A central limit theorem for a random quadratic form of strictly stationary processes (Q1579539)

Let the strictly stationary process \(\{(V_t,e_t)\}\) be \(\alpha\)-mixing, that is, the process satisfies the condition \[ \alpha(t)= \sup \Bigl\{\bigl|P(A\cap B)-P(A)P(B)\bigr |:A\in \Omega^s_1,B \in \Omega^\infty_{s+t} \Bigr\}\to 0\quad (t\to\infty) \] for all \(s\geq 1\), where \(\Omega^b_a =\sigma((V_j,e_j); a\leq j\leq b)\). Let \(z_i(\cdot)\) \((i\geq 1)\) be uniformly bounded continuous functions over \(R^q\) \((q\geq 1)\), \(Z(\cdot)= (z_1 (\cdot), \dots,z_k (\cdot))^\tau\) \((k\) being the truncation parameter) and \(Z= (Z(V_1), \dots, Z(V_T))\). Furthermore, let \(p_{s,t}\) be the \((s,t)\) element of the projection matrix \(P=Z(Z^\tau Z)^+ Z^\tau\), where \((\cdot)^+\) denotes the Moore-Penrose inverse. Put \(S_T=\sum^T_{s,t=1} p_{s,t}e_se_t\). The authors prove, among others, that \[ (2k)^{-1/2} (Ee^2_t)^{-1} (S_T-k(Ee^2_t))\to N(0,1) \quad (T\to \infty). \] An application to nonparametric series regression is also considered.