Subsampling estimates of the Lasso distribution.

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2.2 Convergence in distribution 9 Lemma 2.2.1.6. (Van der Vaart and Wellner, 1996, Lemma 1.3.8) (i) If X n X, then X n is asymptotically measurable. (ii) If X n X, then X n is tight if and only if X n is tight. Definition 2.2.1.7. (i) A vector lattice F ⊂ C b (D) is a vector space that is closed under taking positive parts, that is, if f ∈ F then f + = f ∨ 0 ∈ F. (ii) An algebra F ⊂ C b (D) is a vector space that is closed under taking products, i.e. if f, g ∈ F, then fg : x ↦→ f(x)g(x) ∈ F. (iii) A set of functions on F on D is said to separate points in D if, for every pair x ≠ y, there is a f ∈ F with f(x) ≠ f(y). Lemma 2.2.1.8. (Van der Vaart and Wellner, 1996, Lemma 1.3.12) (i) Let L 1 and L 2 be finite measurable measures on D. If ∫ fdL 1 = ∫ fdL 2 for every f ∈ C b (D), then L 1 = L 2 . (ii) Let L 1 and L 2 be tight Borel probability measures on D. If ∫ fdL 1 = ∫ fdL 2 for every f in a vector laticce F ⊂ C b (D) that contains the constant functions and separates points of D, then L 1 = L 2 . Lemma 2.2.1.9. (Van der Vaart and Wellner, 1996, Lemma 1.3.13) Let {X n : Ω → D} n be a sequence of arbitrary maps. Suppose that {X n } n is asymptotically tight and that E ∗ f (X n ) − E ∗ f (X n ) → 0 (2.2.1.1) for every f in a subalgebra F ⊂ C b (D) that separates points of D. Then {X n } n is asymptotically measurable. Lemma 2.2.1.10. (Van der Vaart and Wellner, 1996, Lemma 1.4.3) Sequences {X n : Ω → D} n and {Y n : Ω n → E} of arbitrary random maps are asymptotically tight if and only if the same is true for (X n , Y n ) : Ω n → D × E. Lemma 2.2.1.11. (Van der Vaart and Wellner, 1996, Lemma 1.4.4) Asymptotically tight sequences {X n : Ω → D} n and {Y n : Ω n → E} are asymptotically measurable if and only if the same is true for (X n , Y n ) : Ω n → D × E. Remark. Both previous results remain true for finitely many or countably many sequences of maps. Theorem 2.2.1.12. (Van der Vaart and Wellner, 1996, Theorem 1.3.9)(Prohorov’s theorem) Let {X n : Ω n → D} be a sequence of arbitrary random maps. If {X n } n is asymptotically tight and asymptotically measurable, then it has a subsequence {X jn } n that converges weakly to a tight Borel law.
10 Minimizers of convex processes 2.2.2 Bounded and locally bounded functions Uniformly bounded functions We are ultimatelly interrested in convex functions which are locally bounded, so from now on we will first focus on (D, d) = (l ∞ (T ), ‖·‖ T ) That is ,the space of bounded functions z : T → R on an arbitrary set T with the supremum norm ‖z‖ T = sup t∈T |z(t)|. Later, we will see that asymptotic tightness of locally bounded processes can be characterized in terms of asymptotic tightness of restrictions on compacta. The next result reformulates in this context the abstract characterization of asymptotic measurability and uniqueness of Borel measures given in Lemma 2.2.1.9 and 2.2.1.8. They give a first hint at the important role played by marginals distributions. Lemma 2.2.2.1. (Van der Vaart and Wellner, 1996, Lemma 1.5.2 and Lemma 1.5.3) (i) Let {X n : Ω n → l ∞ (T )} n be asymptotically tight. Then it is asymptotically measurable if and only if {X n (t)} n is asymptotically measurable for every t ∈ T . (ii) Let X and Y be tight Borel measurable maps into l ∞ (T ). Then X and Y are equal in Borel law if and only if all corresponding marginals of X and Y are equal in law. Proof. One easily verifies that the set F ⊂ C b (l ∞ (T )) of continuous functions f of the form f(z) = g(z(t 1 ), . . . , z(t k )) forms an algebra and a vector lattice, that it contains the constant functions and that it separates points of T . (i) Note that for M > 0, {|X n (t)| ≤ M} ⊂ {‖X n ‖ |T ≤ M} n for every t ∈ T . In particular assymptotic tightness of a sequence {X n : Ω n → l ∞ (T )} implies asymptotic tightness of {X n (t)} n for every t ∈ T . Hence, the necessity of assymptotic tightness for {X n (t)} n for every t ∈ T follows directly from the fact that coordinate maps {π t : l ∞ → R}, t ∈ T are continuous. For the sufficiency of this assymptotic tightness, note first that given the assumption and the fact that F defined above forms a vector laticce containing constants and separating points, it follows from Lemma 2.2.1.11 that every sequence {(X n (t 1 ), . . . , X n (t k )) : Ω n → R k } n is asymptotically measurable. The claim is then a direct consequence of Lemma 2.2.1.9 with F given above. (ii) If X and Y have equal Borel laws on l ∞ (T ), then their marginals are also equal in law since coordinates maps π t1 ,...,t k : l ∞ (T ) → R k are Borel measurables. The converse statement follows from Lemma 2.2.1.8. Theorem 2.2.2.2. (Van der Vaart and Wellner, 1996, Lemma 1.5.4) Let {X n : Ω n → l ∞ (T )} n be a sequence of arbitrary random maps. Then {X n } n converges weakly to a tight limit if and only if {X n } is asymptotically tight and for every finite subset t 1 , . . . , t k
Page 1: ✄✄ ✄ ✄ ✄ ✄✄ ✄ ✄
Page 4 and 5: iv Abstract
Page 6 and 7: vi CONTENTS Contents Notation viii
Page 8 and 9: viii Notation List of Tables 6.1 Mo
Page 10 and 11: Chapter 1 Introduction In this thes
Page 12 and 13: Chapter 2 Minimizers of convex proc
Page 14 and 15: 2.1 Convergence in probability 5 fo
Page 16 and 17: 2.2 Convergence in distribution 7 a
Page 20 and 21: 2.2 Convergence in distribution 11
Page 22 and 23: 2.2 Convergence in distribution 13
Page 24 and 25: 2.3 A continous mapping theorem for
Page 30 and 31: Chapter 3 Application to the Lasso
Page 32 and 33: 3.2 Limit in distribution 23 almost
Page 34 and 35: 3.2 Limit in distribution 25 We als
Page 36 and 37: 3.2 Limit in distribution 27 3.2.2
Page 38 and 39: 3.2 Limit in distribution 29 Figure
Page 40 and 41: Chapter 4 The adaptive Lasso in a h
Page 42 and 43: 33 B.3 (Adaptive irrepresentable co
Page 44 and 45: 35 exists a constant K d depending
Page 46 and 47: 4.1 Variable selection consistency
Page 48 and 49: 4.2 Partial asymptotic normality 39
Page 50 and 51: 4.3 Marginal regressors as initial
Page 52 and 53: Chapter 5 Subsampling In his semina
Page 54 and 55: 5.1 Pointwise consistency for distr
Page 56 and 57: 5.2 Uniform consistency for quantil
Page 64 and 65: Chapter 6 Numerical results Motivat
Page 66 and 67: 6.1 Low dimensinal setting 57 5. De
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6.1 Low dimensinal setting 59 Figur
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6.2 High dimensional setting 61 3.
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6.2 High dimensional setting 63 (d)
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6.2 High dimensional setting 65 Mod
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6.2 High dimensional setting 67 Mod
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6.2 High dimensional setting 69 Fig
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Chapter 7 Concluding remarks We rev
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Bibliography Andersen, P. and R. Gi
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Appendix A R Codes A.1 Simulation c
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A.1 Simulation code for the Lasso i
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A.2 Simulation code for the adaptiv
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A.2 Simulation code for the adaptiv
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Subsampling estimates of the Lasso distribution.

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