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On extreme value asymptotics for increments of renewal processes. (English) Zbl 0831.60060
The authors consider the problem of detecting changes in the intensity of a general renewal counting process. Their main idea is to consider increments of the process rather than the process itself, and to compare the maximum deviation between empirical intensities over subsequent small intervals or between small intervals in the beginning and the whole range, respectively. They prove a number of limit theorems for increments of renewal processes under the null hypothesis of no change in intensity. The paper also gives tables of Monte Carlo estimates of selected percentiles of limiting distribution functions of four sup-functionals proposed for testing change in intensity.

MSC:
60G70 Extreme value theory; extremal stochastic processes
60F05 Central limit and other weak theorems
60K05 Renewal theory
60F17 Functional limit theorems; invariance principles
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[1] Chen, X., Inference in a simple change-point model, Sci. sinica ser. A, 31, 654-667, (1988) · Zbl 0691.62025
[2] Csörgö, M.; Horváth, L., Asymptotic distributions of pontograms, Math. proc. Cambridge philos. soc., 101, 131-139, (1987) · Zbl 0645.60040
[3] Csörgö, M.; Horváth, L.; Steinebach, J., Invariance principles for renewal processes, Ann. probab., 15, 1441-1460, (1987) · Zbl 0635.60032
[4] Deheuvels, P.; Révész, P., Weak laws for the increments of Wiener processes, Brownian bridges, empirical processes and partial sums of i.i.d. r.v.’s, (), 69-88
[5] Eastwood, V.R., Some recent developments concerning asymptotic distributions of pontograms, Math. proc. Cambridge philos. soc., 108, 559-567, (1990) · Zbl 0712.60023
[6] Eastwood, V.R.; Warren, J., Some simulations of kendall—kendall pontograms and related point processes, (), 1-131
[7] Glynn, P.W.; Iglehart, D.L., A new class of strongly consistent variance estimators for steady-state simulations, Stochastic process. appl., 28, 71-80, (1988) · Zbl 0656.62096
[8] Huse, V.R., On some nonparametric methods for changepoint problems, Ph.D. thesis, (1988), Carleton University Ottawa, Canada
[9] Kendall, D.G.; Kendall, W.S., Alignments in two-dimensional random sets of points, Adv. appl. probab., 12, 380-424, (1980) · Zbl 0425.60009
[10] Leadbetter, M.R.; Lindgren, G.; Rootzén, H., Extremes and related properties of random sequences and processes, (1983), Springer New York · Zbl 0518.60021
[11] Mason, D.M.; van Zwet, W.R., A note on the strong approximation to the renewal process, Publ. inst. statist. univ. Paris, 32, 81-93, (1987) · Zbl 0658.60061
[12] Steinebach, J., (), 1-13, Bericht Nr.
[13] Steinebach, J.; Zhang, H., On a weighted embedding for pontograms, Stochastic process. appl., 47, 183-195, (1993) · Zbl 0779.60031
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