Abstract
The paper discusses the stability of suitably-defined maxima of a set of i.i.d. random variables with multidimensional indices.It is shown that theorems of Gnedenko (1943) and Tomkins (1986) concerning relative stability and complete relative stability of maxima remain valid in the new setting.Moreover, a criterion for almost sure relative stability for maxima with multidimensional indices is presented, extending a result of Barndorff-Nielsen (1963).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adler, A., “The fair games problem for multidimensionally indexed random variables,” Bull. Inst. Math. Acad. Sinica 20, 169–185, (1992).
Barndorff-Nielsen, O., “On the limit behaviour of extreme order statistics,” Ann. Math. Statist. 34, 992–1002, (1963).
Chow, Y.S. and Teicher, H., Probability Theory, Springer-Verlag, New York, 1978.
Gnedenko, B.V., “Sur la distribution limite du terme maximum d’une séries aléatoire,” Ann. Math. 44, 423–453, (1943).
Gut, A., “Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices,” Ann. Probab. 6, 469–482, (1978).
Gut, A., “Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices,” Ann. Probab. 8, 298–313, (1980).
Hall, P., “A note on a paper of Barnes and Tucker,” J. London Math. Soc. 19, 170–174, (1979a).
Hall, P., “On the relative stability of large order statistics,” Math. Proc. Cambridge Philos. Soc. 86, 467–475, (1979b).
Hardy, G.H. and Wright, E.M., An Introduction to the Theory of Numbers, Oxford University Press, London, 1954.
Lagodowski, Z.A. and Rychlik, Z., “Complete convergence and convergence rates for randomly indexed sums of random variables with multidimensional indices,” Bull. Polish Acad. Sci. Math. 32, 219–223, (1985).
Li, D. and Wu, Z., “Law of the iterated logarithm for B-valued random variables with multidimensional indices,” Ann. Probab. 17, 760–774, (1989).
Li, Z.F. and Tomkins, R.J., “Complete stability of large order statistics,” J. Theoret. Probab. 4, 213–221, (1991).
Li, D., Rao, M.B. and Wang, X., “The law of the iterated logarithm for independent random variables with multidimensional indices,” Ann. Probab. 20, 660–674, (1992).
Resnick, S.I. and Tomkins, R.J., “Almost sure stability of maxima,” J. Appl. Probab. 10, 387–401, (1973).
Smythe, R.T., “Strong laws of numbers for r-dimensional arrays of random variables,” Ann. Probab. 1, 164–170, (1973).
Smythe, R.T., “Sums of independent random variables on partially ordered sets,” Ann. Probab. 2, 906–917, (1974).
Titchmarsh, E.C. The Theory of the Riemann Zeta-Function, Oxford University Press, New York, 1951.
Tomkins, R.J., “Regular variation and the stability of maxima,” Ann. Probab. 14, 984–995, (1986).
Wichura, M.J., “Some Strassen type laws of the iterated logarithm for multiparameter stochastic processes with independent increments,” Ann. Probab. 1, 272–296 (1973).
Author information
Authors and Affiliations
Corresponding author
Additional information
AMS 2000 Subject Classification. Primary—60F15, 60G60, 62G30, Secondary—10A25, 60G99
Rights and permissions
About this article
Cite this article
Li, M.Z.F., Tomkins, R.J. Stability of Maxima of Random Variables with Multidimensional Indices. Extremes 7, 135–147 (2004). https://doi.org/10.1007/s10687-005-6196-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-005-6196-x