Marc Raimondo - CMI

‘Small’ vs. ‘Large’ Deviations
Rescale X ∼ N(0,In ⊗ Ip)/ √ n so that Eσ1(X) ≈ 1 + √ γ.
Tracy-Widom limit suggests for t ≈ sn −2/3 ,
P {σ1 ≥ Eσ1 + ct} ≈ e −(2/3)nt3/2
.
Lipschitz concentration of measure ⇒ ∀t > 0
P {σ1 ≥ Eσ1 + t} ≤ e −nt2 /2 .
� Not optimal (though simple) for ‘small’ deviations t ≪ 1
� Effective for ’large’ deviations t > 1
(recent: Majumdar-Vargassola)
Luminy 12/08 – p.31