ We study their convergence and characterize it in terms of the density of 1's in the sequences $k_n$ and $j_n.$ We extend our results to series $\sum m_na_n,$ with $m_n∈\{-1,0,1\}$ and apply them to study some associated random series.

    Mathematics Subject Classification: Primary: 40A05; Secondary: 60G50, 65B10, 97I30.


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  • Table 1.  Behavior of the random series $W^{\alpha,\beta},$ with $p = q,$ according to $\alpha$ and $\beta.$ Cases with $(*)$ are covered by the Kolmogorov's Three-Series Theorem but not by our approach

    $0< \alpha<\frac12$ $\alpha=\frac12$ $\frac12< \alpha<1$ $\alpha=1$ $\alpha>1$
    $0\le\beta\le\frac12$ a.s. div. a.s. div. (*) a.s. conv. a.s. conv. conv.
    $\frac12<\beta\le1$ a.s. div. a.s. conv. (*) a.s. conv. a.s. conv. conv.
    $\beta>1$ a.s. div. a.s. conv. a.s. conv. conv. conv.
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