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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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