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The convergence rate of the expectation of the logarithm of the first return time , after being properly normalized, is investigated for ergodic Markov chains. I. Kontoyiannis showed that for any β > 0 we have a.s. for aperiodic cases and A. J. Wyner proved that for any ε >0 we have eventually, a.s., where is the probability of the initial n-block in x. In this paper we prove that converges to a constant depending only on the process where is the modified first return time with...
We give a necessary and sufficient condition such that, for almost all s ∈ ℝ,
||nθ - s|| < ψ(n) for infinitely many n ∈ ℕ,
where θ is fixed and ψ(n) is a positive, non-increasing sequence. This can be seen as a dual result to classical theorems of Khintchine and Szüsz which dealt with the situation where s is fixed and θ is random. Moreover, our result contains several earlier ones as special cases: two old theorems of Kurzweil, a theorem of Tseng and a recent...
The family of cones is one of typical models of non-cylindrical ruled surfaces. Among them, the circular cones are unique in the sense that their Gauss map satisfies a partial differential equation similar, though not identical, to one characterizing the so-called 1-type submanifolds. Specifically, for the Gauss map G of a circular cone, one has ΔG = f(G+C), where Δ is the Laplacian operator, f is a non-zero function and C is a constant vector. We prove that circular cones are characterized by being...
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