Joint asymptotic distribution of exceedances point process and partial sum of stationary Gaussian sequence

Let {Xi}i=1^∞ be a standardized stationary Gaussian sequence with covariance function τ（n） =EX1Xn＋1, Sn =∑i=1^nXi,and X^-n=Sn/n.And let Nn be the point process formed by the exceedances of random level （x/√2 log n＋√2 log n-log（4π log n）/2√log n） √1-τ（n） ＋ X^-n by X1,X2,…, Xn. Under some mild conditions, Nn and Sn are asymptotically independent, and Nn converges weakly to a Poisson process on （0,1].

The Limit Theorems for Maxima of Stationary Gaussian Processes with Random Index

Let {X（t）, t ≥ 0} be a standard（zero-mean, unit-variance） stationary Gaussian process with correlation function r（·） and continuous sample paths. In this paper, we consider the maxima M（T） = max{X（t）, t∈ [0, T ]} with random index TT, where TT /T converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of M（TT） exists under some additional conditions related to the correlation function r（·）.

The asymptotic relation between the first crossing point and the last exit time of Gaussian order statistics sequences

In this paper,we study the asymptotic relation between the first crossing point and the last exit time for Gaussian order statistics which are generated by stationary weakly and strongly dependent Gaussian sequences.It is shown that the first crossing point and the last exit time are asymptotically independent and dependent for weakly and strongly dependent cases,respectively.The asymptotic relations between the first crossing point and the last exit time for stationary weakly and strongly dependent Gaussian sequences are also obtained.

Limit theorems for supremum of Gaussian processes over a random interval

Let {X(t), t ≥ 0} be a centered stationary Gaussian process with correlation r(t)such that 1-r(t) is asymptotic to a regularly varying function. With T being a nonnegative random variable and independent of X(t), the exact asymptotics of P(sup_(t∈[0,T])X(t) > x) is considered, as x → ∞.

A NOTE ON SAMPLE PATH PROPERTIES OF l^p-VALUED GAUSSIAN PROCESSES

The a.s. sample path properties for l p valued Gaussian processes with stationary increments under some more general conditions are established.

BANDWIDTH SELECTION IN NONPARAMETRIC SPECTRAL DENSITY ESTIMATION OF THE STATIONARY GAUSSIAN PROCESS

We propose a method for estimating mean squared error and bandwidth in the windowedspectral density estimation of a stationary Gaussian process, and also provide a method forestimating the second order derivative of the spectral density function. The asymptotic propertiesand the convergence rates of the estimators are given.

Limsup Results and LIL for Partial Sum Processes of a Gaussian Random Field

Let {ξ j ; j ∈ ℤ+ d be a centered stationary Gaussian random field, where ℤ+ d is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝd, having nonnegative integer coordinates. For each j = (j 1 , ..., jd) in ℤ+ d , we denote |j| = j 1 ... j d and for m, n ∈ ℤ+ d , define S(m, n] = Σ m【j≤n ζ j , σ2(|n−m|) = ES 2 (m, n], S n = S(0, n] and S 0 = 0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t 】 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 【 α 【 1. Under some additional conditions, we study limsup results for increments of partial sum processes and prove as well the law of the iterated logarithm for such partial sum processes.

SOME FUNCTIONAL LIMIT THEOREMS FOR THE INFINITE SERIES OF OU PROCESSES

This paper obtains functional modulus of continuity and Strassen's functional LIL of theinfinite series of independent Ornstein-Uhlenbeck processes, which also imply the Levy's exactmodulus of continuity and LIL of this process respectively.