🚀 go-pugleaf

You want a long series with a given autocorrelation structure.

Low order ARMA models are used to model the autocorrelation 
structure of weakly stationary processes. 

(1- a_1L - a_2L^2 - ... - a_pL^p) x_t =
        (1 - b_1L - b_2L^2 - ... - b_qL^q) u_t

u_t ... white noise

If you reverse the procedure and start with a special choice 
of the parameters you obtain a certain covariance structure. 
Cooose the parameters so that you end up with the desired C matrix. 
You find the formula in any times series book.

MA(q) processes have a C matrix with a (2q+1)-band structure,
AR(1) processes an infinte matrix with geometrically decreasing 
coefficients (c_i.) .
The actual generation by

x_t = a_1 x_(t-1) + ... + a_p x_(t-p) - 
           - u_t - b_1 u_(t-1) - ... - b_q u_(t-q)

needs starting values for x_(t-i) ,i=1,...,p  and some u_(t-j) .
The latter are white noise and no problem.

The effect of the x`s, however, should be neglegible.  
How quickly this starting value effect dies out depends on the 
roots of the AR polynomial:
roots close to the unit circle take a long time,
roots far away of the unit circle ( |z| >> 1 ) take only a short time.

E.g. AR(1): corr( x_t,x_(t-i) )= a_1^i  
( 0.1^10 = very small, 0.9^10=0.349 ! )

Remark:
For MA-type correlation structures there is no need to factorize 
the band matrix this method is equally exact.
 
Best wishes Michael

email: hauser@wu-wien.ac.at