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Central limit theorem for degenerate \(U\)-statistics of absolutely regular processes with applications to model specification testing. (English) Zbl 0974.62044
Let \(\{Z_t\}\) be a strictly stationary stochastic process and \({\mathcal M}_s^t\) denote the \(\sigma\)-algebra generated by \((Z_s,\ldots , Z_t)\) for \(s\leq t\). The process \(\{Z_t\}\) is called absolutely regular if \[ E\;\sup_{A\in {\mathcal M}_{s+\tau}}|P(A |{\mathcal M}_{-\infty}^s) - P(A)|\;\to \;0\quad \text{as}\quad \tau \to \infty. \] Denote \[ U_n = \sum_{1\leq s<t\leq n} H_n(Z_t,Z_s), \] where \(H_n\) depends on \(n\) and satisfies \(\int H_n(x,y)dF(x)= 0\) for all \(y\), and \(F(\cdot)\) is the marginal distribution function of \(\{Z_t\}\). Let \(\{\tilde{Z}_t\}\) be an i.i.d. sequence, having the same marginal distribution as \(Z_t\) and define \(\sigma_n^2 = E H_n^2(\tilde{Z}_1,\tilde{Z}_2)\). The main result states, that under some moment conditions on \(H_n\), for strictly stationary and absolutely regular processes \(Z_t\): \[ \sqrt 2 U_n(n\sigma_n)^{-1} \to \;N(0,1)\quad \text{in distribution as}\quad n\to \infty. \] This theorem is used to establish the validity of an asymptotic test for the parametric functional form of a regression model involving time series.

MSC:
62G20 Asymptotic properties of nonparametric inference
60F05 Central limit and other weak theorems
62G10 Nonparametric hypothesis testing
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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