On the complete convergence of randomly weighted sums of random fields

• Autorzy: Gdula A. M. , Krajka A.
• Języki publikacji: EN

Abstrakty

EN

Let (…) be a d-dimensional random field indexed by some subset V of lattice Nd, which are stochastically dominated by a random variable X. Let (…) be a 2d-dimensional random field independent of (…) and such that (…) for some constant M. In this paper, we give conditions under which the following series (…), is convergent for some real t, some fixed p > 0 and all ε > 0. Here |n| is used for (…). The randomly indexed sums of field (…) are considered too.

Słowa kluczowe

EN

complete convergence; rate of convergence; sums of random fields; multidimensional indexing

PL

pełna zbieżność; tempo zbieżności; sumy pól losowych; indeksowanie wielowymiarowe

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2014
• Tom: Vol. 47, nr 1
• Strony: 232--252
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Gdula A. M.

autor

Krajka A.

• Uniwersytet Marii Curie-Skłodowskiej

Bibliografia

• [1] P. Erdos, On a theorem of Hsu and Robbins, Acta. Math. Statist. 20 (1949), 286–291.
• [2] L. E. Baum, M. Katz, Convergence rates in the law of large numbers, Trans. Amer. Math. Soc. 120 (1965), 108–123.
• [3] A. Gut, Complete convergence. Asymptotics statistics. Proceedings of the Fifth Prague Symposium, Physica Verlag held September 4–9, 1993, (1994), 237–247.
• [4] A. Gut, Complete convergence and convergence rates for randomly indexed partial sums with an application to some first passage times, Acta Math. Acad. Sci. Hungar. 42 (1983), 225–232; Correction, ibid. 45 (1985), 235–236.
• [5] A. Gut, Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices, The Annals of Probab. 6(3) (1978), 469–482.
• [6] J. Hoffmann-Jørgensen, Sums of independent Banach space valued random variables, Studia Math. 52 (1974), 159–186.
• [7] P. L. Hsu, H. Robbins, Complete conevergence and the law of large numbers, Proc. Nat. Acad. Sci. USA 33 (1947), 25–31.
• [8] T.-C. Hu, M. O. Cabrera, S. H. Sung, A. Volodin, Complete convergence for arrays of rowwise independent random variables, Commun. Korean Math. Soc. 18 (2003), 375–383.
• [9] T.-C. Hu, F. Móricz, R. L. Taylor, Strong laws of large numbers for arrays of rowwise independent random variables, Acta Math. Hung. 54(1-2) (1989), 153–162.
• [10] T.-C. Hu, D. Szynal, A. I. Volodin, A note on complete convergence for arrays, Statist. Probab. Lett. 38 (1998), 27–31.
• [11] M. Ledoux, M. Talagrand, Probability in Banach Space, Springer, 1991.
• [12] H. P. Rosenthal, On the subspaces of Lp(p > 2) spanned by sequences of independent random variables, Israel J. Math. 8 (1970), 273–303.
• [13] R. T. Smythe, Sums of independent random variables on partially ordered sets, Ann. Probability 2 (1974), 906–917.
• [14] S. H. Sung, Complete convergence for weighted sum of random variables, Statist. Probab. Lett. 77 (2007), 303–311.
• [15] S. H. Sung, A. I. Volodin, T.-C. Hu, More on complete convergence for arrays, Statist. Probab. Lett. 71 (2005), 303–311.
• [16] L. V. Thahn, G. Yin, Almost sure and complete convergence of randomly weighted sums of independent random elements in Banach spaces, Taiwanese Journal of Mathematics 15 (2011), 1759–1781.
• [17] E. T. Whittaker, G. N. Watson, A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions with an account of the principal trnscendental functions, Cambridge University Press, Fourth Edition reprinted 2002.