Abstract: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.
Keywords: almost sure convergence, stopping times, tightness
AMS Subject Classification: 60B10, 60G40