Estimation for non-linear processes with long memory

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Long-memory is characterized by a covariance function which decreases slowly to zero as the lag increases. The decrease is so slow that the corresponding spectral density blows up at very low frequencies, a phenomenon also known as ``long-range dependence'' or ``1/f noise''. Because wavelets are associated with scaling, it is natural to attempt to use wavelets in order to estimate the intensity of long memory in time series. The advantage of wavelets on Fourier methods is that there is no need to difference the time series if these are not stationary. Fourier and wavelet techniques have been applied to processes with long memory that are Gaussian or linear. This study focuses instead on non-linear processes with long memory, for example, outputs of non-linear filters with Gaussian or with linear inputs. The goal is to derive effective techniques to estimate the exponents which characterize the intensity of long-memory, in these more realistic contexts.Time series are a collection of data points collected through time, for instance income or temperature. The dependence between these data points may be weak or strong. When this dependence is strong the time series is said to have long memory. Time series with long memory appear in a number of applications, for example, in economics and in the analysis of traffic in computer networks. The goal of this study is to understand their properties and how to estimate them.