  This paper defines the class of $\mathcal{H}$-valued autoregressive (AR)
processes with a unit root of finite type, where $\mathcal{H}$ is an infinite
dimensional separable Hilbert space, and derives a generalization of the
Granger-Johansen Representation Theorem valid for any integration order
$d=1,2,\dots$. An existence theorem shows that the solution of an AR with a
unit root of finite type is necessarily integrated of some finite integer $d$
and displays a common trends representation with a finite number of common
stochastic trends of the type of (cumulated) bilateral random walks and an
infinite dimensional cointegrating space. A characterization theorem clarifies
the connections between the structure of the AR operators and $(i)$ the order
of integration, $(ii)$ the structure of the attractor space and the
cointegrating space, $(iii)$ the expression of the cointegrating relations, and
$(iv)$ the Triangular representation of the process. Except for the fact that
the number of cointegrating relations that are integrated of order 0 is
infinite, the representation of $\mathcal{H}$-valued ARs with a unit root of
finite type coincides with that of usual finite dimensional VARs, which
corresponds to the special case $\mathcal{H}=\mathbb{R}^p$.
