Strong tightness as a condition of weak and almost sure convergence

Grzegorz Krupa; Wiesław Zieba

Commentationes Mathematicae Universitatis Carolinae (1996)

  • Volume: 37, Issue: 3, page 641-650
  • ISSN: 0010-2628

Abstract

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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 ∈ 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 ∈ ℕ } is said to converge essentially with respect to law to r.e. X if for all sets of continuity of measure P ∘ X - 1 , P lim sup n → ∞ [ X n ∈ A ] = P lim inf n → ∞ [ X n ∈ A ] = P ( [ x ∈ A ] ) . Conditions under which { X n } is essentially w.r.t. law convergent and relations to strong tightness are investigated.

How to cite

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Krupa, Grzegorz, and Zieba, Wiesław. "Strong tightness as a condition of weak and almost sure convergence." Commentationes Mathematicae Universitatis Carolinae 37.3 (1996): 641-650. <http://eudml.org/doc/247901>.

@article{Krupa1996,
abstract = {A sequence of random elements $\lbrace X_j, j\in J\rbrace $ 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 $\lbrace X_n\rbrace $ as well as weak convergence of randomly indexed sequence $\lbrace X_\{\tau \}\rbrace $ assure strong tightness of $\lbrace X_n, n\in \mathbb \{N\}\rbrace $. 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. $\lbrace X_n, n\in \mathbb \{N\}\rbrace $ is said to converge essentially with respect to law to r.e. $X$ if for all sets of continuity of measure $P\circ X^\{-1\}, P\left(\limsup _\{n\rightarrow \infty \}[X_n\in A]\right) =P\left(\liminf _\{n\rightarrow \infty \}[X_n\in A]\right)=P([x\in A])$. Conditions under which $\lbrace X_n\rbrace $ is essentially w.r.t. law convergent and relations to strong tightness are investigated.},
author = {Krupa, Grzegorz, Zieba, Wiesław},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {almost sure convergence; stopping times; tightness; almost sure convergence; stopping times; tightness},
language = {eng},
number = {3},
pages = {641-650},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Strong tightness as a condition of weak and almost sure convergence},
url = {http://eudml.org/doc/247901},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Krupa, Grzegorz
AU - Zieba, Wiesław
TI - Strong tightness as a condition of weak and almost sure convergence
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1996
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 37
IS - 3
SP - 641
EP - 650
AB - A sequence of random elements $\lbrace X_j, j\in J\rbrace $ 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 $\lbrace X_n\rbrace $ as well as weak convergence of randomly indexed sequence $\lbrace X_{\tau }\rbrace $ assure strong tightness of $\lbrace X_n, n\in \mathbb {N}\rbrace $. 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. $\lbrace X_n, n\in \mathbb {N}\rbrace $ is said to converge essentially with respect to law to r.e. $X$ if for all sets of continuity of measure $P\circ X^{-1}, P\left(\limsup _{n\rightarrow \infty }[X_n\in A]\right) =P\left(\liminf _{n\rightarrow \infty }[X_n\in A]\right)=P([x\in A])$. Conditions under which $\lbrace X_n\rbrace $ is essentially w.r.t. law convergent and relations to strong tightness are investigated.
LA - eng
KW - almost sure convergence; stopping times; tightness; almost sure convergence; stopping times; tightness
UR - http://eudml.org/doc/247901
ER -

References

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  5. Edgar G.A., Suchestone L., Amarts: A Class of Asymptotic Martingales. A Discrete Parameter, Journal of Multivariate Analysis 6.2 (1976). (1976) MR0413251
  6. Kruk Ł., Ziȩba W., On tightness of randomly indexed sequences of random elements, Bull. Pol. Ac.: Math. 42 (1994), 237-241. (1994) MR1811853
  7. Neveu J., Discrete-Parameter Martingales, North-Holland Publishing Company (1975). (1975) Zbl0345.60026MR0402915
  8. Szynal D., Ziȩba W., On some characterization of almost sure convergence, Bull. Pol. Acad. Sci. 34 (1986), 9-10. (1986) MR0884212

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