Subseries and signed series

Armengol Gasull; Francesc Mañosas

doi:10.3934/cpaa.2019024

Article Contents

2019, Volume 18, Issue 1: 479-492. Doi: 10.3934/cpaa.2019024

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Subseries and signed series

Armengol Gasull and
Francesc Mañosas^,

Departament de Matemàtiques, Universitat Autònoma de Barcelona, Facultat de Ciències, 08193 Bellaterra, Spain

* Corresponding author
* Corresponding author

Received: January 31, 2018

Revised: January 31, 2018

Published: August 2018

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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Abstract

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.

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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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