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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online