Pointwise Convergence in Probability of General Smoothing Splines

Establishing the convergence of splines can be cast as a variational problem which is amenable to a Γ-convergence approach. We consider the case in which the regularization coefficient scales with the number of observations, n, as λ_n=n^p. Using standard theorems from the Γ-convergence literature, we prove that the general spline model is consistent in that estimators converge in a sense slightly weaker than weak convergence in probability for p≤12. Without further assumptions, we show this rate is sharp. This differs from rates for strong convergence using Hilbert scales where one can often choose p>12.