Chapter Eleven

Convergence Types. Almost Sure Convergence. Lp-Convergence. Convergence in Probability

11.1 Introduction/Purpose of the Chapter

In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and it is a very important application to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence, and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. The different notions of convergence relate to how such a behavior can be characterized. We can talk about a sequence that approaches a random variable exactly, with probability one and looking at the moments of the distribution. In this chapter we will talk about a notion of convergence defined purely by the distribution of random variables.

11.2 Vignette/Historical Notes

In their development of the calculus both Newton and Leibniz used “infinitesimals,” quantities that are infinitely small and yet nonzero. They found it convenient to use these quantities in their computations and their derivations of results. Cauchy, Weierstrass, and Riemann reformulated Calculus in terms of limits rather than infinitesimals. Thus the need for these ...

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