  This paper is composed of two main results concerning chains of infinite
order which are not necessarily continuous. The first one is a decomposition of
the transition probability kernel as a countable mixture of unbounded
probabilistic context trees. This decomposition is used to design a simulation
algorithm which works as a combination of the algorithms given by Comets et al.
(2002) and Gallo (2009). The second main result gives sufficient conditions on
the kernel for this algorithm to stop after an almost surely finite number of
steps. Direct consequences of this last result are existence and uniqueness of
the stationary chain compatible with the kernel.
