非一致双曲动力系统极值过程和记录事件的Weak convergence
Weak convergence to extremal processes and record events for non-uniformly hyperbolic dynamical systems
摘要：
对于保测度动力系统$(\mathcal{X},f, μ)$，我们考虑与过程$X_n=ϕ(f^{n-1}(x))$相关联的极大值$M_n=\max\{X_1,\ldots,X_n\}$的时间序列，其中过程$X_n=ϕ(f^{n-1}(x))$是由动力系统对某个可观测的$ϕ:\mathcal{X}\to\mathbb{R}$产生的.利用点过程方法，我们建立了过程$Y_n(t)=a_n(M_{[nt]}-b_n)$弱收敛到一个极值过程$Y(t)$适当的缩放常数$a_n,b_n\in\mathbb{R}$。这里的收敛发生在Skorokhod空间$\mathbb{D}(0,\infty)$与$J_1$拓扑。我们还建立了相应的极大值过程的记录时间和记录值的分布结果。
英文摘要：
For a measure preserving dynamical system $(\mathcal{X},f, μ)$, we consider the time series of maxima $M_n=\max\{X_1,\ldots,X_n\}$ associated to the process $X_n=ϕ(f^{n-1}(x))$ generated by the dynamical system for some observable $ϕ:\mathcal{X}\to\mathbb{R}$. Using a point process approach we establish weak convergence of the process $Y_n(t)=a_n(M_{[nt]}-b_n)$ to an extremal process $Y(t)$ for suitable scaling constants $a_n,b_n\in\mathbb{R}$. Convergence here taking place in the Skorokhod space $\mathbb{D}(0,\infty)$ with the $J_1$ topology. We also establish distributional results for the record times and record values of the corresponding maxima process.
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