Subsampling estimates of the Lasso distribution.

5.2 Uniform consistency for quantiles appproximation 51
(iii) P (sup x∈R |L n (x, P ) − J n (x, P )| > ε) ≤
√ {
}
1 2π
+ 1 sup |J b (x, P ) − J n (x, P )| > (1 − γ)ε
γε k n x∈R
Proof. We prove (iii), (i) and (ii) can be proved by similar arguments. For arbitrary ε > 0
and 0 < γ < 1, denote by A n (ε, γ) the event {sup x∈R |J b (x, P ) − J n (x, P )| ≤ (1 − γ)ε}, we
then have :
P
(
≤ P
≤ P
sup |L n (x, P ) − J n (x, P )| > ε
x∈R
(
+ ≤ P
≤ P
)
sup{|L n (x, P ) − J b (x, P )| + sup |J b (x, P ) − J n (x, P )|} > ε
x∈R
(
(
x∈R
sup{|L n (x, P ) − J b (x, P )| + |J b (x, P ) − J n (x, P )|} > ε; A n (ε, γ)
x∈R
(
)
sup{|L n (x, P ) − J b (x, P )| + |J b (x, P ) − J n (x, P )|} > ε; A n (ε, γ) c
x∈R
) {
sup |L n (x, P ) − J n (x, P )| > γε
x∈R
+ 1
sup |J b (x, P ) − J n (x, P )| > (1 − γ)ε
x∈R
Applying Lemma 5.2.2.3 to the first term on the right hand side of the last inequality
completes the argument.
Lemma 5.2.2.5. Let X (n) = (X 1 , . . . , X n ) be an i.i.d. sequence of random variables
with distribution P ∈ P. Denote by J n (x, P ) the distribution of a real valued root R n =
R n (X (n) , P ) under P . Let k n = ⌊n/b⌋ and define L n (x, P ) according to . For arbitrary
ε > 0 and 0 < γ < 1, define
δ 1,n (ε, γ, P ) = 1
√ {
}
2π
+ 1 sup{J b (x, P ) − J n (x, P )} > (1 − γ)ε
γε k n x∈R
δ 2,n (ε, γ, P ) = 1
√ {
}
2π
+ 1 sup{J n (x, P ) − J b (x, P )} > (1 − γ)ε
γε k n x∈R
δ 3,n (ε, γ, P ) = 1
√ {
}
2π
+ 1 sup{|J b (x, P ) − J n (x, P )|} > (1 − γ)ε
γε k n x∈R
)
)
}
.
Then we have
(i) P ( R n ≤ L −1
n (1 − α) ) ≥ 1 − (α + ε + δ 1,n (ε, γ, P ))
(ii) P ( R n ≥ L −1
n (α) ) ≥ 1 − (α + ε + δ 2,n (ε, γ, P ))
(
(iii) P
L −1
n
( α 2 ) ≤ R n ≤ L −1
n (1 − α 2 ) )
≥ 1 − (α + ε + δ 3,n (ε, γ, P ))