Strong tightness as a condition of weak and almost sure convergence

Krupa, Grzegorz ; Zieba, Wiesław

Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) no. 3, pp. 641-650.

Résumé

A sequence of random elements $\{X_j, j\in J\}$ is called strongly tight if for an arbitrary $\epsilon >0$ there exists a compact set $K$ such that $P\left(\bigcap_{j\in J}[X_j\in K]\right)>1-\epsilon$. 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 $\{X_{\tau}\}$ assure strong tightness of $\{X_n, n\in \Bbb N\}$. 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. $\{X_n, n\in \Bbb N\}$ is said to converge essentially with respect to law to r.e. $X$ if for all sets of continuity of measure $P\circ X^{-1}, P\left(\limsup_{n\to \infty}[X_n\in A]\right) =P\left(\liminf_{n\to \infty}[X_n\in A]\right)=P([x\in A])$. Conditions under which $\{X_n\}$ is essentially w.r.t. law convergent and relations to strong tightness are investigated.

Détail

Classification : 60B10, 60G40
Keywords: almost sure convergence; stopping times; tightness