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The stochastic geyser problem for first-passage times

Published online by Cambridge University Press:  14 July 2016

Josef Steinebach
Josef Steinebach*
Affiliation:
University of Marburg
*
∗Postal address: Fachbereich Mathematik, Universität Marburg, Lahnberge, 3550 Marburg, W. Germany.
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Abstract

Let X1, X2, · ·· be a sequence of independent, identically distributed (i.i.d.) random variables with positive mean. An analogue of Rényi's (1962) stochastic geyser problem is solved for the associated process of first-passage times. More precisely, it is shown that a single realization of the sequence determines the distribution function (d.f.) of the Xn's almost surely (a.s.), even if the observations are erroneous up to an order o(log n).

Keywords

STOCHASTIC GEYSER PROBLEMFIRST-PASSAGE TIMESERDÖS–RÉNYI STRONG LAWERDÖS–RENYI INCREMENTS
Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 1 , March 1981 , pp. 91 - 103
Copyright
Copyright © Applied Probability Trust 

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References

[1] Bártfai, P. (1966) Die Bestimmung der zu einem wiederkehrenden Prozeß gehörenden Verteilungsfunktion aus den mit Fehlern behafteten Daten einer einzigen Realisation. Studia Sci. Math. Hung. 1, 161–168.Google Scholar
[2] Book, S. A. (1975a) An extension of the Erdös–Rényi new law of large numbers. Proc. Amer. Math. Soc. 48, 438–446.Google Scholar
[3] Book, S. A. (1975b) A version of the Erdös–Rényi law of large numbers for independent random variables. Bull. Inst. Math. Acad. Sinica 3, 199–211.Google Scholar
[4] Book, S. A. (1976) Large deviation probabilities and the Erdös–Rényi law of large numbers. Canad. J. Statist. 4, 185–210.Google Scholar
[5] Erdös, P. and Renyi, A. (1970) On a new law of large numbers. J. Analyse Math. 23, 103–111.Google Scholar
[6] Heyde, C. C. (1967) Asymptotic renewal results for a natural generalization of classical renewal theory. J. R. Statist. Soc. B 29, 141–150.Google Scholar
[7] Lorden, G. (1970) On excess over the boundary. Ann. Math. Statist. 41, 520–527.Google Scholar
[8] Plachky, D. and Steinebach, J. (1975) A theorem about probabilities of large deviations with an application to queuing theory. Period. Math. Hung. 6, 343–345.Google Scholar
[9] Renyi, A. (1962) A sztochasztikus gejzir (Vortragsauszug). Publ. Math. Inst. Hung. Acad. Sci. 7, 643.Google Scholar
[10] Steinebach, J. (1976) Exponentielles Konvergenzverhalten von Wahrscheinlichkeiten großer Abweichungen. Dissertation, University of Düsseldorf.Google Scholar
[11] Steinebach, J. (1978) A strong law of Erdös–Rényi type for cumulative processes in renewal theory. J. Appl. Prob. 15, 96–111.Google Scholar
[12] Steinebach, J. (1979) Erdös–Rényi-Zuwächse bei Erneuerungsprozessen und Partialsummen auf Gittern. Habilitationsschrift, University of Düsseldorf.Google Scholar