Subsampling estimates of the Lasso distribution.

33
B.3 (Adaptive irrepresentable condition) For s n1 defined as
s n1 =
(
) ′
|η n1 | −1 sgn(β 01 ), . . . , |η nkn | −1 sgn(β 0kn )
and some constant κ < 1, it holds that
∣
∣x ′ j X n1C −1
n
n1 s n1∣
≤
κ
|η nj | ,
for every j ∈ {k n + 1, . . . , p n }.
B.4 The constants k n , m n , λ n , M n1 , M n2 and b n1 satisfy
(
log(n) 1{d=1} n −1/2 (log(k n)) 1/d
(
+ n 1/2 λ −1
n (log(m n )) 1/d M n2 + 1 ) ) + M n1λ n
b n1
r n b n1 n → 0
B.5 There is a constant τ 1 > 0 such that τ n1 ≥ τ 1 for all n.
B.6
n −1/2 max
1≤i≤n u′ iu i → 0
Next, we introduce the Orlicz norm and some of its properties which be used later to
bound tail probabilities.
Definition 4.0.2.3. (Orlicz norm) Let ψ d = exp(x d )−1 for d ≥ 1. The ψ d -Orlicz norm
||X|| ψd of a random variable X is defined as
{
( ( ))
∣ |X|
||X|| ψd = inf C > 0∣E
ψ d
C
}
≤ 1
Lemma 4.0.2.4. (Van der Vaart and Wellner, 1996, Lemma 2.2.1) Let X be a random
variable with P (|X| > x) ≤ K exp(−Cx d ) for every x, for constants K and C and for
d ≥ 1. Then its Orlicz norm sasitfies
||X|| ψd ≤ ((1 + K)/C) 1/d
In the next Lemma, let ‖X‖ P,d denote the d-moment of a random variable X and S n the
(partial) sum of the first n random variables in a sequence.
Lemma 4.0.2.5. (Van der Vaart and Wellner, 1996, Proposition A.1.6) Let X 1 , . . . , X n
be independent, mean zero random variables indexed by an arbitrary index set T . Then
(i)
(ii)
‖S n ‖ P,d ≤ K
d
[
∥ ]
‖S n ‖ P,1 +
log(d)
∥ max ∥ ∥ ∥∥∥P,d
∥X i , (d > 1).
1≤i≤n
∥ ]
‖S n ‖ ψd ≤ K p
[‖S n ‖ P,1 +
∥ max ∥ ∥ ∥∥∥ψd
∥X i , (0 < d ≤ 1).
1≤i≤n