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# notes2 - NOTES 2 Stochastic Process and Time Series...

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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 = { 0 , ± 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 0 , Z 1 , Z 2 , . . . } is defined by μ t = E ( Z t ) , t = 0 ± 1 , ± 2 , . . . . Definition : The autocovariance function of { . . . , Z - 1 , Z 0 , 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 0 , 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 = 0 ± 1 , ± 2 , . . . . Some properties : 1. V ar ( Z t ) = γ t,t 2. γ t,s = γ s,t , ρ t,s = ρ s,t 3. | γ t,s | ≤ γ t,t γ s,s , | ρ t,s | ≤ 1 4.

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notes2 - NOTES 2 Stochastic Process and Time Series...

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