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Revision of To prove convergence, find a rapidly converging subsequence from Sat, 09/05/2009 - 07:25

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Any subsequence of a convergent sequence also converges to the limit of the original sequence. One can reverse this idea and study first the convergence of a special subsequence of the original sequence. This study will be simpler than studying the original sequence because one is allowed to choose as well behaving a subsequence as one wishes. If the chosen subsequence converges, this can be useful in several ways. First, if there is no apriori candidate for limit, the limit of the subsequence identifies a unique candidate for the limit of the whole sequence 2) if the chosen subsequence approximates the whole sequence well then this can be used to show that the whole sequence converges to the limit of the subsequence.

Example 1

The proof of the strong law of large numbers. Richard Durrett, Probability: Theory and Examples, Second edition, Chapter 1. This theorem concerns a sequence X_1(\cdot), X_2(\cdot),..., X_n(\cdot) of pairwise independent, identically distributed random variables on a probability space (\Omega, {\mathcal F}, P) with {\mathbb E}|X_i| < \infty. The strong law of large numbers states that

 \frac{1}{n} \sum_{i=1}^n X_i(\omega) \rightarrow {\mathbb E}[X_1] (1)

almost surely. That is, there is a measurable subset \Omega' \subset \Omega with P(\Omega') = 1 and (1) holds for every \omega \in \Omega'.