  In this paper, we define a new type of decoherent quantum random walks with
parameter $0\le p\le 1$, which becomes a unitary quantum random walk (UQRW)
when $p=0$ and an open quantum random walk (OPRW) when $p=1$ respectively. We
call this process a partially open quantum random walk (POQRW). We study the
limiting distribution of a POQRW on $Z^1$ subject to decoherence on coins with
$n$ degrees of freedom, which converges to a convex combination of normal
distributions if the superoperator $\mathcal{L}_{kk}$ satisfies the eigenvalue
condition, that is, 1 is an eigenvalue of $\mathcal{L}_{kk}$ with multiplicity
one and all other eigenvalues have absolute values less than 1. A
Perron-Frobenius type of theorem is provided in determining whether or not the
superoperator satisfies the eigenvalue condition. Moreover, we compute the
limiting distributions of characteristic equations of the position probability
functions for $n=2$ and 3.
