Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-27T07:14:18.187Z Has data issue: false hasContentIssue false
  • English
  • Français

On two classes of reflected autoregressive processes

Part of: Special processes Inference from stochastic processes

Published online by Cambridge University Press:  16 July 2020

Onno Boxma ,
Andreas Löpker  and
Michel Mandjes
Onno Boxma*
Affiliation:
Eindhoven University of Technology
Andreas Löpker*
Affiliation:
HTW Dresden
Michel Mandjes*
Affiliation:
University of Amsterdam
*
*Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven. Email address: o.j.boxma@tue.nl
**Postal address: University of Applied Sciences, Hochschule für Technik und Wirtschaft, Friedrich-List-Platz 1, D-01069 Dresden, Germany. Email address: lopker@htw-dresden.de
***Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands. Email address: m.r.h.mandjes@uva.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce two general classes of reflected autoregressive processes, INGAR+ and GAR+. Here, INGAR+ can be seen as the counterpart of INAR(1) with general thinning and reflection being imposed to keep the process non-negative; GAR+ relates to AR(1) in an analogous manner. The two processes INGAR+ and GAR+ are shown to be connected via a duality relation. We proceed by presenting a detailed analysis of the time-dependent and stationary behavior of the INGAR+ process, and then exploit the duality relation to obtain the time-dependent and stationary behavior of the GAR+ process.

Keywords

INAR(1)AR(1)autoregressive processesreflectiongenerating functionstime-dependent behaviorstationarity

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 62M10: Time series, auto-correlation, regression, etc. 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 2 , June 2020 , pp. 657 - 678
Copyright
© Applied Probability Trust 2020

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

Al-Osh, M. andAlzaid, A. (1987). First-order integer-valued autoregressive (INAR(1)) process. J. Time Ser. Anal. 8, 261–275.CrossRefGoogle Scholar
Asmussen, S. (2008). Applied Probability and Queues. Springer, New York.Google Scholar
Athreya, K. B. andNey, P. E. (1972). Branching Processes. Springer, New York.CrossRefGoogle Scholar
Barreto-Souza, W. (2015). Zero-modified geometric INAR(1) process for modelling count time series with deflation or inflation of zeros. J. Time Ser. Anal. 36, 839–852.CrossRefGoogle Scholar
Bendikov, A. andSaloff-Coste, L. (2012). Random walks on groups and discrete subordination. Math. Nachr. 285, 580–605.CrossRefGoogle Scholar
Berkelmans, W., Cichocka, A. andMandjes, M. (2019). The correlation function of a queue with Lévy and Markov additive input. Stoch. Process. Appl. 130, 1713–1734.CrossRefGoogle Scholar
Boxma, O., Mandjes, M. andReed, J. (2016). On a class of reflected AR(1) processes. J. Appl. Prob. 53, 818–832.CrossRefGoogle Scholar
Brockwell, P. J., Davis, R. A. andCalder, M. V. (2002). Introduction to Time Series and Forecasting. Springer, New York.CrossRefGoogle Scholar
Cohen, J. W. (1982). The Single Server Queue, 2nd edn. North-Holland, Amsterdam.Google Scholar
Davis, R. A., Holan, S. H., Lund, R. andRavishanker, N. (2016). Handbook of Discrete-Valued Time Series. CRC Press, Boca Raton.CrossRefGoogle Scholar
Doukhan, P., Latour, A. andOraichi, D. (2006). A simple integer-valued bilinear time series model. Adv. Appl. Prob. 38, 559–578.CrossRefGoogle Scholar
Glynn, P. W. andMandjes, M. (2011). Simulation-based computation of the workload correlation function in a Lévy-driven queue. J. Appl. Prob. 48, 114–130.CrossRefGoogle Scholar
Grandell, J. (1997). Mixed Poisson Processes. CRC Press, Boca Raton.CrossRefGoogle Scholar
Heathcote, C. (1965). A branching process allowing immigration. J. R. Statist. Soc. B [Statist. Methodology] 27, 138–143.Google Scholar
Heathcote, C. (1966). Corrections and comments on the paper ‘A branching process allowing immigration’. J. R. Statist. Soc. B [Statist. Methodology] 28, 213–217.Google Scholar
Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer Science & Business Media, New York.Google Scholar
Latour, A. (1998). Existence and stochastic structure of a non-negative integer-valued autoregressive process. J. Time Ser. Anal. 19, 439–455.CrossRefGoogle Scholar
McKenzie, E. (1985). Some simple models for discrete variate time series. J. Am. Water Resources Assoc. 21, 645–650.CrossRefGoogle Scholar
McKenzie, E. (1986). Autoregressive moving-average processes with negative-binomial and geometric marginal distributions. Adv. Appl. Prob. 18, 679–705.CrossRefGoogle Scholar
Mimica, A. (2017). On subordinate random walks. In Forum Mathematicum, Vol. 29. De Gruyter, Berlin, pp. 653–664.CrossRefGoogle Scholar
Ristić, M. M., Bakouch, H. S. andNastić, A. S. (2009). A new geometric first-order integer-valued autoregressive (INGAR(1)) process. J. Statist. Planning Infer. 139, 2218–2226.CrossRefGoogle Scholar
Steutel, F. andvan Harn, K. (1979). Discrete analogues of self-decomposability and stability. Ann. Prob. 7, 893–899.CrossRefGoogle Scholar
Weiss, C. H. (2017). An Introduction to Discrete-Valued Time Series. John Wiley, Chichester.Google Scholar