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Existence of moments of a counting process and convergence in multidimensional time

  • Oleg Klesov (a1) and Ulrich Stadtmüller (a2)

Abstract

Starting with independent, identically distributed random variables X 1,X 2... and their partial sums (S n ), together with a nondecreasing sequence (b(n)), we consider the counting variable N=∑ n 1(S n >b(n)) and aim for necessary and sufficient conditions on X 1 in order to obtain the existence of certain moments for N, as well as for generalized counting variables with weights, and multi-index random variables. The existence of the first moment of N when b(n)=εn, i.e. ∑ n=1 ℙ(|S n |>εn)<∞, corresponds to the notion of complete convergence as introduced by Hsu and Robbins in 1947 as a complement to Kolmogorov's strong law.

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Corresponding author

Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine `KPI', Peremogy Avenue 56, 03056 Kyiv, Ukraine. Email address: klesov@matan.kpi.ua
Department of Number and Probability Theory, Ulm University, 89069 Ulm, Germany. Email address: ulrich.stadtmueller@uni-ulm.de

References

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[1] Baum, L. E. and Katz, M. (1965).Convergence rates in the law of large numbers.Trans. Amer. Math. Soc. 120,108123.
[2] Chow, Y. S. and Lai, T. L. (1975).Some one-sided theorems on the tail distribution of sample sums with applications to the last time and largest excess of boundary crossings.Trans. Amer. Math. Soc. 208,5172.
[3] Chow, Y. S. and Lai, T. L. (1978).Paley-type inequalities and convergence rates related to the law of large numbers and extended renewal theory.Z. Wahrscheinlichkeitsth. 45,119.
[4] Fuk, D. H. and Nagaev, S. V. (1971).Probability inequalities for sums of independent random variables.Teor. Veroyat. Primen. 16,660675 (in Russian). English translation: Theory Prob. Appl. 16,643660.
[5] Gut, A. (1978).Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices.Ann. Prob. 6,469482.
[6] Hardy, G. H. and Wright, E. M. (1975).An Introduction to the Theory of Numbers,4th edn.Oxford University Press.
[7] Heyde, C. C. and Rohatgi, V. K. (1967).A pair of complementary theorems on convergence rates in the law of large numbers.Proc. Camb. Phil. Soc. 63,7382.
[8] Hsu, P. L. and Robbins, H. (1947).Complete convergence and the law of large numbers.Proc. Nat. Acad. Sci. USA 33,2531.
[9] Kao, Ch.-S. (1978).On the time and the excess of linear boundary crossings of sample sums.Ann. Statist. 6,191199.
[10] Klesov, O. I. (1985).The strong law of large numbers for multiple sums of independent identically distributed random variables.Mat. Zametki 38,915930 (in Russian). English translation: Math. Notes 38,10061014.
[11] Klesov, O. I. (2014).Limit Theorems for Multi-Indexed Sums of Random Variables(Prob. Theory Stoch. Model. 71).Springer,Berlin.
[12] Lai, T. L. (1974).Summability methods for independent identically distributed random variables.Proc. Amer. Math. Soc. 45,253261.
[13] Petrov, V. V. (1974).The one-sided strong law of large numbers for ruled sums.Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 7,5559 (in Russian).
[14] Sirazhdinov, S. Kh. and Gafurov, M. U. (1987).Method of Series in Boundary Problems for Random Walks.Fan,Tashkent (in Russian).
[15] Smythe, R. T. (1974).Sums of independent random variables on partially ordered sets.Ann. Prob. 2,906917.

Keywords

MSC classification

Existence of moments of a counting process and convergence in multidimensional time

  • Oleg Klesov (a1) and Ulrich Stadtmüller (a2)

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