Abstract
We begin by recalling the Strong Law of Large Numbers (Theorem 20.2): if (X n )n≥1 are i.i.d. with E{X n } = μ and σ 2 Xn < ∞, and if S n = Σ nj=1 X j , then limn→∞ S n /n = μ a.s. Note that since the X n are all independent, the limit must be constant a.s. as a consequence of the tail event zero-one law (Theorem 10.6). It is interesting to study sequences converging to limits that are random variables, not just constant.
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© 2000 Springer-Verlag Berlin Heidelberg
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Jacod, J., Protter, P. (2000). Martingales. In: Probability Essentials. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-51431-9_24
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DOI: https://doi.org/10.1007/978-3-642-51431-9_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66419-2
Online ISBN: 978-3-642-51431-9
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