  The FARIMA models, which have long-range-dependence (LRD), are widely used in
many areas. Through deriving a precise characterisation of the spectrum,
autocovariance function, and variance time function, we show that this family
is very atypical among LRD processes, being extremely close to the fractional
Gaussian noise in a precise sense. Furthermore, we show that this closeness
property is not robust to additive noise. We argue that the use of FARIMA, and
more generally fractionally differenced time series, should be reassessed in
some contexts, in particular when convergence rate under rescaling is important
and noise is expected.
