Abstract
This paper proves the Baum--Katz theorem for sequences of pairwise independent identically distributed random variables with general norming constants under optimal moment conditions. The proof exploits some properties of slowly varying functions and the de Bruijn conjugates, and uses the techniques developed by Rio (1995) to avoid using the maximal type inequalities.
Highlights
Introduction and resultLet 1 ≤ p < 2, αp ≥ 1 and {X, Xn, n ≥ 1} be a sequence of pairwise independent identically distributed (p.i.i.d.) random variables
By using some results related to slowly varying functions and techniques developed by Rio [10], we provide the necessary and sufficient conditions for k nαp−2P max
The result provides the rate of convergence in the Marcinkiewicz–Zygmund strong law of large numbers (SLLN) with regularly varying normalizing constants
Summary
Let 1 ≤ p < 2, αp ≥ 1 and {X , Xn, n ≥ 1} be a sequence of pairwise independent identically distributed (p.i.i.d.) random variables. By using some results related to slowly varying functions and techniques developed by Rio [10], we provide the necessary and sufficient conditions for k nαp−2P max. The result provides the rate of convergence in the Marcinkiewicz–Zygmund strong law of large numbers (SLLN) with regularly varying normalizing constants. When the random variables are i.i.d. with E(X ) = 0, E(|X |p ) < ∞, and L(·) = 1, (1) was obtained by Baum and Katz [2]. The notion of regularly varying function can be found in Seneta [12, Chapter 1]. ISSN (electronic) : 1778-3569 https://comptes- rendus.academie- sciences.fr/mathematique/
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
