Subsampling estimates of the Lasso distribution.

2.3 A continous mapping theorem for argmin functionals 19
and
E ∗ (f( ˜X n )) = E ∗ (f(X n )) (2.3.0.7)
for every f ∈ C b (B loc (R p )) and every n ∈ N. According to the same theorem, we can
assume that the maps ˜X n take the form
˜X n = X n ◦ φ n (2.3.0.8)
for measurable perfect maps φ n : ˜Ω → Ω satisfying P n = ˜P ◦ φ −1
n .
Next, consider the maps ˜t n , ˜t and ˜α n obtained after composittion with φ n . Denote by ˜t
the unique minimizer of ˜X, using a density argument, one can show that ˜t is measurable,
hence has the same law as arg min(X). We need to prove that
E ∗ f(t n ) − Ẽf(˜t) → 0 (2.3.0.9)
for every f ∈ C b (R p ). By perfectness of the maps φ n , the difference is equal to
Ẽ ∗ ( f(˜t n ) − f(˜t) ) ≤ Ẽ∗ (∣∣ f( ˜ t n ) − f(˜t) ∣ ∣ ) .
To prove 2.3.0.9, first define for abitrary δ > 0
{ }
˜∆(δ) = inf ˜X(t) | ‖t − ˜t‖ > δ − ˜X(˜t).
Then, for an arbitrary ε > 0, use tightness of {˜t n } n and uniqueness of ˜t to find R > 0,
η > 0 satisfying :
lim sup
n→∞
˜P ∗ ( ‖˜t n ‖ > R ) < ε
and
˜P
(
‖˜t‖ > R or ˜∆(δ)
)
< η < ε.
The restriction of f on the closed ball B(0; R) is uniformly continuous by Heine-Borel’s
Theorem. Since
E ∗ (f(˜t n ) − f(˜t)) ≤ E ∗ (f(˜t n ) − f(˜t))1 { ‖˜t n ‖, ‖˜t‖ ≤ R } + 2 · 2‖f‖ |B(0;R) ε
for n sufficiently large, it is sufficent to show that
˜P ∗ ( ‖˜t n − ˜t‖ > δ; ‖˜t n ‖, ‖˜t‖ ≤ R ) → 0
for each δ > 0.
According to the Representation theorem, it holds that
2 −R sup
‖t‖≤R
(
‖ ˜X n (t) − ˜X(t)‖
) ∗ (
≤ d( ˜X ˜X)) ∗
n , ≤ ε,
for n sufficently large. Furthermore, it follows from
˜X n (˜t) = ˜X n (t) + ˜X(t) − ˜X n (t)
+ ˜X(˜t) − ˜X n (˜t)
+ ˜X(˜t) − ˜X(t)