Stochastic Convergence

Abstract

Probability theory is founded on an empirical limit concept, and its most characteristic conclusions take the form of limit theorems. Thus, a sequence of r.v.s. 282-1 which one suspects has some kind of limit property for large n is a familiar object. For example, the convergence of the sample average X _n to a common expected value E(X) (in mean square, Exercise 2.8.6; in probability, Exercise 2.9.14 or in distribution, Section 7.3) has been a recurrent theme. Other unforced examples are provided by the convergence of estimates or conditional expectations with increasing size of the observation set upon which they are based (Chapter 14), and the convergence of the standardize sum u _n to normality (Section 7.4). Any infinite sum of r.v.s which we encounter should be construed as a limit, in some sense, of a finite sum. Consider, for instance, the sum 282-2 of Section 6.1, or the formal solution 282-3 of the stochastic difference equation 282-4.