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On some properties of vague convergences

Startek M.

Journal of Mathematics and Applications

2006

Vol. 28

131-135

In this paper is proved that if {X(sub n), n ≥ 1} is a sequence of noncovariant random elements and {X(sub n)} converges vaguely essentionaly or vaguely almost surely to a random element X then X is a degenerated random element. Moreover is given a condition which is necessary and sufficient to equivalence vague essentionaly and vague almost surely convergences.

The constant limits of sequences of defective random elements

Startek M.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

2000

z. 24 [181]

117-123

Let X, X1, X2,... be random elements. It is known, that if the sequence {Xn, n > 1} converges vaguely in probability to the random element X, then this sequence converges also vaguely to X, but not converse. In this paper it is proved, that if the limit random element X is constant, then the converse is true, i.e. if the sequence converges vaguely, then it also converges vaguely in probability. A similar dependence we have for convergences vaguely essentionaly and vaguely almost surely.