  We study "the Wojcik model" which is a discrete-time quantum walk (QW) with
one defect in one dimension, introduced by Wojcik et al.. For the Wojcik model,
we give the weak convergence theorem describing the ballistic behavior of the
walker in the probability distribution in a rescaled position-space. In our
previous studies, we obtained the time-averaged limit and stationary measures
concerning localization for the Wojcik model. As a result, we get the
mathematical expression of the whole picture of the behavior of the walker for
the Wojcik model. Here the coexistence of localization and the ballistic
spreading is one of the peculiar properties of one-dimensional QWs with one
defect. Due to the coexistence, it has been strongly expected to utilize QWs to
quantum search algorithms. In order to derive the weak convergence theorem, we
take advantage of the generating function method. We emphasize that the
time-averaged limit measure is symmetric for the origin, however, the weight
function in the weak limit measure is asymmetric in general, which implies that
the weak convergence theorem represents the asymmetry of the probability
distribution. Furthermore, the weak limit measure heavily depends on the phase
of the defect and initial state of the walker. Comparing with our previous
studies, we also show some numerical results of the probability distribution to
confirm that our result is relevant mathematically, and consider the effect of
changing the phase and initial coin state on the probability distribution, or
the ballistic spreading, which is one of the motivations of our study.
