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 n-E 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 in-equality. 42...
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- Spring '10
- Probability, Probability theory, Sn, WLLN