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.

中文翻译：

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某些非平稳 Markov-Feller 链的 Strassen 不变性原理

我们提出了某些条件，这些条件对于某些一般类别的非平稳马尔可夫 - 费勒链的迭代对数（施特拉森不变性原理）的泛函定律是充分的。这个类可以用以下两个性质来简要说明：首先，所考虑的链的转移算子具有非线性李雅普诺夫型条件，其次，存在适当的马尔可夫耦合，其转移概率函数可以分解为两个部分，其中之一在某种意义上是收缩的和主导的。这种耦合的构造源自 M. Hairer 的论文（Probab. Theory Related Fields, 124(3):345--380, 2002）。我们的标准可以作为一个有用的工具来验证某些随机动力系统的迭代对数的功能规律，例如开发。在分子生物学中。在论文的最后部分，我们展示了我们的主要定理在描述基因表达随机动力学的数学模型中的一个示例应用。