  We consider a discrete-time quantum walk $W_{t,\kappa}$ at time $t$ on a
graph with joined half lines $\mathbb{J}_\kappa$, which is composed of $\kappa$
half lines with the same origin. Our analysis is based on a reduction of the
walk on a half line. The idea plays an important role to analyze the walks on
some class of graphs with \textit{symmetric} initial states. In this paper, we
introduce a quantum walk with an enlarged basis and show that $W_{t,\kappa}$
can be reduced to the walk on a half line even if the initial state is
\textit{asymmetric}. For $W_{t,\kappa}$, we obtain two types of limit theorems.
The first one is an asymptotic behavior of $W_{t,\kappa}$ which corresponds to
localization. For some conditions, we find that the asymptotic behavior
oscillates. The second one is the weak convergence theorem for $W_{t,\kappa}$.
On each half line, $W_{t,\kappa}$ converges to a density function like the case
of the one-dimensional lattice with a scaling order of $t$. The results contain
the cases of quantum walks starting from the general initial state on a half
line with the general coin and homogeneous trees with the Grover coin.
