| Title:
|
Strong tightness as a condition of weak and almost sure convergence (English) |
| Author:
|
Krupa, Grzegorz |
| Author:
|
Zieba, Wiesław |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
37 |
| Issue:
|
3 |
| Year:
|
1996 |
| Pages:
|
641-650 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
almost sure convergence |
| Keyword:
|
stopping times |
| Keyword:
|
tightness |
| MSC:
|
60B10 |
| MSC:
|
60G40 |
| idZBL:
|
Zbl 0881.60003 |
| idMR:
|
MR1426929 |
| . |
| Date available:
|
2009-01-08T18:26:52Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118871 |
| . |
| Reference:
|
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| Reference:
|
[2] Baxter J.R.: Pointwise in terms of weak convergence.Proc. Amer. Math. Soc. 46 (1974), 395-398. Zbl 0329.60029, MR 0380968 |
| Reference:
|
[3] Billingsley P.: Convergence of Probability Measure.Wiley New York (1968). MR 0233396 |
| Reference:
|
[4] Diestel J., Uhl J.J., Jr.: Vector Measures.AMS Mathematical Surveys 15 (1979). |
| Reference:
|
[5] Edgar G.A., Suchestone L.: Amarts: A Class of Asymptotic Martingales. A Discrete Parameter.Journal of Multivariate Analysis 6.2 (1976). MR 0413251 |
| Reference:
|
[6] Kruk Ł., Ziȩba W.: On tightness of randomly indexed sequences of random elements.Bull. Pol. Ac.: Math. 42 (1994), 237-241. MR 1811853 |
| Reference:
|
[7] Neveu J.: Discrete-Parameter Martingales.North-Holland Publishing Company (1975). Zbl 0345.60026, MR 0402915 |
| Reference:
|
[8] Szynal D., Ziȩba W.: On some characterization of almost sure convergence.Bull. Pol. Acad. Sci. 34 (1986), 9-10. MR 0884212 |
| . |
