Hostname: page-component-cb9f654ff-r5d9c Total loading time: 0 Render date: 2025-08-20T16:54:45.275Z Has data issue: false hasContentIssue false
  • English
  • Français

A central limit theorem for random coefficient autoregressive models and ARCH/GARCH models

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Andreas Rudolph
Andreas Rudolph*
Affiliation:
IBM
*
∗ Postal address: IBM WT WZH, Vangerowstr. 18, 69115 Heidelberg, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study the so-called random coeffiecient autoregressive models (RCA models) and (generalized) autoregressive models with conditional heteroscedasticity (ARCH/GARCH models). Both models can be represented as random systems with complete connections. Within this framework we are led (under certain conditions) to CL-regular Markov processes and we will give conditions under which (i) asymptotic stationarity, (ii) a law of large numbers and (iii) a central limit theorem can be shown for the corresponding models.

Keywords

RCA modelsARCH/GARCH modelsrandom systems with complete connectionsmarkov processesCL regularitylaw of large numbersasymptotic stationarity

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 1 , March 1998 , pp. 113 - 121
Copyright
Copyright © Applied Probability Trust 1998 

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

Arras, P. (1995). ARCH- und GARCH-Modelle: Asymptotik, Theorie und Anwendung. Diplomarbeit. Universität Braunschweig.Google Scholar
Doob, J. L. (1953). Stochastic Processes. Wiley, New York.Google Scholar
Herkenrath, U. (1976). Der assoziierte Markov-Prozess verallgemeinerter zufälliger Systeme mit vollständigen Bind-ungen. Dissertation. Universität Bonn.Google Scholar
Herkenrath, U. (1994). Generalized adaptive exponential smoothing procedures. J. Appl. Prob. 31, 673–690.Google Scholar
Iosifescu, M. and Grigorescu, S. (1990). Dependence with Complete Connections and its Applications. Cambridge University Press, Cambridge.Google Scholar
Norman, M. F. (1972). Markov Processes and Learning Models. Academic Press, New York.Google Scholar
Nicholls, D. F. and Quinn, B. G. (1982). Random Coefficient Autoregressive Models: An Introduction. (Lecture Notes in Statistics 11.) Springer, New York.Google Scholar
Ortega, J. M. and Rheinboldt, W. C. (1972). Iterative Solutions of Nonlinear Equations in Several Variables. Academic Press, New York.Google Scholar
Tong, H. (1990). Nonlinear Time Series. Clarendon Press, Oxford.Google Scholar