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Strong laws of large numbers in certain linear spaces

Wojbor A. Woyczynski (1974)

Annales de l'institut Fourier

In this paper we are concerned with the norm almost sure convergence of series of random vectors taking values in some linear metric spaces and strong laws of large numbers for sequences of such random vectors. Section 2 treats the Banach space case where the results depend upon the geometry of the unit cell. Section 3 deals with spaces equipped with a non-necessarily homogeneous F -norm and in Section 4 we restrict our attention to sequences of identically distributed random vectors.

Strong solutions for stochastic differential equations with jumps

Zenghu Li, Leonid Mytnik (2011)

Annales de l'I.H.P. Probabilités et statistiques

General stochastic equations with jumps are studied. We provide criteria for the uniqueness and existence of strong solutions under non-Lipschitz conditions of Yamada–Watanabe type. The results are applied to stochastic equations driven by spectrally positive Lévy processes.

Strong tightness as a condition of weak and almost sure convergence

Grzegorz Krupa, Wiesław Zieba (1996)

Commentationes Mathematicae Universitatis Carolinae

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 with...

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