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

The Strassen Invariance Principle for Certain Non-stationary Markov-Feller Chains