Geodesic
Strong tightness as a condition of weak and almost sure convergence
Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) no. 3, pp. 641-650 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 60B10, 60G40
Keywords: almost sure convergence; stopping times; tightness 
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     title = {Strong tightness as a condition  of weak and almost sure convergence},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {641--650},
     year = {1996},
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     zbl = {0881.60003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1996_37_3_a21/}
}
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Krupa, Grzegorz; Zieba, Wiesław. Strong tightness as a condition  of weak and almost sure convergence. Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) no. 3, pp. 641-650. http://geodesic.mathdoc.fr/item/CMUC_1996_37_3_a21/