The first problem of diffusion on differentiable manifolds

Let $\{\xi_\Delta(k),\ k=0,1,\dots,n=n(\Delta)\}$be a sequence of random walks on a differentiable manifold $M$. In this paper, we obtain the classical conditions for convergence of $\xi_\Delta$ to an inhomogeneous diffusion process $\xi(t)$ in terms of weak convergence of transition probabilities $P_\Delta(t_k,x;t,\Gamma)$ using some modification of Khintchine's idea from [1]. One of many consequences of the result is a limit theorem for convolutions of noncommuting probability measures on Lie groups.