Displaying similar documents to “On a characterization of orthogonality with respect to particular sequences of random variables in L 2 ”

Statistical convergence of a sequence of random variables and limit theorems

Sanjoy Ghosal (2013)

Applications of Mathematics

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In this paper the ideas of three types of statistical convergence of a sequence of random variables, namely, statistical convergence in probability, statistical convergence in mean of order r and statistical convergence in distribution are introduced and the interrelation among them is investigated. Also their certain basic properties are studied.

Strong tightness as a condition of weak and almost sure convergence

Grzegorz Krupa, Wiesław Zieba (1996)

Commentationes Mathematicae Universitatis Carolinae

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A sequence of random elements { X j , j ∈ J } is called strongly tight if for an arbitrary ϵ > 0 there exists a compact set K such that P ⋂ j ∈ J [ X j ∈ K ] > 1 - ϵ . For the Polish space valued sequences of random elements we show that almost sure convergence of { X n } as well as weak convergence of randomly indexed sequence { X τ } assure strong tightness of { X n , n ∈ ℕ } . For L 1 bounded Banach space valued asymptotic martingales strong tightness also turns out to the sufficient condition of convergence. A sequence of r.e. { X n , n ∈ ℕ } is said to converge essentially...