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Unformatted text preview: NOTES 2 Stochastic Process and Time Series : Definition : A stochastic process is a family of random variables { Z t , t ∈ T } . T is thought of as representing time. If T is an interval, then the process { Z t , t ∈ T } is said to be continuous. If T is discrete (for examples, T = { 1 , 2 , ... } or T = { , ± 1 , ± 2 , ... } ), then the process { Z t , t ∈ T } is said to be discrete. Definition : A time series { Z 1 , Z 2 , . . . , Z N } of N successive observations is regarded as a sample realization from an infinite population of such time series that could have been generated by the stochastic process. Note: We do assume the observations are equally spaced in time. Definition : The mean function of a stochastic process { . . . , Z 1 , Z , Z 1 , Z 2 , . . . } is defined by μ t = E ( Z t ) , t = 0 ± 1 , ± 2 , . . . . Definition : The autocovariance function of { . . . , Z 1 , Z , Z 1 , Z 2 , . . . } is defined by γ t,s = Cov ( Z t , Z s ) , t, s = 0 ± 1 , ± 2 , . . . . where Cov ( Z t , Z s ) = E [( Z t μ t )( Z s μ s )] = E ( Z t Z s ) μ t μ s . Definition : The autocorrelation function (denoted by a.c.f.) of { . . . , Z 1 , Z , Z 1 , Z 2 , . . . } is defined by ρ t,s = Corr ( Z t , Z s ) = γ t,s √ γ t,t γ s,s , t, s...
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This note was uploaded on 01/20/2012 for the course STA 4005 taught by Professor ? during the Spring '08 term at CUHK.
 Spring '08
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