Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T19:57:05.320Z Has data issue: false hasContentIssue false
  • English
  • Français

A mixed autoregressive-moving average exponential sequence and point process (EARMA 1,1)

Published online by Cambridge University Press:  01 July 2016

P. A. Jacobs  and
P. A. W. Lewis
P. A. Jacobs
Affiliation:
Stanford University
P. A. W. Lewis
Affiliation:
Naval Postgraduate School, Monterey, California
Get access Rights & Permissions [Opens in a new window]

Abstract

A stationary sequence of random variables with exponential marginal distributions and the correlation structure of an ARMA (1, 1) process is defined. The process is formed as a random linear combination of i.i.d. exponential random variables and is very simple to generate on a computer. Moments and joint distributions for the sequence are obtained, as well as limiting properties of sums of the random variables and of the point process whose intervals have the EARMA (1, 1) structure.

Keywords

POINT PROCESSEXPONENTIAL DISTRIBUTIONPROBABILISTIC LINEAR MODELMOVING AVERAGEAUTOREGRESSIVEVARIANCE-TIME CURVEPOISSON PROCESSDEPENDENT SEQUENCE
Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 1 , March 1977 , pp. 87 - 104
Copyright
Copyright © Applied Probability Trust 1977 

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.)

References

Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Box, G. E. P. and Jenkins, G. M. (1970) Time Series Analysis Forecasting and Control. Holden-Day, San Francisco.Google Scholar
Çinlar, E. (1969) On semi-Markov processes on arbitrary spaces. Proc. Camb. Phil. Soc. B 66, 381–392.Google Scholar
Çinlar, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, New Jersey.Google Scholar
Cox, D. R. and Lewis, P. A. W. (1966) The Statistical Analysis of Series of Events. Methuen, London; Wiley, New York.Google Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Gaver, D. P. and Lewis, P. A. W. (1977) First order autoregressive Gamma sequences and point processes. To appear.Google Scholar
Jacod, J. (1971) Théorème de renouvellement et classification pour les chaînes semi-markoviennes. Ann. Inst. H. Poincaré, 7 (2), 83–129.Google Scholar
Lawrance, A. J. and Lewis, P. A. W. (1977) A moving average exponential point process (EMA1). J. Appl. Prob. 14, 98–113.Google Scholar
Lewis, P. A. W. (1977) Generation of Gamma and mixed exponential time series with controlled dependence. To appear.Google Scholar
Rosenblatt, M. (1971) Markov Processes. Structure and Asymptotic Behaviour. Springer-Verlag, Berlin.Google Scholar