By introducing the sample relative entropy rate as a measure of the deviation between the arbitrary random fields and the Markov chain fields on Cayley trees, a class of small deviation theorems for the frequencies of state ordered couples is established. In the proof a new analytic technique in...

In this paper, the strong limit theorems for arbitrary stochastic sequences are studied. Some convergence theorems for martingale difference sequence and a class of strong limit theorem for countable nonhomogeneous Markov chains are the particular cases of the results of this paper. Finally, the...

Let {Xn,n[greater-or-equal, slanted]0} be a sequence of random variables taking values in S={1,2,...,N}, and let pk(xk x0,...,xk-1)=P(Xk=xk X0=x0,...,Xk-1=xk-1). In this paper a theorem on a.e. convergence for the harmonic mean of the random conditional probabilities {pk(Xk X0,...,Xk-1),...

In this paper, we use an analytic approach to study the limit properties of {inn(Xn)}, the functional of a countable nonhomogeneous Markov chain {Xn}. A class of strong laws of large numbers for these processes, which are different from the usual ones, are obtained. In the theorems of this...

In this paper, we study the strong law of large numbers on the frequencies of states for a Markov chains field on a Bethe tree. In the proof, we apply a new technique in the study of the strong limit theorem.