Random sequences are essential in statistics. The statistical analysis of any experiment usually begins with the words "let X_{1},...,X_{n} be independent random variables...". The easiest way to talk about a situation when you can choose to make new measurements is to assume that an infinite sequence {X_{i}} is given, and that successive stages of the experiment you look at the first N terms of the sequence. Also, the statement of the central limit theorem (essentially that the average of a number of observations converges to the mean value) involves an infinite sequence of independent, identically distributed[?] random variables.
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