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Large deviations for generic stationary processes

Emmanuel Lesigne, Dalibor Volný (2000)

Colloquium Mathematicae

Let (Ω,A,μ,T) be a measure preserving dynamical system. The speed of convergence in probability in the ergodic theorem for a generic function on Ω is arbitrarily slow.

Large sets of integers and hierarchy of mixing properties of measure preserving systems

Vitaly Bergelson, Tomasz Downarowicz (2008)

Colloquium Mathematicae

We consider a hierarchy of notions of largeness for subsets of ℤ (such as thick sets, syndetic sets, IP-sets, etc., as well as some new classes) and study them in conjunction with recurrence in topological dynamics and ergodic theory. We use topological dynamics and topological algebra in βℤ to establish connections between various notions of largeness and apply those results to the study of the sets R A , B ε = n : μ ( A T B ) > μ ( A ) μ ( B ) - ε of times of “fat intersection”. Among other things we show that the sets R A , B ε allow one to distinguish...

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