2 - NOTES 2 Stochastic Process and Time Series Definition A...

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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: In this course, we 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...
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2 - NOTES 2 Stochastic Process and Time Series Definition A...

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