Subsampling estimates of the Lasso distribution.

2.2 Convergence in distribution 11
of T , the sequence of marginals {(X n (t 1 ), . . . , X n (t k ))} n converges weakly to a tight limit. If
{X n } n is asymptotically tight and its marginals converge to the marginals ((X(t 1 ), . . . , X(t k ))
of a stochastic process X, then there is a version of X with uniformly bounded sample paths
and X n X.
Proof. Coordinates maps π t1 ,...,t k
are continuous so it follows from the continuous mapping
Theorem 2.2.1.3 that for every subsequence and every finite subset t 1 , . . . , t k of T {j n } n ,
X jn L jn implies (X jn (t 1 ), . . . , X jn (t k )) L jn ◦ π −1
(t 1 ,...,t k ). In particular, if the whole
sequence converges weakly to a tight limit, the same is true for every sequence of marginals;
asymptotic tightness follows from Lemma 2.2.1.6.
Conversely, suppose that {X n } n is asymptotically tight and that the marginals converge,
then it follows from Lemma 2.2.2.1, trough Lemma 2.2.1.11, that {X n } n is asymptotically
measurable. Asymptotic measurability and tightness are kept under subsequencing, hence,
by Prohorov’s theorem 2.2.1.12, every subsequence {X jn } n converges weakly to a tight
limit, say L jn . It remains to show that the limits are the same. Invoke the continuous
mapping theorem again to show that for every subsequence {j n } n , X jn L jn implies
(X jn (t 1 ), . . . , X jn (t k )) L jn ◦ π −1
(t 1 ,...,t k ) . By assumption, marginals (X j n
(t 1 ), . . . , X jn (t k ))
share the same weak limit, namely the weak limit of (X n (t 1 ), . . . , X n (t k )), uniqueness
now follows from the second part of lemma 2.2.2.1 applied to the measures L jn . Finally,
if {X n } n is asymptotically tight and its marginals converge to the marginals of some
stochastic process X, then {X n } n converge to some Borel measurable process Y : Ω Y →
l ∞ (T ) which has the same marginal distribution as X, again by the second part of lemma
2.2.2.1. This completes the proof.
Definition 2.2.2.3. A sequence {X n : Ω → l ∞ (T )} of arbitrary random maps is asymptotically
uniformly ρ-equicontinuous if for every ɛ, η > 0 there exists a δ > 0 sucht that
)
(
lim sup P ∗ sup |X n (s) − X n (t)|
n→∞ ρ(s,t)
Theorem 2.2.2.4. (Van der Vaart and Wellner, 1996, Theorem 1.5.7)
(i) A sequence {X n : Ω n → l ∞ (T )} n of arbitrary random maps is asymptotically tight
if and only if {X n (t)} n is asymptotically tight in R for every t and there exists a
semimetric ρ on T such that (T, ρ) is totally bounded and {X n } n is asymptotically
uniformly ρ-equicontinous in probability.
(ii) If, moreover, X n X, then almost all paths t ↦→ X(t, ω) are uniformly ρ-continuous;
and the semimetric ρ can without loss of generality be taken equal to any semimetric
ρ for which this is true and (T, ρ) is totally bounded.
Proof. We prove sufficiency of the conditions in i, see the reference for the other direction
and the second part. Suppose that {X n (t)} n is asymptotically tight in R for every t ∈
T and that there exists a semimetric ρ on T such that (T, ρ) is totally bounded and
that {X n } n is asymptotically uniformly ρ-equicontinuous in probability. Using totally
boundedness of (T, ρ), then disjointification of sets we conclude that for every ε, η > 0,
there is a finite partition T = ∪ k i=1 T i such that
)
lim sup
n→∞
P ∗ (sup
i
sup |X n (s) − X n (t)| > ε
s,t∈T i
> η (2.2.2.1)