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

Let Xt=∑j=0∞cjεt−j be a moving average process with GARCH (1, 1) innovations {εt}. In this paper, the asymptotic behavior of the quadratic form Qn=∑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 Qn converges to a stable random variable with index α/2. However, when α≥4, Qn 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.