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[TESTS] – Equations

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Adding Equations

 

P(Xt1x1,Xt2x2,,Xtkxk)=F(xt1,xt2,,xtk)=F(xh+t1,xh+t2,,xh+tk)=P(Xh+t1x1,Xh+t2x2,,Xh+tkxk) Xt

A stationary process 

{Xt,tN}

is said to be strictly or strongly stationary if its statistical distributions remain unchanged after a shift o the time scale. Since the distributions of a stochastic process are defined by the finite-dimensional distribution functions, we can formulate an alternative definition of strict stationarity. If in every n

n

, every choice of times t1,t2,,tnN

t1,t2,,tnN

and every time lag k

k

such that ti+kN

ti+kN

, the n

n

-dimensional random vector (Xt1+k,Xt2+k,,Xtn+k)

(Xt1+k,Xt2+k,,Xtn+k)

has the same distribution as the vector (Xt1,Xt2,,Xtn)

(Xt1,Xt2,,Xtn)

, then the process is strictly stationary. That is for h

h

and xi

xi

 

P(Xt1x1,Xt2x2,,Xtkxk)=F(xt1,xt2,,xtk)=F(xh+t1,xh+t2,,xh+tk)=P(Xh+t1x1,Xh+t2x2,,Xh+tkxk)

P(Xt1x1,Xt2x2,,Xtkxk)=F(xt1,xt2,,xtk)=F(xh+t1,xh+t2,,xh+tk)=P(Xh+t1x1,Xh+t2x2,,Xh+tkxk)

 

for any time shift h

h

and observation xj

xj

. If {Xt,tN}

{Xt,tN}

is strictly stationary, then the marginal distribution of Xt

Xt

is independent of t

t

. Also,the two-dimensional distributions of (Xt1,Xt2)

(Xt1,Xt2)

are independent of the absolute location of t1

t1

and t2

t2

, only the distance t1t2

t1t2

matters. As a consequence, the mean function E(X)

E(X)

is constant, and the covariance Cov(Xt,Xtk)

Cov(Xt,Xtk)

is a function of k

k

only, not of the absolute location of k

k

and t

t

. At higher order moments, like the third order moment, E[XuXtXv]

E[XuXtXv]

remains unchanged if one add a constant time shift to s,t,u

s,t,u

 

 

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