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On the Transition Probability of a Renewal Process

Published online by Cambridge University Press:  22 January 2016

Takeyuki Hida
Takeyuki Hida*
Affiliation:
Mathematical Institute, Aichi-Gakugei University
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J. L. Doob, D. Blackwell, W. Feller and other authors have obtained several results concerning the renewal theorem. Especially Doob [1] has considered the renewal process and has showed that it becomes a stationary Markov process if we add a certain initial random variable to it. In the present note, we shall study this stationary Markov process and try to determine its transition probability by virtue of a pair of partial differential equations.

The author would like to express his hearty thanks to prof. A. Amakusa who has encouraged him with kind discussions throughout the course of preparing the present note.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 11 , February 1957 , pp. 41 - 51
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1957

References

[1] Doob, J. L., Renewal theory from the point of view of the theory of probability. Trans. Amer. Math. Soc. 63 (1948), pp. 422–438.CrossRefGoogle Scholar
[2] Blackwell, D., A renewal theorem. Duke Math. Jour. 15 (1948), pp.145–150.CrossRefGoogle Scholar
[3] Kolmogorov, A., Über die analytische Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104 (1931), pp.415–458.CrossRefGoogle Scholar
[4] Hida, T., On some properties of Poisson process II. (Japanese). Bull. Aichi-Gakugei Univ. 4 (1954), pp.5–9.Google Scholar