×

Limit theorems for record counts and times in the \(F ^{\alpha }\)-scheme. (English) Zbl 1282.60052

The classical law of large numbers and the central limit theorems for the counting process and the record times of iid sequences of continuous random variables are extended to general \(F^{\alpha}\)-schemes, that is, to the class of sequences such that the sequence of occurrences of records, as well as the sequence of interrecord times, are independent.
Some special \(\alpha\)-sequences are investigated for more detailed results. In a final section, some random environment models are considered.

MSC:

60G70 Extreme value theory; extremal stochastic processes
60F15 Strong limit theorems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arnold, B.C., Balakrishnan, N., Nagaraja, H.N.: Records. Wiley, New York (1998) · Zbl 0914.60007
[2] Ballerini, R., Resnick, S.I.: Embedding sequences of successive maxima in extremal processes with applications. J. Appl. Probab. 24, 827-837 (1987) · Zbl 0633.60046
[3] Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge, MA (1987) · Zbl 0617.26001
[4] Borovkov, K., Pfeifer, D.: On record indices and record times. J. Stat. Plan. Inference 45, 65-79 (1995) · Zbl 0826.60039
[5] Chow, Y.S., Teicher, H.: Probability Theory. Springer, New York (1978) · Zbl 0399.60001
[6] Deheuvels, P., and Nevzorov, V.B.: Records in the Fα-scheme. II. Limit laws. J. Math. Sci. 88, 29-35 (1998)
[7] Gut, A.: Probability: A Graduate Course. Springer, Berlin (2005) · Zbl 1076.60001
[8] Gut, A., Stadtmüller, U.: Records in subsets of a random field. Preprint, Uppsala University and University of Ulm, 11 pp. (2012) · Zbl 1267.60032
[9] Hall, P., and Heyde, C.C.: Martingale Limit Theory and its Application. Academic, New York (1980) · Zbl 0462.60045
[10] Nevzorov, V.B.: Record and inter-record times for sequences of nonidentically distributed random variables. Zapiski Nauchm. Semin. LOMI 142, 109-118 (1985) (Engl. Transl.: J. Soviet Math. 36, 510-516, 1987) · Zbl 0565.62032
[11] Nevzorov, V.B.: The number of records in a sequence of non-identically distributed random variables. In: Probability Distributions and Mathematical Statistics, Part 2, pp. 373-388. Fan, Tashkent (1986) (Engl. transl.: J. Soviet Math., 38, 2375-2382, 1987) · Zbl 0667.60039
[12] Nevzorov, V.B.: Asymptotic distributions of records in nonstationary schemes. J. Stat. Plan. Infer. 45, 261-273 (1995) · Zbl 0818.62016
[13] Nevzorov, V.B.: Records: Mathematical Theory. Amer. Math. Soc., Providence RI (2001) · Zbl 0989.62029
[14] Pakes, A.G.: Convergence and divergence of random series. Aust. N.Z. J. Stat. 46, 29-40 (2004) · Zbl 1082.60026
[15] Weissman, I.: Records from a power model of independent observations. J. Appl. Probab. 32, 982-990 (1995) · Zbl 0840.60049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.