Abstract
Baum and Katz (Trans. Am. Math. Soc. 120:108-123, 1965) obtained convergence rates in the Marcinkiewicz-Zygmund law of large numbers. Their result has already been extended to the short-range dependent linear processes by many authors. In this paper, we extend the result of Baum and Katz to the long-range dependent linear processes. As a corollary, we obtain convergence rates in the Marcinkiewicz-Zygmund law of large numbers for short-range dependent linear processes.
Highlights
There are many literature works concerning the convergence rates in the MarcinkiewiczZygmund law of large numbers
In the short-range dependent case, Koopmans [ ] showed that if ζ has the moment generating function, the strong law of large numbers for the linear process holds with exponential convergence rate
We extend Theorem . to the long-range dependent linear processes
Summary
There are many literature works concerning the convergence rates in the MarcinkiewiczZygmund law of large numbers. Baum and Katz [ ] obtained the following convergence rates in the MarcinkiewiczZygmund law of large numbers. In the short-range dependent case, Koopmans [ ] showed that if ζ has the moment generating function, the strong law of large numbers for the linear process holds with exponential convergence rate. Theorem to the setting of short-range dependent linear processes. With r > to the short-range dependent linear process of a sequence of identically distributed φ-mixing random variables. We obtain convergence rates in the Marcinkiewicz-Zygmund law of large numbers for long-range dependent linear processes of i.i.d. random variables. The above theorem shows a convergence rate in the Marcinkiewicz-Zygmund weak law of large numbers with the norming sequence Wn(p).
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