A limit property of random conditional probabilities

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), 1[less-than-or-equals, slant]k[less-than-or-equals, slant]n} is obtained by using the tool of the conditional moment generating function and the differentiation on a net.