We analyze the long-time behavior of discrete-time quantum walks on the integers with space- and time-homogeneous unitary evolution. Using a matrix-function representation, we study the pointwise convergence of characteristic functions under ballistic rescaling. By combining explicit bounds on this convergence with a modified version of a classical estimate due to Zolotarev, we obtain rates of convergence for the associated position distributions, measured in terms of the Levy metric. Our results apply to the general class of unitary evolutions whose Fourier transforms are analytic, and provide quantitative insight into how fast the walk approaches its asymptotic distribution. As an application we combine these estimates with a stationary phase analysis for the finite time wavefront amplitudes for the Hadamard Walk to show an analogue to the classical Berry-Esseen theorem.