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STA213lecture20.notes

STA213lecture20.notes - X 1,X 2 convergence in probability...

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Today’s outline: pp 232-240 Convergence concepts 1. Almost sure convergence, convergence in probability, convergence in distribution. 2. Weak and strong law of large numbers 3. Central Limit Theorem 4. Slutsky’s Theorem Next time: delta method Please go through Taylor series! Convergence concepts in decreasing strength Almost sure convergence for X 1 , X 2 , ... if P ( lim n →∞ | X n - X | < ) eg Example 5.5.7 Convergence in probability if lim n →∞ P ( | X n - X | < ) eg Example 5.5.3: S 2 n Convergence in distribution lim n →∞ F X n ( x ) = F X ( x ) eg Example 5.5.11: maximum of uniforms. Weak Law of Large Numbers (in probability) lim n →∞ P ( | ¯ X n - μ | < ) = 1 in other words ¯ X n converges in probability to μ . Strong Law of Large Numbers (almost sure) P ( lim n →∞ | ¯ X n - μ | < ) = 1 in other words ¯ X n converges almost surely to μ . Theorem 5.5.14: Convergence to a constant The sequence of random variables X
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Unformatted text preview: X 1 ,X 2 ,... convergence in probability to a constant μ iff the sequence also converges in distribution to μ . Central Limit Theorem Let X 1 ,X 2 ... with existing MGF, with EX = μ and V ar ( X ) = σ 2 > 0. Let G n ( x ) denote the cdf of Z = √ n ( ¯ X n-μ ) σ . Then lim n →∞ G n ( x ) = Z x-∞ 1 √ 2 π e-y 2 / 2 dy, in other words Z has limiting N (0 , 1) distribution. 1 The CLT is actually also true for ANY distribution with EX = μ and 0 < V ar ( X ) < ∞ . Example 5.5.16: Normal approx to NegBin Slutsky’s Theorem 5.5.17 If X n → X in distribution and Y n → a , a constant, in probability, then • Y n X n → aX in distribution • X n + Y n → X + a in distribution Example 5.5.18: Normal approximation with estimated variance 2...
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