An Extension of Theorems of Hechner and Heinkel

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

$$\displaystyle{\left \{\begin{array}{ll} \mbox{ $\int _{0}^{\infty }\mathbb{P}^{q/p}\left (\|X\|^{q}> t\right )dt <\infty $}&\mbox{ if $0 <q <p$,}\\ & \\ \mbox{ $\mathbb{E}\|X\|^{p}\ln (1 +\| X\|) <\infty $} &\mbox{ if $q = p$,}\\ & \\ \mbox{ $\mathbb{E}\|X\|^{q} <\infty $} &\mbox{ if $q> p$.} \end{array} \right.}$$