  We consider 2-state quantum walks (QWs) on the line, which are defined by two
matrices. One of the matrices operates the walk at only half-time. In the usual
QWs, localization does not occur at all. However, our walk can be localized
around the origin. In this paper, we present two limit theorems, that is, one
is a stationary distribution and the other is a convergence theorem in
distribution.
