Hostname: page-component-7dd5485656-wlg5v Total loading time: 0 Render date: 2025-10-28T21:21:45.828Z Has data issue: false hasContentIssue false
  • English
  • Français

A use of complex probabilities in the theory of stochastic processes

Published online by Cambridge University Press:  24 October 2008

D. R. Cox
D. R. Cox
Affiliation:
Statistical Laboratory Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

The exponential distribution is very important in the theory of stochastic processes with discrete states in continuous time. A. K. Erlang suggested a method of extending to other distributions methods that apply in the first instance only to exponential distributions. His idea is generalized to cover all distributions with rational Laplace transforms; this involves the formal use of complex transition probabilities. Properties of the method are considered.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 51 , Issue 2 , April 1955 , pp. 313 - 319
Copyright
Copyright © Cambridge Philosophical Society 1955

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

(1) Bartlett, M. S. Proc. Camb. phil. Soc. 41 (1945), 71.Google Scholar
(2) Benson, F. and Cox, D. R. J.R. statist. Soc. B, 13 (1951), 65.Google Scholar
(3) Brockmeyer, E., HalstrØm, H. L. and Jensen, A. The life and works of A. K. Erlang (Copenhagen, 1948).Google Scholar
(4) Feller, W. An introduction to probability theory and its applications (New York, 1950).Google Scholar
(5) Jensen, A. A distribution model (Copenhagen, 1954).Google Scholar
(6) Kendall, D. G. Biometrika, 35 (1948), 316.Google Scholar
(7) Kendall, D. G. Ann. Inst. Poincaré, 13 (1952), 43.Google Scholar
(8) Kendall, D. G. J.R. statist. Soc. B, 14 (1952), 41.Google Scholar
(9) Lukacs, E. and SzÁsz, O. J. Res. nat. Bur. Stand. 48 (1952), 139.Google Scholar
(10) Lukacs, E. and SzÁsz, O. J. Res. nat. Bur. Stand. 52 (1954), 153.Google Scholar
(11) Pollaczek, F. Math. Z. 32 (1930), 64 and 729.Google Scholar
(12) Smith, W. L. Proc. Camb. phil. Soc. 49 (1953), 449.Google Scholar
(13) Titchmarsh, E. C. The theory of functions, 2nd ed. (Oxford, 1939).Google Scholar
(14) Walsh, J. L. Interpolation and approximation by rational functions in the complex domain (Colloq. Publ. Amer. math. Soc., New York, 1935).Google Scholar
(15) Whittaker, E. and Robinson, G. The calculus of observations, 4th ed. (London, 1944).Google Scholar