Convergence of integrated processes of arbitrary Hermite rank
Convergence of integrated processes of arbitrary Hermite rank
复制标题
DOI：
10.1007/bf00535674
复制
发表时间：
1979
期刊：
影响因子：
--
通讯作者：
M. Taqqu
中科院分区：
文献类型：
--
作者：
M. Taqqu
令 {X(s), −∞<s<∞} 为具有长程相关性的归一化平稳高斯过程。研究了积分过程 $$Z_x \left( t \right) = \frac{1}{{d\left( x \right)}}\mathop \smallint \limits_0^{xt} G\left( {X\left( s \right)} \right)ds,{\text{ }}x \to \infty$$ 的 C[0,1] 中的弱极限。 Hered(x) = xHL(x) 且 <H<1 且 L(x) 是无穷远处缓慢变化的函数。函数G满足EG(X(s))=0，EG2(X(s))<∞并且具有任意Hermite秩m≥1。 （Gi 的 Hermite 秩是 Gin Hermite 多项式展开式中第一个非零系数的索引。）结果表明，对于所有 m≧1，Zx(t) 收敛到本质上依赖于 m 的某个过程 Zm(t)。极限过程 ¯Zm(t) 通过涉及多个 Itô 积分的各种表示来表征。这些表示在有限维分布意义上都是等价的。当 m≧2 时，过程 ¯Zm(t) 是非高斯的。它们是自相似的，即，对于 alla>0， ˆZm(at) 和 HˆZm(t) 具有相同的有限维分布。
Let {X(s), −∞<s<∞} be a normalized stationary Gaussian process with a long-range correlation. The weak limit in C[0,1] of the integrated process $$Z_x \left( t \right) = \frac{1}{{d\left( x \right)}}\mathop \smallint \limits_0^{xt} G\left( {X\left( s \right)} \right)ds,{\text{ }}x \to \infty$$ , is investigated. Hered(x) = xHL(x) with<H<1 andL(x)is a slowly varying function at infinity. The functionGsatisfiesEG(X(s))=0,EG2(X(s))<∞ and has arbitrary Hermite rankm≧1. (TheHermite rankofGis the index of the first non-zero coefficient in the expansion ofGin Hermite polynomials.) It is shown thatZx(t)converges for allm≧1 to some process¯Zm(t)that depends essentially onm. The limiting process¯Zm(t)is characterized through various representations involving multiple Itô integrals. These representations are all equivalent in the finite-dimensional distributions sense. The processes¯Zm(t)are non-Gaussian whenm≧2. They are self-similar, that is,¯Zm(at)andaH¯Zm(t)have the same finite-dimensional distributions for alla>0.