Let the martingale Wn = Zn mn , where Zn is a weighted branching process and m = E ∑ i Ti is the expected sum of the = factors Ti, converge to some limiting random variable W . We give conditions in terms of the factors such that W belongs to the domain of attraction or to the domain of normal attraction of an = α–stable distribution with 1 < α ≤ 2. Further the rate of convergence of Wn to W is...

We give a strong approximation of Rosenblatt process via transport processes and we give the rate of convergence.

This paper contains an asymptotic analysis of a fluid model for a heavily loaded processor sharing queue. Specifically, we consider the behavior of solutions of critical fluid models as time approaches ∞. The main theorems of the paper provide sufficient conditions for a fluid model solution to converge to an invariant state and, under slightly more restrictive assumptions, provide a rate of co...

We prove sharp convergence rates for a class of domain embedding methods for elliptic boundary value problems. The theory is established for Dirichlet, Neumann, and Robin boundary conditions, and is unified in a new formulation for mixed boundary condition. On the continuous level, the methods for these boundary value problems perform equally well.

We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of any even order converge in the almost sure central limit theorem for martingales. A conjecture about almost sure upper bounds under wider hypotheses is formul...

We obtain rates of convergence and asymptotic expansions in limit theorems for powers of reciprocals of random variables. Our results improve on recent work of Shapiro [18], who obtained Berry-Esseen bounds of order n~ l/9+c (e > 0). Under weaker conditions than Shapiro imposed we obtain asymptotic expansions whose first term is of order n~ i (logn) 2 .

One of the generalizations of statistical convergence is I-convergence which was introduced by Kostyrko et al. [12]. In this paper, we define and study the concept of I-convergence, I∗-convergence, I-limit points and I-cluster points of double sequences in probabilistic normed space. We discuss the relationship between I2-convergence and I ∗ 2 -convergence, i.e., we show that I ∗ 2 -convergence...

we have by construction μ(En) = an. An important observation is that a subset of consecutive intervals En, En+1, . . . , En+k are either pairwise disjoint, or they cover the whole interval [0, 1]. Now for all n, let fn = 1En . Then we have • ‖fn‖1 = ∫ X |1En |dμ = μ(En) = an. • for all x ∈ [0, 1], (fn(x))n does not converge. Indeed, if we suppose by contradiction that (fn(x))n converges for som...

The law of large numbers for the empirical density for the pairs of uniformly distributed integers with a given greatest common divisor is a classic result in number theory. In this paper, we study the large deviations of the empirical density. We will also obtain a sharp rate of convergence to the normal distribution for the central limit theorem. Some generalizations are provided.