  We study a series of one-dimensional discrete-time quantum-walk models
labeled by half integers $j=1/2, 1, 3/2, ...$, introduced by Miyazaki {\it et
al.}, each of which the walker's wave function has $2j+1$ components and
hopping range at each time step is $2j$. In long-time limit the density
functions of pseudovelocity-distributions are generally given by superposition
of appropriately scaled Konno's density function. Since Konno's density
function has a finite open support and it diverges at the boundaries of
support, limit distribution of pseudovelocities in the $(2j+1)$-component model
can have $2j+1$ pikes, when $2j+1$ is even. When $j$ becomes very large,
however, we found that these pikes vanish and a universal and monotone convex
structure appears around the origin in limit distributions. We discuss a
possible route from quantum walks to classical diffusion associated with the $j
\to \infty$ limit.
