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

Let X t = ∑ j = 0 ∞ c j ε t − j be a moving average process with GARCH (1, 1) innovations { ε t } . In this paper, the asymptotic behavior of the quadratic form Q n = ∑ j = 1 n ∑ s = 1 n b ( t − s ) X t X s is derived when the innovation { ε t } 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.