Abstract
Suppose that X1, X2, X3,... are scalar-valued random variables, independently and identically distributed (IID). There is an enormous literature on the behaviour of the sums \({S_{n = }}\sum\nolimits_{j = 0}^n {{X_j}} \) and averages \({{\bar X}_n} = {S_n}/n\) of such variables as n becomes large. The motivation is clear from the discussion of Section 1.2. Sample averages of actual data are observed to ‘converge’ with increasing sample size, and it was this ‘limit’ which we idealized to provide the concept of an expectation. There is then interest in confirming whether such behaviour can be reproduced within the theory, in that one can demonstrate convergence of \({\bar X_n}\) in some well-defined sense, under appropriate assumptions.
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© 2000 Springer Science+Business Media New York
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Whittle, P. (2000). The Two Basic Limit Theorems. In: Probability via Expectation. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0509-8_7
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DOI: https://doi.org/10.1007/978-1-4612-0509-8_7
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