In this paper,we study a class of stochastic processes{X_(t)}t∈N,where X_(t)=φ■T_(s)^(t)(X_(0))is obtained from the iterations of the transformation T_(s),invariant for an ergodic probabilityμ_(s)on[0,1]and a cert...In this paper,we study a class of stochastic processes{X_(t)}t∈N,where X_(t)=φ■T_(s)^(t)(X_(0))is obtained from the iterations of the transformation T_(s),invariant for an ergodic probabilityμ_(s)on[0,1]and a certain constant by partial functionφ:[0,1]→R.We consider here the family of transformations T_(s):[0,1]→[0,1]indexed by a parameters>0,known as the Manneville–Pomeau family of transformations.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,the smoothed periodogram function,the variance of the partial sum,and the wavelet theory.To obtain our results we analyzed the properties of the spectral density function and the associated Fourier series.展开更多
基金supported by CNPq-Brazil(Grant No.310053/2020-0)Silvia Regina Lopes was partially supported by CNPq-Brazil(Grant No.303453/2018-4).
文摘In this paper,we study a class of stochastic processes{X_(t)}t∈N,where X_(t)=φ■T_(s)^(t)(X_(0))is obtained from the iterations of the transformation T_(s),invariant for an ergodic probabilityμ_(s)on[0,1]and a certain constant by partial functionφ:[0,1]→R.We consider here the family of transformations T_(s):[0,1]→[0,1]indexed by a parameters>0,known as the Manneville–Pomeau family of transformations.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,the smoothed periodogram function,the variance of the partial sum,and the wavelet theory.To obtain our results we analyzed the properties of the spectral density function and the associated Fourier series.
