【作者】 孙丽雅；

【导师】 闫理坦；

【作者基本信息】 东华大学 ， 应用数学， 2011， 硕士

【摘要】 本文主要研究Rosenblatt过程及其更一般的Hermite过程的鞅差逼近.所谓指数为H∈（1/2,1）的k-阶Hermite过程是指由如下积分所确定的过程：Zkh（t）其中W为一个标准布朗运动,而核KH由如下定义：Hermite过程具有以下性质：（1）对任意的c>0,（Zkh（ct））和（cHZkh（t））有同分布,知Z是H阶自相似过程；（2）对h>0,联合分布（Zkh（t+h）—Zkh（t）,t∈[O,T]）是独立的,知它有稳定增量；（3）由结合Kolmogorov连续性质,知ZHk有δ<H阶Hǒlder连续路径;（4）由知过程Z有长相依性；（当H>1/2时,fBM和Rosenblatt过程有此性质）（5）协方差函数为：当κ=1时,它是众所周知的分数布朗运动；κ=2时,Hermite过程被称为Rosenblatt过程.值得注意的是这类过程既不是Gaussian过程也不是Markov过程或半鞅除非H=1/2,即它是Brownian运动,但是它却具有与是Gaussian过程的分数布朗运动完全相同的相依结构!首先,我们考虑Rosenblatt过程的鞅差逼近,我们证明过程列在n→∞时,弱收敛于Rosenblatt过程ZH的,这里{ξ（n）,n≥1}是满足适当条件的鞅差序列.其次,对于满足适当条件的鞅差序列我们构造了随机过程列的递推累加和其中,Q（n）（t,i₁/n,...,i_k/n）为：我们证明过程列{Zn,n=1,2,...}在n→∞时,弱收敛于Hermite过程ZH.更多还原

【Abstract】 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.更多还原