Diffusion approximation of non-Markov random walks on differentiable manifolds

The present paper considers limit theorems for sequences of non-Markov random walks on a differentiable manifold of $C^3$-class. The result obtained is a generalization of the classic theorem for sums of dependent random variables (theorem 1). This theorem is applied then to investigation of some special random walks on a Lie group $\mathfrak G$ admitting the “polar” factorization $\mathfrak G=\mathfrak R\cdot\mathfrak U$ where $\mathfrak U$ is a compact subgroup of $\mathfrak G$. Similarly to the well-known method of N. N. Bogolyubov for differential equations with a small parameter, it may be called the principle of (compact) averaging for triangle systems of random elements on Lie groups.