  A prevalent issue in practical applications of quantum key distribution (QKD)
is the emergence of correlations among the emitted signals. Although recent
works have proved the security of QKD in the presence of this imperfection,
they rest on the premise that pulse correlations are of finite length. However,
this assumption is not necessarily met in practice, since the length of these
correlations could be potentially unbounded. Indeed, the first emitted pulse
could be correlated with the last one, even if very faintly. Still,
intuitively, there should exist a pulse separation threshold after which these
correlations become so small as to be essentially negligible, rendering them
inconsequential from a security standpoint. Building on this insight, we
introduce a general formalism designed to extend existing security proofs to
the practically relevant scenario in which pulse correlations have an unbounded
length. This approach significantly enhances the applicability of these proofs
and the robustness of QKD's implementation security.
