# chap2 - Stationary Processes (Chapter 2) 2.1 Basic...

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1 { X t } stationary time series Mean: µ = Ε X t for all t. ACVF: γ( h ) = cov( X t+h , X t ), h=0, +1, . . . ACF: ρ( h )= 2.1 Basic Properties Stationary Processes (Chapter 2) γ γ () h 0 2 Problem: predict X n+h from X n . (Assume { X t } is a stationary Gaussian time series. Soln: X n+h | X n has a normal distribution with conditional mean: E( X n+h | X n ) = Ε X n+h + ( X n - Ε X n ), = µ + ρ( h)( X n - µ) Var( X n ) DEFINITION. { X t } is a Gaussian time series if all of its joint distributions are multivariate normal, i.e. ( X 1 , . . ., X n ) is multivarite normal for all n. Cov( X n+h , X n )

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3 conditional variance: Var( X n+h | X n ) = γ( 0)(1 - ρ 2 ( h)). The “best” mean square predictor of X n+h in terms of X n is then µ + ρ( h)( X n - µ) Remark: For Gaussian time series, best MS predictor = best linear predictor 4 Result: γ( ) is the ACVF of a stationary time series iff (i) γ( ) is an even function (ii) γ( ) is non-negative definite (nnd) a i γ( i-j) a j > 0 for all n . (i) γ( 0) 0, (ii) | γ( h)| γ( 0) , (iii) γ( h) = γ(− h) , ( γ( ) is an even function) Basic Properties of γ (h)=Cov( X t+h , X t ) : ij n , = 1
5 Ex. Is γ( h):= cos ( ω h) an ACVF? Ans. Yes. Verification of nnd property is difficult. Easier to check that cos ( ω h) is the ACVF of X t = A cos ( ω h) + B sin ( ω h) where A & B are uncorrelated (0,1) rv’s. Ex 2.1.1. The function γ( h) := is an ACVF iff |ρ| < .5 1, if h = 0, ρ, if h = +1, 0, otherwise.

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## This note was uploaded on 06/16/2009 for the course STAT 525 taught by Professor Brockwell during the Spring '09 term at Colorado State.

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chap2 - Stationary Processes (Chapter 2) 2.1 Basic...

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