Before revision:  Primarily, if we have a time series, using a GARCH(1,1) model with a double gaussian conditional probability (a GARCH-double-normal model) we cannot necessarily fit the statistical moments of the time series. Highlighting, that the GARCH-double-normal model only allows fitting of specific lengths of time series .
After revision:  It is seen for this model that we can only fit data collected within specific time window lengths and only with certain parameters of the conditional double gaussian distribution .