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Previously unrecognized links between statistical dependence and some common modes of convergence. (English) Zbl 1229.00007

Most introductory courses in probability include the definitions of, and relations between, convergence in distribution, probability and mean square. Surprisingly, there are also links between the apparently unrelated notion of statistical dependence and two of these modes of convergence. These links are virtually absent from both the research literature and probability texts. The authors explain these links between statistical dependence and three modes of convergence, filling a gap in the literature. They establish the ultimate dependence between members of a convergent sequence of random variables and its nondegenerate limit when convergence is according to one of the latter two modes. Through simple examples and graphical displays, they convey the heuristics that underly the main results.
The references contain 5 sources:
[M. Scarsini, Riv. Mat. Sci. Econ. Soc. 7, 39–44 (1984; Zbl 0554.60003); P. Billingsley, Probability and measure. 2nd ed. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York etc.: John Wiley&Sons, Inc. XIV, 622 p. (1986; Zbl 0649.60001); D. Williams, Probability with martingales. Cambridge etc.: Cambridge University Press. xv, 251 p. (1991; Zbl 0722.60001); R. C. Mittelhammer, Mathematical statistics for economics and business. New York, NY: Springer. xviii, 723 p. (1996; Zbl 0851.62001) and R. M. Dudley, Real analysis and probability. Repr. Cambridge Studies in Advanced Mathematics. 74. Cambridge: Cambridge University Press. x, 555 p. (2002; Zbl 1023.60001)].

MSC:

00A35 Methodology of mathematics
97K70 Foundations and methodology of statistics (educational aspects)
97K50 Probability theory (educational aspects)
60B10 Convergence of probability measures
62A09 Graphical methods in statistics
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