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 $$\Gamma $$ Γ -convergence approach. We consider the case in which the regularization coefficient scales with the number of observations, n, as $$\lambda _n=n^{-p}$$ λ n = n - p . Using standard theorems from the $$\Gamma $$ 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 $$p\le \frac{1}{2}$$ p ≤ 1 2 . Without further assumptions, we show this rate is sharp. This differs from rates for strong convergence using Hilbert scales where one can often choose $$p>\frac{1}{2}$$ p > 1 2 .