  We propose certain conditions which are sufficient for the functional law of
the iterated logarithm (the Strassen invariance principle) for some general
class of non-stationary Markov-Feller chains. This class may be briefly
specified by the following two properties: firstly, the transition operator of
the chain under consideration enjoys a non-linear Lyapunov-type condition, and
secondly, there exists an appropriate Markovian coupling whose transition
probability function can be decomposed into two parts, one of which is
contractive and dominant in some sense. The construction of such a coupling
derives from the paper of M. Hairer (Probab. Theory Related Fields,
124(3):345--380, 2002). Our criterion may serve as a useful tool in verifying
the functional law of the iterated logarithm for certain random dynamical
systems, developed eg. in molecular biology. In the final part of the paper we
present an example application of our main theorem to the mathematical model
describing stochastic dynamics of gene expression.
