  Within both slightly non--extensive statistics and related numerical model, a
picture is elaborated to treat self--similar time series as a thermodynamic
system. Thermodynamic--type characteristics relevant to temperature, pressure,
entropy, internal and free energies are introduced and tested. Predictability
conditions of time series analysis are discussed on the basis of Van der Waals
model. Maximal magnitude for time interval and minimal resolution scale of the
value under consideration are found and analyzed in details. The statistics
developed is shown to be governed by effective temperature being exponential
measure of the fractal dimension of the time series. Testing of the analytical
consideration is based on numerical scheme of non--extensive random walk. A
statistical scheme is introduces to present numerical model as a grand
canonical ensemble for which entropy and internal energy are calculated as
functions of particle number. Effective temperature is found numerically to
show that its value is reduced to averaged energy per one degree of freedom.
