We prove convergence in probability for particular sequences defined in terms of the digits appearing in Oppenheim Series expansions and Oppenheim Continued Fractions expansions of real numbers. Our results are obtained by first proving a general theorem (Theorem 2.2) having both kinds of expansion as particular cases
In this paper we study some sequences of weighted means of continuous real valued Gaussian processes. More precisely we consider suitable generalizations of both arithmetic and logarithmic means of a Gaussian process with covariance function which satisfies either an exponential decay condition or a power decay condition. Our aim is to provide limits of variances of functionals of such weighted means which allow the application of some large deviation results in the literature
We show that the Bernoulli part extraction method can be used to obtain approximate forms of the local limit theorem for sums of independent lattice valued random variables, with effective error term, that is with explicit parameters and universal constants. We also show that our estimates allow to recover Gnedenko and Gamkrelidze local limit theorems. We further establish by this method a local limit theorem with effective remainder for random walks in random scenery.2010 Mathematics Subject Classification: Primary: 60F15, 60G50 ; Secondary: 60F05.
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