  Quantum walks subject to decoherence generically suffer the loss of their
genuine quantum feature, a quadratically faster spreading compared to classical
random walks. This intuitive statement has been verified analytically for
certain models and is also supported by numerical studies of a variety of
examples. In this paper we analyze the long-time behavior of a particular class
of decoherent quantum walks, which, to the best of our knowledge, was only
studied at the level of numerical simulations before. We consider a local coin
operation which is randomly and independently chosen for each time step and
each lattice site and prove that, under rather mild conditions, this leads to
classical behavior: With the same scaling as needed for a classical diffusion
the position distribution converges to a Gaussian, which is independent of the
initial state. Our method is based on non-degenerate perturbation theory and
yields an explicit expression for the covariance matrix of the asymptotic
Gaussian in terms of the randomness parameters.
