  This contribution deals with the generalized symmetric FastICA algorithm in
the domain of Independent Component Analysis (ICA). The generalized symmetric
version of FastICA has been shown to have the potential to achieve the
Cram\'er-Rao Bound (CRB) by allowing the usage of different nonlinearity
functions in its parallel implementations of one-unit FastICA. In spite of this
appealing property, a rigorous study of the asymptotic error of the generalized
symmetric FastICA algorithm is still missing in the community. In fact, all the
existing results exhibit certain limitations, such as ignoring the impact of
data standardization on the asymptotic statistics or being based on a heuristic
approach.
  In this work, we aim at filling this blank.
  The first result of this contribution is the characterization of the limits
of the generalized symmetric FastICA. It is shown that the algorithm optimizes
a function that is a sum of the contrast functions used by traditional one-unit
FastICA with a correction of the sign. Based on this characterization, we
derive a closed-form analytic expression of the asymptotic covariance matrix of
the generalized symmetric FastICA estimator using the method of estimating
equation and M-estimator.
