Subsampling estimates of the Lasso distribution.

18 Minimizers of convex processes
(v) We show that ˜X n → au ˜X∞ . It follows from the construction of the maps ˜X n =
X n ◦ φ n and of the balls B (εn)
i
that d( ˜X n , ˜X ∞ ) on {˜ω : X ∞ /∈ B o
(εn) , ξ ≤ 1 − ε}. Set
A k = ∪ m≥k {˜ω : X ∞ ∈ B 1/m
0 or ξ > 1 − 1/k}. Note that ˜P (A k ) ≤ 1/k, so for every
ε > 0 there exists some k with ˜P (A k ) ≤ ε. For ˜ω ∈ ˜Ω\A k and n such that ε n ≤ 1/k,
it holds that d( ˜X n , ˜X ∞ ) ≤ ε n . This completes the argument.
(vi) Let T : Ω n → R be bounded, and let T ∗ be its minimal measurable cover with respect
to P n . Write
k
∑ εn
T ◦ φ n = 1 πξ ≤1−ε n
T |A n
i
i=0
◦ π n,i 1 X∞◦π ∞∈B (εn)
i
+ 1 πξ >1−ε n
T ◦ π n .
The coordinate projections are perfect so that the minimal measurable cover of (T ◦
φ n ) with respect to ˜P can be computed as follows
k
(T ◦ φ n ) ∗ ∑ εn
= 1 πξ ≤1−ε n
(T |A n
i
◦ π n,i ) ∗ 1 X∞◦π∞∈B (εn) + 1 πξ >1−ε n
(T ◦ π n ) ∗
i=0
k
∑ εn
= 1 πξ ≤1−ε n
(T |A n
i
) ∗Pn[·|An i ] ◦ π n,i 1 X∞◦π∞∈B (εn) + 1 πξ >1−ε n
(T ) ∗µn ◦ π n
i=0
Now note that P n and P n (·|A n i ) are equivalent on (An i , A n∩A n i ), P n and µ n are equivalent
on (Ω n , A n ), hence the measurable covers in the last formula can be computed
under P n . It follows that (T ◦ φ n ) ∗ = T ∗ ◦ φ n . This completes the proof.
i
i
In the following, denote by B loc (R p ) the space of locally bounded functions on R p equipped
with the topology of uniform convergence on compacta; denote by C min (R p ) the separable
subset of continuous functions z(·) with minimum achieved at a unique point in R p and
satisfying z(t) → ∞ as |t| → ∞.
Dudley’s Almost sure representation theorem is used to prove
Theorem 2.3.0.11. (Argmin continuous mapping theorem)
Let {X n : Ω n → B loc (R p )} n and {t n : Ω n → R p } n be sequences of random maps. If
(i) X n X for X Borel measureable and concentrated on C min (R p );
(ii) t n = O P (1);
(iii) X n (t n ) ≤ inf t∈R p X n (t) + α n for random variables {α n } n of order o P (1);
then t n arg min(X).
Proof. Invoke Dudley’s almost sure representation Theorem 2.3.0.10 to find a probability
space (˜Ω, Ã, ˜P ) and maps ˜X n : ˜Ω → B loc (R p ), ˜X : ˜Ω → Bloc (R p ) satisfying
˜X n → as∗ ˜X (2.3.0.6)