  Recently, Freud-Rodriguez proposed a new counting process which is called the
Bell-Touchard process and based on the Bell-Touchard probability distribution.
This process was developed to solve the problem of rare events hypothesis which
is one of the limitations of the Poisson process. In this paper, we consider
the discrete degenerate Bell distributions and the degenerate Bell process
which are 'degenerate versions' of the Bell-Touchard probability distributions
and the Bell-Touchard process, respectively. We investigate several properties
of the degenerate Bell distribution. We introduce the degenerate Bell process
by giving two equivalent definitions and show one method of constructing a new
infinite family of degenerate Bell process out of a given infinite family of
degenerate Bell process.
