# Article

Full entry | PDF   (0.2 MB)
Keywords:
almost sure convergence; stopping times; tightness
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.
References:
[1] Austin D.G., Edgar G.A., Ionescu Tulcea A.: Pointwise convergence in terms of expectations. Z. Wahrscheinlichkeitsteorie verw. Gebiete 30 (1974), 17-26. MR 0358945 | Zbl 0276.60034
[2] Baxter J.R.: Pointwise in terms of weak convergence. Proc. Amer. Math. Soc. 46 (1974), 395-398. MR 0380968 | Zbl 0329.60029
[3] Billingsley P.: Convergence of Probability Measure. Wiley New York (1968). MR 0233396
[4] Diestel J., Uhl J.J., Jr.: Vector Measures. AMS Mathematical Surveys 15 (1979).
[5] Edgar G.A., Suchestone L.: Amarts: A Class of Asymptotic Martingales. A Discrete Parameter. Journal of Multivariate Analysis 6.2 (1976). MR 0413251
[6] Kruk Ł., Ziȩba W.: On tightness of randomly indexed sequences of random elements. Bull. Pol. Ac.: Math. 42 (1994), 237-241. MR 1811853
[7] Neveu J.: Discrete-Parameter Martingales. North-Holland Publishing Company (1975). MR 0402915 | Zbl 0345.60026
[8] Szynal D., Ziȩba W.: On some characterization of almost sure convergence. Bull. Pol. Acad. Sci. 34 (1986), 9-10. MR 0884212

Partner of