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Functional limit theorem for the empirical process of a class of Bernoulli shifts with long memory. (English) Zbl 1073.60036

Let \(X_t=Y_t+V_t,t\in Z=\{0,\pm1,\pm2,\dots\}\), be a strictly stationary process where \(Y_t\) is a linear long memory process and \(V_t\) is a nonlinear short memory process. More precisely, \(Y_t=\sum_{i=0}^\infty b_i\zeta_{t-i}\) is a moving average process in independent and identically distributed random variables \(\zeta_i,i\in Z\), with mean zero and variance one and with hyperbolically decaying coefficients \(b_i\sim c_0i^{d-1}\) for some \(d\in(0,1/2)\) and \(c_0\not=0\). The short memory process \(V_t\) is a Bernoulli shift \(V_t=V(\zeta_t,\zeta_{t-1},\dots)\) where \(V(z_0,z_1,\dots)\) is a Borel function defined on \(R^{Z_+}\) with \(Z_+=\{0,1,\dots\}\) which satisfies \[ E^{1/2}[V(\zeta_0,\dots,\zeta_{-n},0,0,\dots)- V(\zeta_0,\dots,\zeta_{-n+1},0,0,\dots)]^2\leq Cn^{-\rho} \] for some finite constant \(C\) and \(\rho>\max\{24-22d,13-11d+3(1-2d)/(4d)\}\). For every integer \(N\geq1\), set \(\overline{Y}_N=N^{-1}\sum_{t=1}^NY_t\) and \(\widehat{F}_N(x)=N^{-1}\sum_{t=1}^NI(X_t\leq x),x\in R\). Let \(F\) denote the marginal distribution function of \(X_0\) and \(f\) its desity. Under mild assumptions on the distribution of \(\zeta_0\) it is shown that \(\sup_{x\in R}N^{1/2-d}| \widehat{F}_N(x)-F(x)+f(x)\overline{Y}_N| =o_p(1)\). As a consequence, the processes \(N^{1/2-d}(\widehat{F}_N-F)\) converge in distribution in the Skorokhod space \(D[-\infty,\infty]\) to the degenerate process \(\widetilde{c}fZ\), where \(\widetilde{c}=(c_0^2B(d,2-2d)/ d(1+2d))^{1/2}\), with \(B\) being the beta function, and where \(Z\) is a standard normal random variable. To elucidate the role of the weakly dependent Bernoulli shift \(V_t\), a discussion of several concrete examples is also included.

MSC:

60F17 Functional limit theorems; invariance principles
60G18 Self-similar stochastic processes
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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