Asymptotic theory for the regularized system identification has received increasing interests in recent years. In this paper, for the finite impulse response (FIR) model and filtered white noise inputs, we show the convergence in distribution of the Steins unbiased risk estimator (SURE) based hyper-parameter estimator and find factors that influence its convergence properties. In particular, we consider the ridge regression case to obtain closed-form expressions of the limit of the regression matrix and the variance of the limiting distribution of the SURE based hyper-parameter estimator, and then demonstrate their relation numerically.