Maxima of long memory stationary symmetric $alpha$-stable processes, and self-similar processes with stationary max-increments
Maxima of long memory stationary symmetric $alpha$-stable processes, and self-similar processes with stationary max-increments
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长记忆平稳对称 $alpha$ 稳定过程的最大值，以及具有平稳最大增量的自相似过程
DOI：
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发表时间：
2013
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影响因子：
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通讯作者：
G. Samorodnitsky
中科院分区：
文献类型：
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作者：
Takashi Owada;G. Samorodnitsky
基于长记忆平稳$ α $稳定过程，导出了部分极大值过程的泛函极限定理。稳定过程的记忆长度由过程的积分表示中的某个遍历理论参数来参数化。极限过程不再是经典的极值Fr'{e}chet过程。它是一个具有$alpha$-Fr'{e}chet边际的自相似过程，并且它具有平稳的最大增量，本文引入了这一性质。利用Skorohod $M_1$-拓扑，在空间$D[0, inty] $上建立了泛函极限定理；在某些特殊情况下，拓扑可以增强为Skorohod $J_1$-拓扑。
We derive a functional limit theorem for the partial maxima process based on a long memory stationary $alpha$-stable process. The length of memory in the stable process is parameterized by a certain ergodic-theoretical parameter in an integral representation of the process. The limiting process is no longer a classical extremal Fr'{e}chet process. It is a self-similar process with $alpha$-Fr'{e}chet marginals, and it has stationary max-increments, a property which we introduce in this paper. The functional limit theorem is established in the space $D[0,infty)$ equipped with the Skorohod $M_1$-topology; in certain special cases the topology can be strengthened to the Skorohod $J_1$-topology.