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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....
The authors provide a correction to “Some mean convergence and complete convergence theorems for sequences of -linearly negative quadrant dependent random variables”.
We prove a law of the iterated logarithm for sums of the form where the satisfy a Hadamard gap condition. Here we assume that f is a Dini continuous function on ℝⁿ which has the property that for every cube Q of sidelength 1 with corners in the lattice ℤⁿ, f vanishes on ∂Q and has mean value zero on Q.
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.
We prove a lower bound in a law of the iterated logarithm for sums of the form where f satisfies certain conditions and the satisfy the Hadamard gap condition .
We prove the central limit theorem for the multisequence
where , are reals, are partially hyperbolic commuting s × s matrices, and x is a uniformly distributed random variable in . The main tool is the S-unit theorem.
The present article studies the
conditions under which the almost
everywhere convergence and the
convergence in measure coincide.
An application in the statistical
estimation theory is outlined as
well.
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