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On a characterization of orthogonality with respect to particular sequences of random variables in L 2

Umberto Triacca, Andrei Volodin (2010)

Applications of Mathematics

This note deals with the orthogonality between sequences of random variables. The main idea of the note is to apply the results on equidistant systems of points in a Hilbert space to the case of the space L 2 ( Ω , ℱ , ℙ ) of real square integrable random variables. The main result gives a necessary and sufficient condition for a particular sequence of random variables (elements of which are taken from sets of equidistant elements of L 2 ( Ω , ℱ , ℙ ) ) to be orthogonal to some other sequence in L 2 ( Ω , ℱ , ℙ ) . The result obtained is interesting...

On asymptotic behaviors and convergence rates related to weak limiting distributions of geometric random sums

Tran Loc Hung, Phan Tri Kien, Nguyen Tan Nhut (2019)

Kybernetika

Geometric random sums arise in various applied problems like physics, biology, economics, risk processes, stochastic finance, queuing theory, reliability models, regenerative models, etc. Their asymptotic behaviors with convergence rates become a big subject of interest. The main purpose of this paper is to study the asymptotic behaviors of normalized geometric random sums of independent and identically distributed random variables via Gnedenko's Transfer Theorem. Moreover, using the Zolotarev probability...

On Billard's Theorem for Random Fourier Series

Guy Cohen, Christophe Cuny (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

We show that Billard's theorem on a.s. uniform convergence of random Fourier series with independent symmetric coefficients is not true when the coefficients are only assumed to be centered independent. We give some necessary or sufficient conditions to ensure the validity of Billard's theorem in the centered case.

On concentrated probabilities

Wojciech Bartoszek (1995)

Annales Polonici Mathematici

Let G be a locally compact Polish group with an invariant metric. We provide sufficient and necessary conditions for the existence of a compact set A ⊆ G and a sequence g n ∈ G such that μ ∗ n ( g n A ) ≡ 1 for all n. It is noticed that such measures μ form a meager subset of all probabilities on G in the weak measure topology. If for some k the convolution power μ ∗ k has nontrivial absolutely continuous component then a similar characterization is obtained for any locally compact, σ-compact, unimodular, Hausdorff topological...

On fully coupled continuous time random walks

W. Szczotka, P. Żebrowski (2012)

Applicationes Mathematicae

Continuous time random walks with jump sizes equal to the corresponding waiting times for jumps are considered. Sufficient conditions for the weak convergence of such processes are established and the limiting processes are identified. Furthermore one-dimensional distributions of the limiting processes are given under an additional assumption.

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