Approximations of fractional Brownian motion using Poisson processes whose
parameter sets have the same dimensions as the approximated processes have been
studied in the literature. In this paper, a special approximation to the
one-parameter fractional Brownian motion is constructed using a two-parameter
Poisson process. The proof involves the tightness and identification of
finite-dimensional distributions.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ319 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Let {X n , n ≥ 1} be a sequence of i.i.d. random variables with partial sums {S n , n ≥ 1}. Based on the classical Baum-Katz theorem, a paper by Heyde in 1975 initiated the precise asymptotics for the sum n≥1 P(|S n | ≥ ǫn) as ǫ goes to zero. Later, Klesov determined the convergence rate in Heyde's theorem. The aim of this paper is to extend Klesov's result to the precise asymptotics for Davis law of large numbers, a theorem in Gut and Spȃtaru [2000a].Keywords: convergence rate precise asymptotics law of large numbers MSC(2010): 60F15 60G50
In this paper, we consider a three-dimensional Brownian-driven tandem queue with intermediate inputs, which corresponds to a three-dimensional semimartingale reflecting Brownian motion whose reflection matrix is triangular. For this three-node tandem queue, no closed form formula is known, not only for its stationary distribution but also for the corresponding transform. We are interested in exact tail asymptotics for stationary distributions. By generalizing the kernel method, and using extreme value theory and copula, we obtain exact tail asymptotics for the marginal stationary distribution of the buffer content in the third buffer and for the joint stationary distribution.
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