Subsampling estimates of the Lasso distribution.

2.3 A continous mapping theorem for argmin functionals 15
(iii) A bounded convex function x on K ε is automatically Lipschitz on K with Lipschitz
constant (2/ε)‖x‖ K ε.
Indeed, for arbitrary y 1 , y 2 ∈ K, set z = y 1 + η(y 1 − y 2 )/‖y 1 − y 2 ‖ with η < ε. Then
z ∈ K ε and y 1 = λz + (1 − λ)y 2 , λ = ‖y 1 − y 2 ‖/(‖y 1 − y 2 ‖ + η) ∈ [0, 1]. By convexity
of x, we have x(y 1 ) − x(y 2 ) ≤ λ(f(z) − f(y 2 )) ≤ 2λ‖x‖ K ε. Since z ∈ K ε and x ∈ K
it follows from the triangle inequality that λ ≤ 1/ε.
(iv) We can now show that {X n } n is asymptotically equicontinuous in l ∞ (K).
For arbitrary ε equi > 0, η equi > 0, choose M > 0 with
lim sup P (‖X n ‖ K ε > M) < η equi
n→∞
by invoking asymptotical tightness of ‖X n ‖ K ε. On {‖X n ‖ K ε ≤ M} it holds that
|X n (t) − X n (s)| ≤ (2/ε)M‖t − s‖. Set δ = ε equi ((2/ε)M) −1 . Then,
{
}
sup |X n (t) − X n (s)| > ε equi ⊆ {‖X n ‖ K ε > M} .
‖t−s‖