Stationary Stochastic Process Process Lecture 12

Stationary Stochastic Process Process 

Lecture 12

Stationary Stochastic Process

A stochastic process is said to be stationary if its joint distribution does not change by the change over time or in state space. Thus, the statistical characteristics, if they exist, do not change over time or position.

Let the joint distribution of Xt1, Xt2, ..., Xtn and Xt1+τ, Xt2+τ, ..., Xtn+τ be independent of all t1, t2, ..., tn and t1+τ, t2+τ, ..., tn+τ in all T.

That's

F(Xt1, X2t,⋯,Xtn) = F(Xt1+τ, Xt2+τ,⋯,Xt2+τ,⋯,Xtn+τ)

Or we can say the probability distribution is independent of the time increment/decrement.

 F(Xt1) = F(Xt1+τ)

Consider the time series random process given below:

Statistical Parameters of Stationary Stochastic Process

i. The mean of a stationary stochastic process (Xt) is constant over time.

Consider a stationary stochastic process Xt

The mean is constant with respect to time.

ii. The variance of a stationary stochastic process (Xt) is constant over time.

iii. The covariance depends only on the distance between the two periods and not on time. Let Xt1+i and Xt2+j be stochastic processes. then

Covariance (Xt+i, Xt+j) = Covariance (Xi, Xj)

iv. The autocorrelation depends on the gap of time, not on time.

ρXX(t,t + τ) = ρXX(τ)

Classification of Stationary Stochastic Process

The stationary stochastic process can be further divided into strongly and weakly stationary.

 Strictly (Strongly) Stationary Stochastic Process

(SSS)

If the joint distribution function of a time series stochastic process remains unchanged over time, it is called strongly or strictly stationary.

Let the joint distribution of Xt1, Xt2, ..., Xtn be the same as the joint distribution of the new set of shifted random variables Xt1+τ, Xt2+τ, ..., Xtn+τ for all t1, t2, ..., tn and τ > 0.

F(Xt1,Xt2,⋯,Xtn) = F(Xt1+τ, Xt2+τ,⋯,Xt2+τ,⋯,Xtn+τ)

In strictly stationary stochastic process

The first-order distribution function is invariant over time

F(Xt1) = F(Xt1+τ)

The second-order distribution function is invariant over time

F(Xt1,Xt2) = F(Xt1+τ, Xt2+τ)

The kth-order distribution function is invariant over time

F(Xt1,Xt2,⋯,Xtk) = F(Xt1+τ, Xt2+τ,⋯,Xt2+τ,⋯,Xtk+τ)

The stochastic process will be jointly stationary if

X(t1), X(t2), ..., X(ti), ..., X(tn) and Y(t1), Y(t2), ..., Y(tj), ..., Y(tn) are invariant w.r.t. location t = 0 for all i and j.

Fxi(ti)Yj(tj)(Xi,Yj)=Fxi(ti+τ)Yj(tj+τ)(Xi,Yj) 

Statistical Parameters of a Strictly Stationary Stochastic Process

i. The mean of the strictly stationary stochastic process is invariant of time and constant.

μX(t) = E(Xt) = μ

ii. The variance of the strictly stationary stochastic process is invariant of time and constant.

Var(Xt) = σ²

iii. The autocorrelation of the strictly stationary stochastic process is invariant by time.

iv. The autocovariance of a strictly stationary stochastic process depending on time shift but independent of time.

Weakly (Wide) Stationary Stochastic Process

(WSS)

A stochastic process will be a weak stationary stochastic process if it satisfies the properties of a strongly stochastic process for the first- and second-order cumulative distribution functions.

That’s

The first-order distribution function is invariant over time.

F(Xt1) = F(Xt1+τ) = F(X)

The second-order distribution function is invariant over time.

F(Xt1,Xt2) = F(Xt1+τ, Xt2+τ)

i. The mean of the weakly stationary stochastic process is constant for first- and second-order distribution functions .

μX(t) = E(Xt) = μ

ii. The variance of the weakly stationary stochastic process is constant for first- and second-order distribution functions.

Var(Xt) = σ²

iii. The autocorrelation of the weakly stationary stochastic process is invariant by time for first- and second-order distribution functions.

iv. The autocovariance of the weakly stationary stochastic process depends on the time shift but is independent of time for first- and second-order distribution functions.

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