Subsampling estimates of the Lasso distribution.

12 Minimizers of convex processes
For such a partition and an arbitrary choice of points t i ∈ T i , i = 1, . . . , k, we have
(
)
(
)
lim inf P ∗ ‖X n ‖
n→∞ |T ≤ max |X n(t i )| + ε ≥ lim inf P ∗ sup sup |X n (s) − X n (t)| ≤ ε
i=1,...,k n→∞ i s,t∈T i
≥ 1 − η.
The maximum of finitely many asymptotically tight sequences of real valued maps is
asymptotically tight (consider the finite union of compacts), so it follows that {‖X n ‖} n is
asymptotically tight in R.
Fix ζ > 0 and consider a sequence ε m ↘ 0. Take a constant M > 0 with
(
lim sup P ∗ ‖X n ‖ |T > M
n→∞
)
< ζ
by invoking asymptotic tightness, and for each ε = ε m and η = 2 −m ζ, take a partition
T = ∪ km
1=1 t i satisfying 2.2.2.1. Fix m for the moment and consider the set
Z m =
⋂ {
}
z ∈ l ∞ (T ) | z |Ti ≡ jε m for some j ∈ {−⌊M/ε⌋, . . . , ⌊M/ε⌋} .
i=1,...,k m
Z m is obviously finite, denote its elements by z 1 , . . . , z p and define
K m =
⋃
B(z i , ε m ).
Then the two conditions
i=1,...,p
‖X n ‖ |T ≤ M and sup
i
sup |X n (s) − X n (t)| ≤ ε m
s,t∈T i
imply that X n ∈ K m . An explicit z ∈ Z m with ‖X n − z‖ T ≤ ε m is given by setting
z |Ti ≡ c for some c ∈ ( inf s∈Ti |X n (s)|, sup s∈Ti |X n (s)| ) ∩ {0, ±ε m ± ⌊M/ε m ⌋ε m }. This is
true for every m. Let K = ∩ ∞ i=1 K m, K is closed, it is totally bounded since ε m ↘ 0; we
are working in a metric space so it follows that K is compact. Now we show that for every
δ > 0, there is a m with K δ ⊃ ∩ m i=1 K i. Indeed, if this is not true then there exists a
sequence {z m } m not contained in K δ but with z m ∈ ∩ m 1=1 for every m. This sequence has
a first subsequence contained in one of the balls making up K 1 , then a further subsequence
contained in one of the balls making ip K 2 . By the usual diagonal argument it follows
that there is a subsequence which is contained in a single ball of radius ε m for every m,
hence Cauchy. Since K is closed its limit is contained in K, which stands in contradiction
to the fact that {z m } m ⊂ l ∞ \ K δ . It follows that
{
}
m⋃
{X n /∈ K δ } ⊂ X n /∈ K m
for some fixed m. For this m, we have
lim sup
n→∞
( P ∗ X n /∈ K δ) ≤ lim sup P
(X ∗ n /∈
n→∞
which concludes the proof.
≤ lim sup P ∗ (‖X n ‖ > M) +
n→∞
m∑
≤ ζ + ζ2 −m < 2ζ,
i=1
i=1
)
m⋂
K i
i=1
m∑
i=1
P ∗ (sup
i
sup |X n (s) − X n (t)| > ε
s,t∈T i
)