  We prove an extended Granger-Johansen representation theorem (GJRT) for
finite or infinite order integrated autoregressive time series on Banach space.
We assume only that the resolvent of the autoregressive polynomial for the
series is analytic on and inside the unit circle except for an isolated
singularity at unity. If the singularity is a pole of finite order the time
series is integrated of the same order. If the singularity is an essential
singularity the time series is integrated of order infinity. When there is no
deterministic forcing the value of the series at each time is the sum of an
almost surely convergent stochastic trend, a deterministic term depending on
the initial conditions and a finite sum of embedded white noise terms in the
prior observations. This is the extended GJRT. In each case the original series
is the sum of two separate autoregressive time series on complementary
subspaces--a singular component which is integrated of the same order as the
original series and a regular component which is not integrated. The extended
GJRT applies to all integrated autoregressive processes irrespective of the
spatial dimension, the number of stochastic trends and cointegrating relations
in the system, and the order of integration.
