Almost periodically unitary stochastic processes
Almost periodically unitary stochastic processes
复制标题
几乎周期性酉随机过程
DOI：
10.1016/0304-4149(92)90078-5
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发表时间：
1992
期刊：
影响因子：
--
通讯作者：
H. Hurd
中科院分区：
文献类型：
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作者：
H. Hurd
定义在概率空间（Ω，F，P）上的连续二阶复过程{X（t），t <$R}称为几乎周期酉过程（APU），如果存在一个强连续的单参数酉算子群{U（τ），τ <$R}，使得集合S（ε，X，U）={τ：对于每一个ε> 0，supt <$R <$X（t+ τ）− U（τ）X（t）<$L 2< ε是相对稠密的（有界间隙）。这些过程包括S（ε，X，U）= R的连续平稳过程，S（ε，X，U）<$R {jT，j <$Z}的连续周期相关过程，以及U（τ）= R的L2值一致概周期函数。证明了X（t）是APU的充要条件是X（t）= U（t）[P（t）]，其中P（t）是L2值一致概周期函数.给出了例子，并提供了一致概周期函数理论的基本性质。这些过程被证明是一致的几乎周期性相关，因此几乎周期性的Gladyshev意义上的相关。我们表示的基础上的酉群的谱理论和调和分析的一致概周期函数的过程。最后，我们给出了相关函数的表示理论的分析，并证明了每一个APU过程是一个序列的强协调过程的一致极限。
A continuous second order complex process {X (t), tϵ R} defined on a probability space (Ω, F, P) is called almost periodically unitary (APU) if there exists a strongly continuous one parameter group of unitary operators {U (τ), τϵ R} for which the set S (ε, X, U)={τ: sup tϵ R‖ X (t+ τ)− U (τ) X (t)‖ L 2< ε relatively dense (has bounded gaps) for every ε> 0. These processes include continuous stationary processes for which S (ε, X, U)= R, continuous periodically correlated processes for which S (ε, X, U)⊃ R {jT, jϵ Z} for some real T, and the L 2-valued uniformly almost periodic functions for which U (τ)= In this paper, we show that X (t) is APU if and only if X (t)= U (t)[P (t)] where P (t) is an L 2-valued uniformly almost periodic function. Examples are given and basic properties motivated by the theory of uniformly almost periodic functions are provided. These processes are shown to be uniformly almost periodically correlated and hence almost periodically correlated in the sense of Gladyshev. We give representations for the processes based on the spectral theory for unitary groups and on the harmonic analysis of uniformly almost periodic functions. Finally, we give an analysis of the correlation functions in terms of the representation theory, and show that every APU process is a uniform limit of a sequence of strongly harmonizable processes.