Subsampling estimates of the Lasso distribution.

Notation
Spaces and sets
A
A σ-algebra.
B o The Borel σ-algebra on [0, 1].
B loc (R p ) Set of locally bounded functions on R p .
B(x, ε) Open ball of radius ε around a point x in a metric space.
C b (D) Bounded continous functions on a metric space D.
C min (R p ) Set of continuous functions uniquely minimized on R p .
K δ
δ-enlargement of a compact set K in a metric space.
l ∞ (T ) Space of uniformly bounded functions on a set T .
l ∞ (T 2 , T 2 , . . . ) Space of functions uniformly bounded on each T i , i ∈ N.
R
The extended real line.
Ω
Sample space.
Modes of convergence
→
→ P
→ as∗
→ au
Deterministic or almost sure convergence.
Convergence in probability.
Outer almost sure convergence.
Almost uniform convergence.
Weak convergence.
O notation
X n = o P (a n ) Convergence in probability of {X n /a n } n to zero.
X n = O P (a n ) Uniform tightness of {X n /a n } n .
Further notation
∆J(x) Jump height at x of a right continous function J with left limit.
E, E ∗ , E ∗ Expectation, outer expectation and inner expection, respectively.
f |T Restriction of a function f on a subset T .
P, P ∗ , P ∗ Probability, outer probability and inner probability, respectively.
T ∗ Minimal measurable majorant of a random map T .
T ∗ Maximal measurable minorant of a random map T .
‖X‖ ψd Orlicz norm of a random variable X.
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