A stationary process has the long memory property,
if for its autocorrelation
function
holds:
With respect to (14.1), note that the classical
expression for the variance of the sample mean,
, for independent and
identically distributed
,
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(14.3) |
Note that (14.1) holds in particular if
the autocorrelation is approximately
with
a constant
and a parameter
.
If we know the autocorrelations we also know the spectral
density
, defined as
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(14.4) |
To connect the long memory property with the Hurst coefficient, we
introduce self similar processes. A stochastic process is called
self similar with self similarity parameter
, if for any positive
stretching factor
, the rescaled process
has the same distribution as the original process
.
If the increments
are stationary, there
autocorrelation function is given by
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In this section, we introduce a particular self similar process with stationary increments, namely the fractional Brownian motion (FBM) and fractional Gaussian noise (FGN), Mandelbrot and van Ness (1968), Beran (1994).
For the process has long range dependence, and for
the process has short range dependence.
Figures 14.2 and 14.3 show two simulated paths of
observations of FGN with parameter
and
using an algorithm proposed by
Davies and Harte (1987). For
, the FBM path is much more jagged and the range of
the
-axis is about ten times smaller than for
which is due to the reverting
behavior of the time series.
The estimated autocorrelation function (ACF) for the path
simulated with along with the
-confidence
band is shown in Figure 14.4. For comparison the ACF
used to simulate the process given by (14.5) is
superimposed (dashed line). The slow decay of correlations can be
seen clearly.
Applying R/S analysis we can retrieve the Hurst coefficient used
to simulate the process. Figure 14.5 displays the
estimated regression line
and the data points used in the regression. We simulate the
process with and the R/S statistic yields
.
Finally, we mention that fractional Brownian motion is not the only stationary process revealing properties of systems with long memory. Fractional ARIMA processes are an alternative to FBM, Beran (1994). As well, there are non stationary processes with infinite second moments that can be used to model long range dependence, Samrodnitsky and Taqqu (1994).
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