On the Marcinkiewicz–Zygmund strong laws for arbitrary dependent sequences
Section snippets
Result
Let be a sequence of non-negative random variables defined on a probability space and set . The main result is as follows.
Theorem 1 Suppose is non-negative and non-decreasing sequence such that in . If then converges almost surely. Moreover, if then, (1.1) can be replaced by
Let , be a random sequence defined on and set , . Let be a slowly varying
Proof
We have the following Hájek–Rényi type inequality (cf. Hájek and Rényi, 1955).
Theorem 2 Let , . Then, for any , and ,
Proof of Theorem 2 Assume . For set Clearly . Suppose first that , then
Application
Suppose , and is slowly varying. Without loss of generality we can assume that is increasing and differentiable (cf. Proposition B1.9(3), p.366 in de Haan and Fereira, 2006). Define the generalized inverse of . Further, , where is a slowly varying function, with , , for some (cf. Proposition B1.9(9), p.367 in de Haan and Fereira, 2006). Assume that random variables , are
Acknowledgments
I would like to thank the editor and referee for comments and suggestions.
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