A limit property of random conditional probabilities
This paper provides a strong law of large numbers for the harmonic mean of random conditional probabilities \(\{p_k(X_k\mid X_0,\dots, X_{k-1}): k= 1,\dots, n\}\), where \(p_k(x_k\mid x_0,\dots, x_{k-1})= P(X_k= x_k\mid X_0= x_0,\dots, X_{k-1}= x_{k-1})\), \(k\geq 1\), and \(\{X_n:n\geq 0\}\) is a sequence of random variables taking values in \(S= \{1,\dots, N\}\). The proof makes use of conditional moment generating functions and differentiation on a net. As an application, almost sure limiting properties of the arithmetic mean of the (random) transition probabilities in an \(m\)th order nonhomogeneous Markov chain are discussed.
scientific article; zbMATH DE number 1517226