If no characteristic roots share the same value, the solution of the homogeneous linear difference equation x _ t = a _ 1x _ { t-1 } + \ cdots + a _ nx _ { t-n } can be written in terms of the characteristic roots as
12.
In the solution equation x _ t = c _ 1 \ lambda _ 1 ^ t + \ cdots + c _ n \ lambda _ n ^ t, a term with real characteristic roots converges to 0 as " t " grows indefinitely large if the absolute value of the characteristic root is less than 1.
13.
In the solution equation x _ t = c _ 1 \ lambda _ 1 ^ t + \ cdots + c _ n \ lambda _ n ^ t, a term with real characteristic roots converges to 0 as " t " grows indefinitely large if the absolute value of the characteristic root is less than 1.
14.
A pair of terms with complex conjugate characteristic roots will converge to 0 with dampening fluctuations if the absolute value of the modulus " M " of the roots is less than 1; if the modulus equals 1 then constant amplitude fluctuations in the combined terms will persist; and if the modulus is greater than 1, the combined terms will show fluctuations of ever-increasing magnitude.
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