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Functional central limit theorem for ARCH(∞) models with weakly dependent innovations
Journal of the Korean Data & Information Science Society 2021;32:417-26
Published online March 31, 2021;
© 2021 Korean Data and Information Science Society.

Oesook Lee1

1Department of Statistics, Ewha Womans University
Correspondence to: This research was supported by Basic Science Research Program through the NRF funded by the Ministry of Education, Science and Technology ( 2018R1D1A1B07044546).
1Professor, Department of Statistics, Ewha Womans University, Ewhayeodaegil 52, Seoul 03760, Korea. E-mail:
Received January 22, 2021; Revised February 25, 2021; Accepted February 26, 2021.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Since the seminal work of Engle (1982) and Bollerslev (1986), many ARCH-type models have been suggested and examined to explain a variety of stylized facts of financial and economic time series. Various popular ARCH-type models can be expressed as ARCH(∞) models. In this paper, we study the stationarity and functional central limit theorem (FCLT) for ARCH(∞) models, because statistical inferences for ARCH(∞) sequences require the study of asymptotics of various statistics concerned. Most previous results are obtained under independent and identically distributed (i.i.d.) innovation processes. But the i.i.d. assumption on innovations substantially restricts the flexibility of the models. In addition, many authors have shown that the i.i.d. assumption can be weakened to mild conditions. We consider the ARCH(1) model where the innovation processes are strictly stationary and λ-weakly dependent instead of inde- pendent and identically distributed.We provide sufficient conditions for the existence of a unique stationary and λ-weakly dependent Volterra series type solution to the given ARCH(∞) process. The FCLT for the stationary and λ-weakly dependent solution is also obtained by adding weak dependence coefficients condition on innovations and condition on ARCH(∞) parameters. The FCLT for GARCH(p, q) model with λ-weakly dependent innovations is considered as an example.
Keywords : Functional central limit theorem, λ-weakly dependent process, stationarity.