The Approximation To Rosenblatt Process And Related Analysis

In this article, we study the martingale difference approximation of Rosenblatt process and Hermite process. The so-called Hermite process of order k with index H∈(1/2,1) is the process given by the integralZkh(t)where W is a standard Brownian motion, and the kernel KH has the expression:Hermite process admits the following properties:(1) the process Z is H-selfsimilar:for any c> 0, (ZkH(ct)) and (cHZkH(t)) have the same distribution;(2) the process has the stationarity of increments:for any h>0, the joint distribu-tion(ZkH(t+h)-ZkH(t),t∈[O,T]) is independent;(3) the mean square of the increment is given by as a consequence, it follows will little extra effort from Kolmogorov's continuity criterion that the Hermite process ZkH has Holder continuous paths of orderδ< H: 6(4) it exhibits long-range dependence in the sense that (This property is identical to that of fBm since the processes share the same covariance structure, and the property is well-known for f Bm with H>1/2.)(5) the covariance function is:For k=1, it is well known Brownian motion; for k=2, the Hermite process is known as the Rosenblatt process.It should be noted that this type of process is neither Gaussian process is not Markov process or semi-martingale unless H=1/2, that it is a Brownian motion, but it is a Gaussian process with fractional Brownian motion in exactly the same dependence structure!Firstly, we consider the martingale difference approximation of the Rosenblatt process, We prove that the sequence of the process converges weakly to the Rosenblatt process ZH, as n tends to infinity, where{ξ(n),n≥1} is a sequence of martingale-differences satisfying.appropriate conditions.Secondly, for the sequence of martingale-differences satisfying appropriate condi-tions, we construct the sums of product'of a sequence of the stochastic process where, Q(n) (t,i₁/n,...,i_k/n)is an approximation of Q(t, u, v): and we prove that the processes{Zn,n= 1,2,...} converges weakly to the Hermite process ZH, as n tends to infinity.