Inference for the tail parameters of a linear process with heavy tail innovations

Consider a linear process X-t = Sigma(i=0)(infinity) c(i)Z(t-i), where the innovations Z's are i.i.d. satisfying a standard tail regularity and balance condition, viz., P(Z > z) similar to rz(-alpha) L-1 (z), P(Z < -z) similar to sz(-alpha) L-1 (z), as z --> infinity, where r + s = 1, r, s greater than or equal to 0, alpha > 0 and L-1 is a slowly varying function. It turns out that in this setup, P(X > x) similar to px(-alpha) L(x), P(X < -x) similar to qx(-alpha) L(x), as x --> infinity where ct is the same as above, p is a convex combination of r and s, p + q = 1, p,q greater than or equal to 0 and L = //c//(alpha)(alpha) L-1 where //C//alpha = (Sigma/c/(alpha))(1/alpha). The quantities alpha and beta = 2p - 1 can be regarded as tail parameters of the marginal distribution of Xt. We estimate alpha and beta based on a finite realization X-1,...,X-n of the time series. Consistency and asymptotic normality of the estimators are established. As a further application, we estimate a tail probability under the marginal distribution of the X-t. A small simulation study is included to indicate the finite sample behavior of the estimators.