In this paper, we study the limit properties of countable nonhomogeneous Markov chains in the generalized gambling system by means of constructing compatible distributions and martingales. By allowing random selection functions to take values in arbitrary intervals, the concept of random selection is generalized. As corollaries, some strong limit theorems and the asymptotic equipartition property (AEP) theorems for countable nonhomogeneous Markov chains in the generalized gambling system are established. Some results obtained are extended.
In this paper we establish a new local convergence theorem for partial sums of arbitrary stochastic adapted sequences. As corollaries, we generalize some recently obtained results and prove a limit theorem for the entropy density of an arbitrary information source, which is an extension of case of nonhomogeneous Markov chains.
Assume that $X$, $Y$ are continuous-path martingales taking values in $\mathbb R^\nu $, $\nu \geq 1$, such that $Y$ is differentially subordinate to $X$. The paper contains the proof of the maximal inequality $$ \|\sup _{t\geq 0} |Y_t| \|_1\leq 2\|\sup _{t\geq 0} |X_t| \|_1. $$ The constant $2$ is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder's method and rests on the construction of an appropriate special function.