Probability-theoretical analog of the vector lyapunov function method

The main ideas of the vector Lyapunov function (VLF) method were advanced in 1962 by Bellman and Matrosov. In this method, a Lyapunov function and a comparison equation are constructed for each subsystem. Then the dependences between the subsystems and the effect of external noise are allowed for by constructing a vector Lyapunov function (as a collection of the scalar Lyapunov functions of the subsystems) and an aggregate comparison function for the entire complex system. A probability-theoretical analog of this method for convergence analysis of stochastic approximation processes has been developed. The abstract approach proposed elsewhere eliminates all restrictions on the system phase space, the system trajectories, the class of Lyapunov functions, etc. The analysis focuses only on the conditions that relate sequences of Lyapunov function values with the derivative and ensure a particular type (mode, character) of stability. In our article, we extend this approach to the VLF method for discrete stochastic dynamic systems.