Our next interest is in whether a sequence of distribution functions converges weakly. To be more specific, subsequential convergence of distribution functions are is the topic of this subsection. Helly’s selection theorem shows there always exists a vaguely convergent subsequence. Uniform tightness of a sequence strengthen this result to be weakly convergent.
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Now that we covered convergence of point estimates (specifically, the sample mean) our next interest is in weaker concept of convergence where convergence in probability is not guaranteed. In undergraduate statistics, we call it convergence in distribution. Here we prefer borrowing terminology from measure theory and call it weak convergence and write $X_n \overset{w}{\to} X$.
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This is a quick review of convergence concepts and their relations.
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As the last section in chapter 2, we cover convergence of random series. Especially, since I already explained what tail $\sigma$-fields and tail events are, our focus will be on Kolmogorov’s maximal inequality and the three series theorem.
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Putting together all the topics we have covered so far, we now move on to one of the most impactful theorem in probability theory: the strong law of large numbers (SLLN).
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