Long memory time series models

For a long time the most frequently used models in time series analysis were the AR, MA and ARMA processes. Their spectral densities are continuous and therefore bounded functions on [ — n, it]. If the periodogram of real data reached significantly high values, it was considered as an indication of the trend or of a periodic component. The bias arising after trend removal in the spectral density estimators was corrected using special factors (see [7] and [19]). However, the statistical analysis of many hydrological time series has led in the last time to the conclusion that the peak of the periodogram near to the origin should be rather explained by a model with a spectral density, which is not bounded in the neighbourhood of the zero frequency. From this reason models with long memory have been investigated, because they appear to be suitable for applications of such kind. Their definition reads as follows. Let {X,} be a stationary (discrete) process with a covariance function Rk. Then {Xt} is called a process with long memory, if Y\k\ = °°In the c a s e t n a t Xli^l < °° we say that the process {X,} has short memory. From practical point of view we restrict ourselves to the processes with R0 4= 0. Then the above definitions can be formulated in the same way using the correlation function. The definition itself was proposed in [15].