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An Extension of Theorems of Hechner and Heinkel

  • Deli LiEmail author
  • Yongcheng Qi
  • Andrew Rosalsky
Part of the Fields Institute Communications book series (FIC, volume 76)

Abstract

Let 0 < p < 2 and \(0 <q <\infty\). Let {X n ;  n ≥ 1} be a sequence of independent copies of a random variable X taking values in a real separable Banach space \((\mathbf{B},\|\cdot \|)\) and set \(S_{n} = X_{1} + \cdots + X_{n},\ n \geq 1\). This paper is devoted to extending recent theorems of Hechner (Lois Fortes des Grands Nombres et Martingales Asymptotiques. Doctoral thesis, l’Université de Strasbourg, France, 2009) and Hechner and Heinkel (J Theor Probab 23:509–522, 2010). By applying the new versions of the classical Lévy (Théorie de L’addition des Variables Aléatoires. Gauthier-Villars, Paris, 1937), Ottaviani (1939), and Hoffmann-Jørgensen (Studia Math 52:159–186, 1974) inequalities obtained by Li and Rosalsky (Stoch Anal Appl 31:62–79, 2013), we show that
$$\displaystyle{\sum _{n=1}^{\infty }\frac{1} {n}\mathbb{E}\left ( \frac{\left \|S_{n}\right \|} {n^{1/p}}\right )^{q} <\infty }$$
if and only if
$$\displaystyle{\sum _{n=1}^{\infty }\frac{1} {n}\left ( \frac{\|S_{n}\|} {n^{1/p}}\right )^{q} <\infty \ \ \text{a.s.}}$$
and
$$\displaystyle{\left \{\begin{array}{ll} \mbox{ $\int _{0}^{\infty }\mathbb{P}^{q/p}\left (\|X\|^{q}> t\right )dt <\infty $}&\mbox{ if $0 <q <p$,}\\ & \\ \mbox{ $\mathbb{E}\|X\|^{p}\ln (1 +\| X\|) <\infty $} &\mbox{ if $q = p$,}\\ & \\ \mbox{ $\mathbb{E}\|X\|^{q} <\infty $} &\mbox{ if $q> p$.} \end{array} \right.}$$

Notes

Acknowledgements

The authors are grateful to the referees for carefully reading the manuscript and for providing many constructive comments and suggestions which enabled them to improve the paper. In particular, one of the referees so kindly pointed out to us the relationship of our Theorem 3 to Theorem 2.4.1 of the Doctoral Thesis of Florian Hechner [6] prepared for l’Université de Strasbourg, France. The research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada and the research of Yongcheng Qi was partially supported by NSF Grant DMS-1005345.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematical SciencesLakehead UniversityThunder BayCanada
  2. 2.Department of Mathematics and StatisticsUniversity of Minnesota DuluthDuluthUSA
  3. 3.Department of StatisticsUniversity of FloridaGainesvilleUSA

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