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JohnNichols

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08-28-2020
02:34 PM

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Interesting problem to code in Fortran - for non-stationary process

Consider X1,X2 i.i.d. with P(Xi=1)=P(Xi=0)=0.5. Now define X3=X1+X2(mod 2)

Notice that X3 is independent from X1 and from X2 individually. However the three variables X1,X2,X3 are not jointly independent.

Now consider a sequence X1,X2,X3,…such that each triplet (X3n+1,X3n+2,X3n+3) has the same distribution as (X1,X2,X3) and all these triplets are independent.

With this one has that all Xn have the same distribution as X1, so that the process is 1 stationary. Also, all pairs Xi,Xj with i≠j are independent and hence have the same distribution (this gives 2 stationarity). However the triplet (X2,X3,X4) has a different distribution from (X1,X2,X3) (in fact, X2,X3 and X4 are jointly independent).

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