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Regular variation of GARCH processes. (English) Zbl 1060.60033
Let a $d$-dimensional time series $\left({𝐗}_{t}\right)$ be defined by a stochastic recurrence equation (SRE) ${𝐗}_{t}={𝐀}_{t}{𝐗}_{t-1}+{𝐁}_{t}$ where ${𝐀}_{t}$ are $d×d$ random matrices and ${𝐁}_{t}$ are $d$-dimensional vectors such that $\left(\left({𝐀}_{t},{𝐁}_{t}\right)\right)$ is an iid sequence. A generalized autoregressive conditionally heteroscedastic process $\left({X}_{t}\right)$ of order $\left(p,q\right)$ (GARCH$\left(p,q\right)$) is given by ${X}_{t}={\sigma }_{t}{Z}_{t}$, ${\sigma }_{t}^{2}={\alpha }_{0}+{\sum }_{i=1}^{p}{\alpha }_{i}{X}_{t-i}^{2}+{\sum }_{j=1}^{q}{\beta }_{j}{\sigma }_{t-j}^{2}$ where $\left({Z}_{t}\right)$ is an iid sequence and ${\alpha }_{i}$’s, ${\beta }_{j}$’s are nonnegative constants. The squared processes $\left({X}_{t}^{2}\right)$ and $\left({\sigma }_{t}^{2}\right)$ satisfy a SRE ${𝐗}_{t}={𝐀}_{t}{𝐗}_{t-1}+{𝐁}_{t}$ with ${𝐗}_{t}={\left({\sigma }_{t+1}^{2},\cdots ,{\sigma }_{t-q+2}^{2},{X}_{t}^{2},\cdots ,{X}_{t-p+2}^{2}\right)}^{\text{'}}$. The authors start with some introduction to basic theory for SRE. This part is a valuable review of results concerning SRE. Using the connection between GARCH and SRE the authors show that the finite-dimensional distributions of a GARCH process are regularly varying. This implies the moment properties of the process and the dependence structure between neighboring observations. Regular variation enables to establish the large sample behavior of statistics from GARCH process such as the sample mean, sample autocovariance and sample autocorrelation function. For example, if the fourth moment of the process does not exist, then the rate of convergence of the sample autocorrelations is extremely slow.

MSC:
 60G10 Stationary processes 62M10 Time series, auto-correlation, regression, etc. (statistics) 60G55 Point processes 91B28 Finance etc. (MSC2000) 62G20 Nonparametric asymptotic efficiency 62P05 Applications of statistics to actuarial sciences and financial mathematics