Subsampling estimates of the Lasso distribution.

20 Minimizers of convex processes
that
{
‖˜t‖, ‖t‖ ≤ R; ‖t − ˜t‖ > δ; ˜∆(δ)
} {
≤ η ⊆ ˜Xn (˜t) ≤ ˜X
}
n (t) + 2 R+1 ε − η . (2.3.0.10)
The event
{
}
η > ˜α n + 2 R+1 ε
has probability tending to one as n → ∞, since ˜α n = o ˜P
(1). Note that by perfectness of
the maps φ n , one also has
˜X n (˜t n ) ≤ inf
t
˜X n (t) + ˜α n
for every n ∈ N. Thus, 2.3.0.10 forces ˜t n , either to be at a distance greater than R from
zero, or to stay within a δ neighborhood of ˜t. We finally obtain
˜P ∗ ( ‖˜t n − ˜t‖ > δ; ‖˜t n ‖, ‖˜t‖ ≤ R ) ≤ 3ε
for n large enough. This completes the proof.