Manneville-Pomeau过程的参数估计
Parameter Estimation in Manneville-Pomeau Processes
摘要：
本文研究了一类随机过程$\{X_t\}_{t\in\N}$，其中$X_t = (ϕ\circ T_s^t)(X_0)$由变换T_s的迭代得到，T_s对[0，1]上的遍历概率μ_s不变，且是按部分连续的函数$ϕ:[0,1] \to \R$ .这里我们考虑$T_s:[0,1]\to [0,1]$的Manneville-Pomeau变换。由此产生的过程的自相关函数双曲线（或多项式）衰减，我们得到有效的方法来估计参数s从一个有限的时间序列。因此，我们也估计这些过程的自相关衰减的收敛收敛速率。我们比较了基于周期图函数、平滑周期图函数、部分和方差和小波理论的不同估计方法。
英文摘要：
In this work we study a class of stochastic processes $\{X_t\}_{t\in\N}$, where $X_t = (ϕ\circ T_s^t)(X_0)$ is obtained from the iterations of the transformation T_s, invariant for an ergodic probability μ_s on [0,1] and a continuous by part function $ϕ:[0,1] \to \R$. We consider here $T_s:[0,1]\to [0,1]$ the Manneville-Pomeau transformation. The autocorrelation function of the resulting process decays hyperbolically (or polynomially) and we obtain efficient methods to estimate the parameter s from a finite time series. As a consequence we also estimate the rate of convergence of the autocorrelation decay of these processes. We compare different estimation methods based on the periodogram function, on the smoothed periodogram function, on the variance of the partial sum and on the wavelet theory.
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