Subsampling estimates of the Lasso distribution.

3.2 Limit in distribution 23
almost surely. Finally we have
Z n (φ) → P (φ − β) ′ C(φ − β) ′ + σ 2 + λ 0
as claimed.
p∑
|φ j |.
j=1
3.2 Limit in distribution
Asymptotic normality of independent, but not necessarily identically distributed random
variables is usually proved by verifying the so called Lindeberg-Feller condition.
Theorem 3.2.0.14. (Van der Vaart, 2000, Proposition 2.27) For each n ∈ N, let
ξ n,1 , . . . , ξ n,kn
be independent random vectors with finite variances Σ n,1 , . . . , Σ n,kn
such that
k n ∑
i=1
(
)
E ‖ξ n,i ‖ 2 1 {‖ξn,i ‖>ε} → 0 for every ε > 0
and
k n ∑
i=1
Σ n,i → Σ
Then
k n ∑
i=1
(
)
ξ n,i − E(ξ n,i ) N (0, Σ)
The Lindeberg-Feller condition will also be used later to prove a partial asymptotic normality
result for the adaptive Lasso.
Theorem 3.2.0.15. (Convergence in distribution) If λ n / √ n → λ 0 ≥ 0 and C is
nonsingular then
√ n( ˆβ − β) → arg min(V )
where
V (u) = −2u ′ W + u ′ Cu + λ 0
for W ∼ N (0, σ 2 C).
p∑
(sgn(β j )I(β j ≠ 0) + |u j |I(β j = 0))
j=1
Proof. Define V n : R p → R as
V n (u) =
n∑
i=1
(
(ε 2 i − u ′ x i / √ ) p∑
n) 2 − ε 2 (
i + λ n |βj + u j / √ n| − |β j | ) ,
j=1