  It is well established that gene expression can be modeled as a Markovian
stochastic process and hence proper observables might be subjected to large
fluctuations and rare events. Since dynamics is often more than statics, one
can work with ensembles of trajectories for long but fixed times, instead of
states or configurations, to study dynamics of these Markovian stochastic
processes and glean more information. In this paper we aim to show that the
concept of ensemble of trajectories can be applied to a variety of stochastic
models of gene expression ranging from a simple birth-death process to a more
sophisticate model containing burst and switch. By considering the protein
numbers as a relevant dynamical observable, apart from asymptotic behavior of
remote tails of probability distribution, generating function for the cumulants
of this observable can also be obtained. We discuss the unconditional
stochastic Markov processes which generate the statistics of rare events in
these models.
