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We obtain an optimal bound for a Gaussian approximation of a large class of vector-valued random processes. Our results provide a substantial generalization of earlier results that assume independence and/or stationarity. Based on the decay rate of the functional dependence measure, we quantify the error bound of the Gaussian approximation using the sample size n and the moment condition. Under the assumption of pth finite moment, with p > 2, this can range from a worst case rate of n1/2 to the best case rate of n1/p. [ABSTRACT FROM AUTHOR]

The paper considers suprema of empirical processes for linear time series indexed by functional classes. We derive an upper bound for the tail probability of the suprema under conditions on the size of the function class, the sample size, temporal dependence and the moment conditions of the underlying time series. Due to the dependence and heavy-tailness, our tail probability bound is substantially different from those classical exponential bounds obtained under the independence assumption in that it involves an extra polynomial decaying term. We allow both short- and long-range dependent processes. For empirical processes indexed by half intervals, our tail probability inequality is sharp up to a multiplicative constant. [ABSTRACT FROM AUTHOR]

We consider the problem of predicting aggregates or sums of future values of a process based on its past values. In contrast with the conventional prediction problem in which one predicts a future value given past values of the process, in our setting the number of aggregates can go to infinity with respect to the number of available observations. Consistency and Bahadur representations of the prediction estimators are established. A simulation study is carried out to assess the performance of different prediction estimators. [ABSTRACT FROM AUTHOR]

We study asymptotic properties of the local Whittle estimator of the long memory parameter for a wide class of fractionally integrated nonlinear time series models. In particular, we solve the conjecture posed by Phillips and Shimotsu (2004, Annals of Statistics32, 656–692) for Type I processes under our framework, which requires a global smoothness condition on the spectral density of the short memory component. The formulation allows the widely used fractional autoregressive integrated moving average (FARIMA) models with generalized autoregressive conditionally heteroskedastic (GARCH) innovations of various forms, and our asymptotic results provide a theoretical justification of the findings in simulations that the local Whittle estimator is robust to conditional heteroskedasticity. Additionally, our conditions are easily verifiable and are satisfied for many nonlinear time series models.We thank Liudas Giraitis for providing the manuscript by Dalla, Giraitis, and Hidalgo (2006). We are grateful to the two referees and the editor for their detailed comments, which led to substantial improvements. We also thank Michael Stein for helpful comments on an earlier version. The work is supported in part by NSF grant DMS-0478704. [ABSTRACT FROM AUTHOR]

We consider quadratic forms of martingale differences and establish a central limit theorem under mild and easily verifiable conditions. By approximating Fourier transforms of stationary processes by martingales, our central limit theorem is applied to the smoothed periodogram estimate of spectral density functions. Our results go beyond earlier ones by allowing a variety of nonlinear time series and by avoiding strong mixing andor summability conditions on joint cumulants.We thank the two reviewers for their detailed comments, which led to substantial improvements. The work is supported in part by NSF grant DMS-0478704. [ABSTRACT FROM AUTHOR]