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Gibbs 分布的动力系统假设检验与决策理论AI 解读

We consider the problem of testing for two Gibbs probabilities $μ_0$ and $μ_1$ defined for a dynamical system $(Ω,T)$. Due to the fact that in general full orbits are not observable or computable, one needs to restrict to subclasses of tests defined by a finite time series $h(x_0), h(x_1)=h(T(x_0)),..., h(x_n)=h(T^n(x_0))$, $x_0\in Ω$, $n\ge 0$, where $h:Ω\to\mathbb R$ denotes a suitable measurable function. We determine in each class the Neyman-Pearson tests, the minimax…

Ornstein–Uhlenbeck 过程的推广：理论、模拟与参数估计AI 解读

In this work, we study the class of stochastic process that generalizes the Ornstein-Uhlenbeck processes, hereafter called by \emph{Generalized Ornstein-Uhlenbeck Type Process} and denoted by GOU type process. We consider them driven by the class of noise processes such as Brownian motion, symmetric $α$-stable Lévy process, a Lévy process, and even a Poisson process. We give necessary and sufficient conditions under the memory kernel function for the time-stationary and the…