Subsampling estimates of the Lasso distribution.

8 Minimizers of convex processes
Remark. Weak convergence is also defined for convergence toward a Borel measurable
map X defined on a probability space (Ω, A, P ) by taking L := P ◦ X −1 , this is denoted
by X n X. Note that, while measurability of the maps X n has been dropped, it remains
mandatory for the limit X.
The Portmanteau theorem and the continuous mapping are essential tools in asymptotic
statistics. They remain valid in this general framework. Their proof are omitted but can
be found in the references.
Theorem 2.2.1.2. (Van der Vaart and Wellner, 1996, Theorem 1.3.4) Let {X n } n be a
sequence of arbitrary random maps and let L be a Borel measure. The following statements
are equivalent :
(i) X n L;
(ii) lim inf P ∗ (X n ∈ G) ≥ L(G) for every open G;
(iii) lim sup P ∗ (X n ∈ F ) ≤ L(F ) for every closed F;
(iv) lim P ∗ (X n ∈ B) = lim P ∗ (X n ∈ B) = L(B) for every Borel L-continuous set B, that
is B with L(∂B) = 0 ;
(v) lim inf E ∗ f(X n ) ≥ ∫ fdL for every bonded, Lipschitz continuous, nonnegative function
f.
Theorem 2.2.1.3. (Van der Vaart and Wellner, 1996, Theorem 1.3.6) Let g : D → E be
continous at every point in a set D 0 ⊂ D. If X n X and X takes its values in D 0 , then
g(X n ) g(X).
Definition 2.2.1.4. A Borel probability measure L is called tight if for every ε > 0, there
exists a compact set K ⊂ D with L(K) ≥ 1 − ε. A Borel measurable map X : Ω → D is
called tight if its law P ◦ X −1 is tight.
Next, we want to state Prohorov’s theorem which is a fundamental result in weak convergence
theory, it associates sequential compactness (with respect to weak convergence) to
the concept of tightness, hence giving conditions for the existence of a weak limit. In the
present setting, the statement of the result requires introducing the concept of asymptotic
measurability first.
Definition 2.2.1.5. (Asymptotic measurability and tightness) A sequence of arbitrary
random maps {X n : Ω n → D} n is asymptotically measurable if and only if
E ∗ f(X n ) − E ∗ f(X n ) → 0,
for every f ∈ C b (D).
The sequence {X n } n is called asymptotically tight if for every ε > 0, there exists a compact
set K such that
lim inf
n→∞ P ∗(X n ∈ K δ ) ≥ 1 − ε, for every δ > 0.
Here, K δ = {y ∈ D | d(x, K) < δ} is the δ-enlargement around K, an open set.
Remark. One can show that for a sequence {X n } n of Borel measurable maps, asymptotic
tightness and the usual uniform tighness, that is, the existence for every ε > 0 of a compact
set K ⊂ D satisfying P (X n ∈ K) ≥ 1 − ε for every n ∈ N, are equivalent.