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A class of strong limit theorems for countable nonhomogeneous Markov chains on the generalized gambling system

Kangkang Wang (2009)

Czechoslovak Mathematical Journal

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....

A law of the iterated logarithm for general lacunary series

Charles N. Moore, Xiaojing Zhang (2012)

Colloquium Mathematicae

We prove a law of the iterated logarithm for sums of the form ∑ k = 1 N a k f ( n k x ) where the n k 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.

A local convergence theorem for partial sums of stochastic adapted sequences

Wei Guo Yang, Zhong Xing Ye, Liu, Wen (2006)

Czechoslovak Mathematical Journal

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.

A multiparameter variant of the Salem-Zygmund central limit theorem on lacunary trigonometric series

Mordechay B. Levin (2013)

Colloquium Mathematicae

We prove the central limit theorem for the multisequence ∑ 1 ≤ n ₁ ≤ N ₁ ⋯ ∑ 1 ≤ n d ≤ N d a n ₁ , . . . , n d c o s ( ⟨ 2 π m , A ₁ n ₁ . . . A d n d x ⟩ ) where m ∈ ℤ s , a n ₁ , . . . , n d are reals, A ₁ , . . . , A d are partially hyperbolic commuting s × s matrices, and x is a uniformly distributed random variable in [ 0 , 1 ] s . The main tool is the S-unit theorem.

A note on almost sure convergence and convergence in measure

P. Kříž, Josef Štěpán (2014)

Commentationes Mathematicae Universitatis Carolinae

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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