Limit Properties For A Class Of Stationary Processes

The first part of this paper concerns some limit properties of a class of stationaryprocesses (i.e. X_n = g(…,ε_n-1,ε_n)), including strong invariance principlesfor partial sums, the maximum of periodograms and asymptotics of the spectral density estimation. These subjects are very important in probability and statistics, and have been explored in many classical textbooks. Because of a surge of interest in nonlinear time series, people proposed many new questions on the subjects referred above. For example, can we consider similar problems for nonlinear time series ? It is well known that martingale approximation is an effective method to deal with stationary processes. However, martingale approximationseems not very suitable for the problems above. For this reason, we use m dependence approximation and obtain the optimal rates for strong invarianceprinciples, the asymptotic distributions of the maxima of periodograms of nonlinear time series and the maximum deviation of the spectral density estimation.Meanwhile, we solve some open questions proposed by some previous papers. The second part of this paper concerns the test on independence between components of a high dimensional vector. The high dimension problem is very popular recently. Since the independence is usually assumed in many statistical problem, test on independence is an important problem. Based on some previous work, we propose a new statistic to test whether components are independent. We also prove that the limit distribution of this statistic is the extreme distributionof type I with a rate of convergence O((logn)^5/2/(?)). This is much fasterthan O(1/ log n), a typical convergence rate for this type of extreme distribution. The third part of this paper considers LIL for independent B valued random variables.Based on the previous literature, we prove some LIL for independent B valued random variables when their variances are infinite. Some previous results are extended.