Displaying similar documents to “On the convergence of certain sums of independent random elements”

A note on k-c-semistratifiable spaces and strong β -spaces

Li-Xia Wang, Liang-Xue Peng (2011)

Mathematica Bohemica


Recall that a space X is a c-semistratifiable (CSS) space, if the compact sets of X are G δ -sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a T 2 -space X is a k-c-semistratifiable space if and only if X has a g function which satisfies the following conditions: (1) For each x X , { x } = { g ( x , n ) : n } and g ( x , n + 1 ) g ( x , n ) for each...

On the classes of hereditarily p Banach spaces

Parviz Azimi, A. A. Ledari (2006)

Czechoslovak Mathematical Journal


Let X denote a specific space of the class of X α , p Banach sequence spaces which were constructed by Hagler and the first named author as classes of hereditarily p Banach spaces. We show that for p > 1 the Banach space X contains asymptotically isometric copies of p . It is known that any member of the class is a dual space. We show that the predual of X contains isometric copies of q where 1 p + 1 q = 1 . For p = 1 it is known that the predual of the Banach space X contains asymptotically isometric copies of...

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 } . 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 } is said to converge essentially...