  In this paper, a fourth moment bound for partial sums of functional of
strongly ergodic Markov chain is established. This type of inequality plays an
important role in the study of empirical process invariance principle. This one
is specially adapted to the technique of Dehling, Durieu and Voln\'y (2008).
The same moment bound can be proved for dynamical system whose transfer
operator has some spectral properties. Examples of applications are given.
