The Contraction Method on C([0,1]) and Donsker's ... - Eurandom

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$Continuous-time random walks, fractional calculus and ... - Eurandom$

Donsker’s <strong>The</strong>orem Let X1, X2, . . . be iid r.v. with EX1 = 0, EX 2 1 = 1 and E|X1| 2+ε < ∞ for some ε > 0. with S n t = 1 √ n ⎛ ⌊nt⌋ ⎝ k=1 S n = (S n t ) t∈[0,1] Xk + (nt − ⌊nt⌋)X ⌊nt⌋+1 ⎞ ⎠ , t ∈ [0, 1] Henning Sulzbach J. W. Goethe-Universität Frankfurt a. M. <strong>The</strong> <strong>Contraction</strong> <strong>Method</strong> on C([0, 1]) and Donsker’s <strong>The</strong>orem
Donsker’s <strong>The</strong>orem Let X1, X2, . . . be iid r.v. with EX1 = 0, EX 2 1 = 1 and E|X1| 2+ε < ∞ for some ε > 0. with S −4 −2 0 2 4 S n t = 1 √ n ● ● ● ● ● ● ● ● ● ⎛ ⌊nt⌋ ⎝ ● ● k=1 ● ● ● S n = (S n t ) t∈[0,1] Xk + (nt − ⌊nt⌋)X ⌊nt⌋+1 ● ● ● ● ● ● ● S^n −1.0 −0.5 0.0 0.5 1.0 ⎞ ⎠ , t ∈ [0, 1] Henning Sulzbach J. W. Goethe-Universität Frankfurt a. M. <strong>The</strong> <strong>Contraction</strong> <strong>Method</strong> on C([0, 1]) and Donsker’s <strong>The</strong>orem
Page 1 and 2: The Contra
Page 3 and 4: Quickselect / Find Task: Given a li
Page 5 and 6: Quickselect / Find Task: Given a li
Page 7 and 8: For simplicity k = 1. Quickselect /
Page 9 and 10: For simplicity k = 1. Quickselect /
Page 11 and 12: Recursive equation Yn := Xn n A pos
Page 13 and 14: Contraction Hennin
Page 15 and 16: For µ, ν ∈ M1(R) let Co
Page 17 and 18: Contraction Proof:
Page 19 and 20: Contraction Proof:
Page 21 and 22: The stochastic fix
Page 23: Donsker’s Theore
Page 27 and 28: S n t = 1 √ n ⎛ ⌊nt⌋ ⎝ D
Page 29 and 30: In space: Decomposition of Brownian
Page 31 and 32: In space: B −1.5 −0.5 0.5 1.0 1
Page 33 and 34: It holds Decomposition of Brownian
Page 35 and 36: In time: B −1.5 −0.5 0.5 1.0 1.
Page 37 and 38: (Bt) t Decomposition of Brownian Mo
Page 39 and 40: Theorem (Uniquenes
Page 41 and 42: S_1 Time-decomposition of the rando
Page 43 and 44: Time-decomposition of the random wa
Page 45 and 46: The Zolotarev metr
Page 47 and 48: The Zolotarev metr
Page 49 and 50: Properties of the Zolotarev distanc
Page 51 and 52: Lemma Properties of the Zolotarev d
Page 53 and 54: Proof of Donsker’s The</s
Page 55 and 56: Proof of Donsker’s The</s
Page 57 and 58: Properties of ζs in C([0, 1]) Henn
Page 59 and 60: Theorem Properties

The Contraction Method on C([0,1]) and Donsker's ... - Eurandom

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