A rate of convergence for the arcsine law by Stein's method

Using Stein’s method for the Beta distributions and a recent technique by Goldstein and Reinert of comparing the Stein characterization of the target distribution with that of the approximating distribution we prove a rate of convergence in the classical arcsine law, which states that the distribution of the relative time spent positive by a symmetric random walk on $\mathbb{Z}$ converges weakly to the arcsine distribution on $[0,1]$.