22ergodic - EECS 501 RESULTS SUMMARY: ERGODICITY Fall 2001...

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ISSUE: Let { x i ,i = 1 , 2 ... } be a sequence of id rvs. Does the sample mean M n = 1 n n i =1 x i converge to the ensemble mean E [ x i ] = μ , and in what sense? ”id”=”identically distributed”; assume E [ x i ] 2 x i < . 1. Weak Law of Large Numbers: { x i } are independent ( M n μ in probability ). PROOF: Lecture in Oct.; ”Convergence of RVs” handout. Equivalent to: ” M n is a weakly consistent estimator of μ .” 2. Mean Ergodic Theorem: { x i } are independent ( M n μ in mean square ): PROOF: ”Convergence of RVs” handout. L.I.M. n →∞ M n = μ . 3. Strong Law of Large Numbers: { x i } are independent ( M n μ with probability one ). PROOF: See ”Strong Law of Large Numbers” handout. 4. { x i } have finite correlation length : K x ( i,j ) = 0 if | i - j | > M for some M < ( M n μ in probability ). PROOF: Exam #2, Fall 1998. 5. { x i } has LIM n →∞ 1 n n i =1 K x ( i,n ) = 0. ( M n μ in mean square ): L.I.M. n →∞ M n = μ . PROOF: Problem Set #8 (adapted to a nonzero mean μ ). 6. { x i } asymptotically uncorrelated: LIM | n |→∞ K x ( n ) = 0. ( M n μ in mean square ): L.I.M. n
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22ergodic - EECS 501 RESULTS SUMMARY: ERGODICITY Fall 2001...

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