13convg - EECS 501 CONVERGENCE OF SEQUENCES OF RVs Fall...

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EECS 501 CONVERGENCE OF SEQUENCES OF RVs Fall 2001 Recall: lim n →∞ x n = x For any ² > 0 , N suchthat | x n - x | < ² n > N . Given: A sequence of random variables { x 1 ,x 2 ... } . Need not be iidrv. DEF: x n x in probability lim n →∞ Pr [ | x n - x | > ² ] = 0 stochastic convergence . EX1: If { x n } iidrv, then sample mean ˆ M n = 1 n n i =1 x i E [ x ] in prob. Proof: See Estimators handout. Requires both E [ x ] and σ 2 x to be finite. Note: This is weak law of large numbers , since convergence in prob. is weak . EX2: If { x n } iidrv, f x i = 1 A , 0 < X < A , then max[ x 1 ...x n ] A in prob. Note: Each of these shows consistency of an estimator (#1 of prob. set #7). DEF: x n x in mean square lim n →∞ E [( x n - x ) 2 ] = 0 L . I . M . n →∞ x n = x . EX: If { x n } iidrv, then ˆ M n E [ x ] in mean square=in quadratic mean. Note:
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13convg - EECS 501 CONVERGENCE OF SEQUENCES OF RVs Fall...

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