Limit theory of quadratic forms of long-memory linear processes with heavy-tailed GARCH innovations

Let X_t=∑_j=0^∞c_jε_t-j be a moving average process with GARCH (1, 1) innovations {_εt} In this paper, the asymptotic behavior of the quadratic form Q_n=∑j=1n∑s=1nb(t-s)XtXs is derived when the innovation {_εt} is a long-memory and heavy-tailed process with tail index α, where {b (i) } is a sequence of constants. In particular, it is shown that when 1 < α < 4 and under certain regularity conditions, the limit distribution of _{Q n} converges to a stable random variable with index α / 2. However, when α ≥ 4, _{Q n} has an asymptotic normal distribution. These results not only shed light on the singular behavior of the quadratic forms when both long-memory and heavy-tailed properties are present, but also have applications in the inference for general linear processes driven by heavy-tailed GARCH innovations. © 2013 Elsevier Inc.