  One-parameter family of discrete-time quantum-walk models on the square
lattice, which includes the Grover-walk model as a special case, is
analytically studied. Convergence in the long-time limit $t \to \infty$ of all
joint moments of two components of walker's pseudovelocity, $X_t/t$ and
$Y_t/t$, is proved and the probability density of limit distribution is
derived. Dependence of the two-dimensional limit density function on the
parameter of quantum coin and initial four-component qudit of quantum walker is
determined. Symmetry of limit distribution on a plane and localization around
the origin are completely controlled. Comparison with numerical results of
direct computer-simulations is also shown.
