Asymptotic behavior of the geometric sequence U n + 1 = Q U n in a probability problemWe know the asymptotic behavior of the scalar geometric sequence u n + 1 = q u n with respect to the values of q. That is, | q | &lt; 1 implies lim n u n = 0, q = 1 implies u n is constant, etc.Now I am just asking myself the same question about the geometric sequence U n + 1 = Q U n , where Q is a 2 × 2 matrix and U n is a two dimensional vector.

Question

Asymptotic behavior of the geometric sequence       U          n      +      1        =  Q      U    n   in a probability problemWe know the asymptotic behavior of the scalar geometric sequence       u          n      +      1        =  q      u    n   with respect to the values of q. That is,       |    q      |    &amp;lt;  1 implies       lim          n            u    n    =  0,   q  =  1 implies       u    n   is constant, etc.Now I am just asking myself the same question about the geometric sequence       U          n      +      1        =  Q      U    n  , where Q is a   2  ×  2 matrix and       U    n   is a two dimensional vector.

Tamoni5e · Accepted Answer

Step 1If matrix and vector norms are compatible in the way that   ‖  Q  x  ‖  ≤  ‖  Q  ‖  ⋅  ‖  x  ‖  , then   ‖  Q  ‖  &amp;lt;  1 implies       Q    n    x  →  0 for all x.Step 2If   ‖  Q  ‖  &amp;gt;  1 then Q could be the identity or a rotation matrix, with the obvious consequences on convergence of       Q    n    x.If   ‖  Q  ‖  &amp;gt;  1 then       Q    n    x is not guaranteed to be unbounded. Take   Q  =      (                            2                          0                                      1                          0                      )   and   x  =      (                            1                                      0                      )   or   x  =      (                            0                                      1                      )