IJPAM: Volume 35, No. 3 (2007)


E. Gonçalves$^1$, N. Mendes-Lopes$^2$
$^{1,2}$Department of Mathematics
Faculty of Science and Technology
University of Coimbra
Ap. 3008, Coimbra, 3001-454, PORTUGAL
$^{1}$e-mail: esmerald@mat.uc.pt
$^{2}$e-mail: nazare@mat.uc.pt

Abstract.We study the marginal distribution function of a GTARCH process $\varepsilon =(\varepsilon _{t},t\in Z$) for which we obtain bounds based on the distribution function of the independent white noise, $Z$, associated to process $\varepsilon $. We point out that even if the marginal law of $\varepsilon $ presents effectivelly different characteristics from the marginal law of $Z$ it is, in some regions, strongly controlled by the law of the white noise associated. Those regions are evaluated for noise distributions particularly useful in applications, and with different properties, namely in what concerns the behaviour of the corresponding tails. The laws of finite dimension of the absolute value of the process $\varepsilon $ are also evaluated in terms of the $Z$ law; the bounds obtained for these laws are related to the run length of control charts.

Received: January 20, 2007

AMS Subject Classification: 62G20, 62M10

Key Words and Phrases: nonlinear time series, GTARCH models, marginal distribution

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 35
Issue: 3