  In this paper, we study the memory properties of transformations of linear
processes. Dittmann and Granger (2002) studied the polynomial transformations
of Gaussian FARIMA(0,d,0) processes by applying the orthonormality of the
Hermite polynomials under the measure for the standard normal distribution.
Nevertheless, the orthogonality does not hold for transformations of
non-Gaussian linear processes. Instead, we use the decomposition developed by
Ho and Hsing (1996, 1997) to study the memory properties of nonlinear
transformations of linear processes, which include the FARIMA(p,d,q) processes,
and obtain consistent results as in the Gaussian case. In particular, for
stationary processes, the transformations of short-memory time series still
have short-memory and the transformation of long-memory time series may have
different weaker memory parameters which depend on the power rank of the
transformation. On the other hand, the memory properties of transformations of
non-stationary time series may not depend on the power ranks of the
transformations. This study has application in econometrics and financial data
analysis when the time series observations have non-Gaussian heavy tails. As an
example, the memory properties of call option processes at different strike
prices are discussed in details.
