The asymptotic behavior of quadratic forms in -mixing random variables

Let { Z i , i ≥ 1 } be a linear process defined by Z t = ∑ j = 0 ∞ d j ξ t − j with { d j , j ≥ 0 } being a regular varying sequence of real numbers and { ξ t , − ∞ < t < ∞ } being a sequence of ϕ -mixing random variables. The present paper studies the asymptotic behavior of the quadratic form ∑ k , l = 1 n μ ̂ ( k − l ) Z k Z l under some mild assumptions on d j and ξ t . Meanwhile, the similar results of α -mixing random variables are presented.