Marc Raimondo - CMI

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Tracy Widom Limits Theorem For {real, complex}, {single, double } Wishart matrices, if n/p → γ, [or (n1/p,n2/p) → (γ1,γ2),] then P {n ˆ ℓ1 ≤ µnp + σnps|H0} → Fβ(s) – “universality” in RMT (Deift, 06 ICM) Single Wishart: β = 2 (Johansson, 00), β = 1 (J, 01) “Microarray case” (El Karoui, 03) β = 1, 2, n,p → ∞, n/p → 0, ∞. Non Gaussian: (Soshnikov, 02, 06; S. + Fyodorov, 06) � subGaussian: n − p = O(p 1/3 ) ⇒ TW limit � heavy tails (e.g. Cauchy) ⇒ Poisson process limit Luminy 12/08 – p.18
Second order accuracy For {real, complex}, {single, double } Wishart matrices, if n/p → γ, [or (n1/p,n2/p) → (γ1,γ2),] then |P {n ˆ ℓ1 ≤ µnp + σnps|H0} − Fβ(s)| ≤ Ce −cs p −2/3 . Single Wishart Complex: El Karoui (2006). Real: Ma (2008) µnp = σnp = �� n−1 2 + �� n−1 2 + � � p− 1 2 p− 1 2 � 2 � � 1 √n− 1 2 + 1 √ p− 1 2 � 1/3 . Luminy 12/08 – p.19
Page 1 and 2: Marc Raimondo Luminy 12/08 - p.1
Page 3 and 4: The Annals of Statistics 2004, Vol.
Page 5 and 6: Effect of Ageing on Morphologic and
Page 7 and 8: Marc Raimondo Memorial Session: - N
Page 9 and 10: Theme � Many problems of classica
Page 11 and 12: Joint density of eigenvalues Single
Page 13 and 14: Symmetric Gaussian matrices A - sym
Page 15 and 16: f(x) 0.35 0.3 0.25 0.2 0.15 0.1 0.0
Page 17: I. Setting Outline 1. Single & Doub
Page 21 and 22: I. Setting Outline 1. Single & Doub
Page 23 and 24: The Tracy-Widom approximation Let W
Page 25 and 26: Accuracy of approximate α th perce
Page 27 and 28: θ α log 10 n 1 0.5 p = 4 at 90th
Page 29 and 30: I. Setting Outline 1. Single & Doub
Page 31 and 32: ‘Small’ vs. ‘Large’ Deviati
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Page 35 and 36: Marc Raimondo The Talented Student
Page 37 and 38: Determinant representation: Correla
Page 39 and 40: After rescaling: Sτ(s,t) = 1 2 Uni
Page 41 and 42: Orthogonal Case - Convergence P { m
Page 43 and 44: Tail p−values: s = 2 variables F

Marc Raimondo - CMI

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