Parameter Estimation in Manneville-Pomeau Processes

Abstract

In this work we study a class of stochastic processes $\{X_t\}_{t\in\N}$, where $X_t = (\phi \circ T_s^t)(X_0)$ is obtained from the iterations of the transformation T_s, invariant for an ergodic probability \mu_s on [0,1] and a continuous by part function $\phi:[0,1] \to \R$. 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.