Unformatted text preview: n is called the partial sum process. Theorem 17.1 E  X i  < ∞ and let μ = E X i . Then S n n → μ. This is known as the strong law of large numbers (SLLN). The convergence here means that S n ( ω ) /n → μ for every ω ∈ S , where S is the probability space, except possibly for a set of ω of probability 0. The proof of Theorem 13.1 is quite hard, and we prove a weaker version, the weak law of large numbers (WLLN). The WLLN states that for every a > 0, P ±² ² ² S n nE X 1 ² ² ² > a ³ → as n → ∞ . It is not even that easy to give an example of random variables that satisfy the WLLN but not the SLLN. Before proving the WLLN, we need an inequality called Chebyshev’s inequality. 42...
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 Spring '10
 ansan
 Probability, Probability theory, Sn, WLLN

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