Why can't ARIMA model large lags and/or long range dependence? ARIMA cannot model large lags (obtained from autocorrelation plot) and long range dependency (hurst exponent $H > 0.5$). Why is it so?
 A: As Pr. Hyndman explains in this blog post, there's nothing in the mathematics of ARMA models that would restrict forecasting long seasonal periods. The reason you can't forecast very long periods is the fact that most software tools (including R packages) have a threshold on the allowed seasonal lags due to the high computational demands of the estimation process. I can imagine that the estimation of MA(q) processes, in particular, can become extremely expensive since its complexity increases in a high polynomial degree with seasonal order (if I'm not mistaken). 
Besides the restrictions on computational demands, there is also little point in trying to forecast such a large series. As Hyndman points out, "seasonal differencing of very high order does not make a lot of sense - for daily data it involves comparing what happened today with what happened exactly a year ago and there is no constraint that the seasonal pattern is smooth". These long seasonal terms could result to overfitting and the best way to deal with this scenario would be to use a Fourier series.  
