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A note on the waiting times between record observations

Published online by Cambridge University Press:  14 July 2016

Paul T. Holmes  and
William E. Strawderman
Paul T. Holmes
Affiliation:
Rutgers University
William E. Strawderman
Affiliation:
Rutgers University
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Extract

Let X1,X2,X3,··· be independent, identically distributed random variables with a continuous distribution function and let the sequence of indices {Vr} be defined as follows: and for r ≧ 1, Vr is the trial on which the rth (upper) record observation occurs. {Vr} will be an infinite sequence of random variables since the underlying distribution function of the X's is continuous. It is well known that the expected value of Vr. is infinite for every r (see, for example, Feller [1], page 15). Also define and for r > 1 δr is the number of trials between the (r - l)th and the rth record. The distributions of the random variables Vr and δr do not depend on the distribution of the original random variables. It can be shown (see Neuts [2], page 206 or Tata 1[4], page 26) that The following theorem is due to Neuts [2].

Type
Short Communications
Information
Journal of Applied Probability , Volume 6 , Issue 3 , December 1969 , pp. 711 - 714
Copyright
Copyright © Applied Probability Trust 

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References

[1] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. Wiley, New York.Google Scholar
[2] Neuts, M. F. (1967) Waitingtimes between record observations. J. Appl. Prob. 4, 206–208.Google Scholar
[3] Renyi, A. (1962) Théorie des elements saillants d'une suite d'observations. Colloquium on Combinatorial Methods in Probability Theory. Aarhus University. 104–115.Google Scholar
[4] Tata, M. N. (1969) On outstanding values in a sequence of random variables. Z. Wahrscheinlichkeitsth. 12, 9–20.CrossRefGoogle Scholar