An Outlook on Further Topics
Probability theory is, of course, much more than what one will find in this book (so far). In this chapter we provide an outlook on some extensions and further areas and concepts in probability theory. For more we refer to the more advanced literature cited in Appendix A. We begin, in the first section, by presenting some extensions of the classical limit theorems, that is, the law of large numbers and the central limit theorem, to cases where one relaxes the assumptions of independence and equidistribution. Another question in this context is whether there exist (other) limit distributions if the variance of the summands does not exist (is infinite). This leads, in the case of i.i.d. summands, to the class of stable distributions and their, what is called, domains of attraction. Sections 2 and 3 are devoted to this problem.
KeywordsLimit Theorem Central Limit Theorem Independent Random Variable Stable Distribution Complete Convergence
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