  We study the long time behaviour of a Markov process evolving in $\mathbb{N}$
and conditioned not to hit 0. Assuming that the process comes back quickly from
infinity, we prove that the process admits a unique quasi-stationary
distribution (in particular, the distribution of the conditioned process admits
a limit when time goes to infinity). Moreover, we prove that the distribution
of the process converges exponentially fast in total variation norm to its
quasi-stationary distribution and we provide an explicit rate of convergence.
  As a first application of our result, we bring a new insight on the speed of
convergence to the quasi-stationary distribution for birth and death processes:
we prove that these processes converge exponentially fast to a quasi-stationary
distribution if and only if they have a unique quasi-stationary distribution.
Also, considering the lack of results on quasi-stationary distributions for
non-irreducible processes on countable spaces, we show, as a second application
of our result, the existence and uniqueness of a quasi-stationary distribution
for a class of possibly non-irreducible processes.
