On the Hsu-Robbins-Erdős-Spitzer-Baum-Katz theorem for random fields
2012 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 387, no 1, 447-463 p.Article in journal (Refereed) Published
The by now classical results on convergence rates in the law of large numbers involving the sums Sigma(infinity)(n=1) n(alpha r-2)P(vertical bar S(n)vertical bar > n(alpha)epsilon), where r > 0, alpha > 1/2, such that alpha r >= 1 has been extended to the case alpha = 1/2 by adding additional logarithms. All of this has been generalized to random fields by the first named author in [A. Gut, Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices, Ann. Probab. 6 (1978) 469-482; A. Gut, Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices, Ann. Probab. 8 (1980) 298-313]. The purpose of the present paper is to treat the case when the alpha's differ in the different directions of the field, as well as mixed cases with some alpha's equal to 1/2 with added logarithms and/or iterated ones.
Place, publisher, year, edition, pages
2012. Vol. 387, no 1, 447-463 p.
Sums of i.i.d. random variables, Random fields, Law of large numbers, Law of the iterated logarithm, Convergence rates, Last exit time
IdentifiersURN: urn:nbn:se:uu:diva-161422DOI: 10.1016/j.jmaa.2011.09.010ISI: 000296115100037OAI: oai:DiVA.org:uu-161422DiVA: diva2:457554