Theory of Probability and Mathematical Statistics
Strong laws of large numbers for weighted sums of -dimensional arrays of random variables and applications to marked point processes
Ta Cong Son, Tran Manh Cuong, Le Quang Dung and Le Van Dung
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Abstract: The aim of this paper is twofold. Firstly, we give equivalent conditions for the convergence of each of three series in the Kolmogorov-type three-series theorem for random variables with multidimensional indices. Our results extend the results of Sung (2011) for multi-indexed sums. Secondly, by using these results, we develop almost sure convergence for weighted sums of d-dimensional arrays of random variables. We obtain the Jajte type strong law of large number for these arrays of random variables. As an application, the almost sure asymptotic behavior for sums generated by marked point processes is investigated.
Keywords: Marcinkiewicz laws of large numbers, strong laws of large numbers, weighted sums, multidimensional index, marked point processes
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