r t r t rst - - PowerPoint PPT Presentation
trt rsts t t r r t r t rst
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
❲❛♥❣ ❩❤♦✉✶ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥✶
✶❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s ❛♥❞ ❆♣♣❧✐❡❞ Pr♦❜❛❜✐❧✐t②
◆❛t✐♦♥❛❧ ❯♥✐✈❡rs✐t② ♦❢ ❙✐♥❣❛♣♦r❡
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ▼❛✐♥ r❡s✉❧ts ✸ ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
❊✐❣❡♥✈❛❧✉❡s ♦❢ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ♣❧❛② ✈✐t❛❧ r♦❧❡s ✐♥ ✈❛r✐♦✉s st❛t✐st✐❝❛❧ ♣r♦❜❧❡♠s✳ P❈❆✱ ❋❛❝t♦r ♠♦❞❡❧✱ ❙♣✐❦❡❞ ❝♦✈❛r✐❛♥❝❡ ♠♦❞❡❧ ❏♦❤♥st♦♥❡ ✭✷✵✵✶✮✱ ❡t❝✳ ❚r❛❞✐t✐♦♥❛❧❧②✱ ❡✐❣❡♥✈❛❧✉❡s ♦❢ t❤❡ s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ❛r❡ ✉s❡❞ ❛s t❤❡ ❡st✐♠❛t♦rs ♦❢ t❤❡✐r ♣♦♣✉❧❛t✐♦♥ ❝♦✉♥t❡r♣❛rts✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
▲❡t y1, . . . , yN ❜❡ ✐✳✐✳❞✳ M✲✈❛r✐❛t❡ r❛♥❞♦♠ ✈❡❝t♦rs ✇✐t❤ ♠❡❛♥ 0 ❛♥❞ ✐❞❡♥t✐t② ♣♦♣✉❧❛t✐♦♥ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✱ ✐✳❡✳ Ey1y1 = I✳ ▲❡t Σ ❜❡ ❛ M × M s②♠♠❡tr✐❝ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ♠❛tr✐①✳ ■♥ ♠❛♥② st❛t✐st✐❝❛❧ ♣r♦❜❧❡♠s✱ ✇❡ ♦❜s❡r✈❡ ✐✳✐✳❞✳ ❞❛t❛ Σ1/2y1, . . . , Σ1/2yN✳ ❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ♠❛❦❡ ✐♥❢❡r❡♥❝❡ ♦♥ t❤❡ ✉♥❦♥♦✇♥ ♠❛tr✐① Σ ❢r♦♠ t❤❡ s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① W = N−1 N
i=1 Σ1/2yiy∗ i Σ1/2
✇❤❡r❡ ∗ ✐s t❤❡ tr❛♥s♣♦s❡ ♦❢ ♠❛tr✐❝❡s✳ ▲❡t λ1 ❜❡ t❤❡ ❧❛r❣❡st ❡✐❣❡♥✈❛❧✉❡ ♦❢ W✳ ■t ✐s ❦♥♦✇♥ t❤❛t ✐❢ ❡❛❝❤ yi ❝♦♥s✐sts ♦❢ ✐✳✐✳❞✳ ❝♦♠♣♦♥❡♥ts ✇✐t❤ ❛r❜✐tr❛r② ✜♥✐t❡ ♦r❞❡r ♠♦♠❡♥t ❛♥❞ M, N → ∞ ✇✐t❤ M/N → φ > 0✱ t❤❡♥ t❤❡ ♣r♦♣❡r❧② ♥♦r♠❛❧✐③❡❞ λ1 ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ ❚r❛❝②✲❲✐❞♦♠ ❞✐str✐❜✉t✐♦♥✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
◗✉❡st✐♦♥ ❲❤❛t ✐❢ t❤❡ ✐✳✐✳❞✳ ❝♦♠♣♦♥❡♥ts ❛ss✉♠♣t✐♦♥ ♦♥ yi ✐s ✈✐♦❧❛t❡❞✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❡❧❧✐♣t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❞❛t❛ x = ξΣ1/2u ✇❤❡r❡ ξ ✐s ❛ s❝❛❧❡r r❛♥❞♦♠ ✈❛r✐❛❜❧❡ r❡♣r❡s❡♥t✐♥❣ t❤❡ r❛❞✐✉s ♦❢ x ❛♥❞ u ✐s ❛ r❛♥❞♦♠ ✈❡❝t♦r ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ ♦♥ t❤❡ M − 1 ❞✐♠❡♥s✐♦♥❛❧ ✉♥✐t s♣❤❡r❡ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ✇✐t❤ ξ✳ ❘❡s✉❧t ❙✉♣♣♦s❡ ✇❡ ♦❜s❡r✈❡ ✐✳✐✳❞✳ ❞❛t❛ x1, . . . , xN ✇❤❡r❡ xi = ξiΣ1/2ui ❢♦r i = 1, . . . , N ❛r❡ ❡❧❧✐♣t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ r❛♥❞♦♠ ✈❡❝t♦rs✳ ❯♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s ♦❢ ξi✱ t❤❡ ♥♦r♠❛❧✐③❡❞ ❧❛r❣❡st ❡✐❣❡♥✈❛❧✉❡ ♦❢ t❤❡ s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① W = N−1 N
i=1 xix∗ i st✐❧❧ ❝♦♥✈❡r❣❡s
✇❡❛❦❧② t♦ ❚r❛❝②✲❲✐❞♦♠ ❞✐str✐❜✉t✐♦♥✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
▲❡t XN, YN ❜❡ t✇♦ s❡q✉❡♥❝❡s ♦❢ ♥♦♥♥❡❣❛t✐✈❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ❲❡ s❛② t❤❛t XN ✐s st♦❝❤❛st✐❝❛❧❧② ❞♦♠✐♥❛t❡❞ ❜② YN✱ ❞❡♥♦t❡❞ ❛s XN ≺ YN ✐❢ ❢♦r ❛♥② ✭s♠❛❧❧✮ ε > 0 ❛♥❞ ✭❧❛r❣❡✮ D > 0✱ t❤❡r❡ ❡①✐sts N0(ε, D) s✉❝❤ t❤❛t P(XN ≥ NεYN) ≤ N−D ❢♦r ❛❧❧ N ≥ N0(ε, D)✳ ■t ❝❛♥ ❜❡ ✐♥t✉✐t✐✈❡❧② ✐♥t❡r♣r❡t❡❞ ❛s ✏✇✐t❤ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✱ XN ✐s ♥♦ ❣r❡❛t❡r t❤❛♥ YN ✉♣ t♦ ❛ s♠❛❧❧ ♠✉❧t✐♣❧❡ ❞❡♣❡♥❞✐♥❣ ♦♥ N✑✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
❋♦r ❛♥② ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ρ✱ ✐ts ❙t✐❡❧t❥❡s tr❛♥s❢♦r♠ ✐s ❞❡✜♥❡❞ ❛s mρ(z) =
x − z ρ(dx), ∀z ∈ C+. ❚❤❡r❡ ✐s ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ❙t✐❡❧t❥❡s tr❛♥s❢♦r♠ ❛♥❞ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡✳ ❯s✐♥❣ t❤❡ ✐♥✈❡rs✐♦♥ ❢♦r♠✉❧❛✱ ♦♥❡ ❝❛♥ r❡❝♦✈❡r t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ❢r♦♠ ✐ts ❙t✐❡❧t❥❡s tr❛♥s❢♦r♠✳ ▲❡t X = (x1, . . . , xN) ❜❡ t❤❡ M × N ♠❛tr✐①✳ ❉❡♥♦t❡ W = N−1X ∗X. ▲❡t λ1, . . . , λN ❜❡ t❤❡ s❡t ♦❢ ❡✐❣❡♥✈❛❧✉❡s ♦❢ W ✳ ❖♥❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ❡♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ λ1, . . . , λN✱ ρN(x) := N−1 N
i=1 1{λi ≤ x}✱ ✐s ✐♥ ❢❛❝t ❛ ✭r❛♥❞♦♠✮ ♣r♦❜❛❜✐❧✐t②
♠❡❛s✉r❡✳ ❖✉r ❛♥❛❧②s✐s ✇✐❧❧ ❢♦❝✉s ♦♥ ✐ts ❙t✐❡❧t❥❡s tr❛♥s❢♦r♠ mρN(z)✳ ❲❡ ♥♦t❡ t❤❛t t❤❡ s❡t ♦❢ ❡✐❣❡♥✈❛❧✉❡s ♦❢ W ❛♥❞ W ❛r❡ t❤❡ s❛♠❡ ✉♣ t♦ |N − M| ③❡r♦s✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
❋♦r z ∈ C+✱ ❞❡✜♥❡ G(z) = (W − zI)−1 ❛♥❞ G(z) = (W − zI)−1✳ G(z) ❛♥❞ G(z) ❛r❡ r❡❢❡rr❡❞ t♦ ❛s t❤❡ ●r❡❡♥ ❢✉♥❝t✐♦♥s✳ ❲❡ ♥♦t❡ t❤❛t mρN(z) = N−1TrG(z). ▲❡t σ1, . . . , σM ❜❡ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ Σ✳ ❙✉♣♣♦s❡ t❤❡ ❡♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥ πN ♦❢ σ1, . . . , σM ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ ❛ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ π ❛s ❛s M, N → ∞ ✇✐t❤ M/N → φ > 0✳ ❆ ✇❡❧❧✲❦♥♦✇♥ r❡s✉❧t ✐♥ r❛♥❞♦♠ ♠❛tr✐① t❤❡♦r② ✐s t❤❛t ❢♦r ❛♥② ✜①❡❞ z ∈ C+✱ mρN(z) ❝♦♥✈❡r❣❡s t♦ ❛ ❧✐♠✐t m(z) ❚❤❡ ❧✐♠✐t m(z) ✐s t❤❡ ❙t✐❡❧t❥❡s tr❛♥s❢♦r♠ ♦❢ ❛ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✱ s❛② ρ✳ ❈♦rr❡s♣♦♥❞✐♥❣❧②✱ ρN ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ ρ✳ ▼♦r❡♦✈❡r✱ t❤❡ q✉❛♥t✐t❛t✐✈❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ρ ❛♥❞ π ✐s ❞❡s❝r✐❜❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ✐♥✈♦❧✈✐♥❣ t❤❡ ❙t✐❡❧t❥❡s tr❛♥s❢♦r♠ z = − 1 m(z) + φ
1 + xm(z) ∀z ∈ C+. ✭✶✮
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
❚❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ❡st❛❜❧✐s❤❡s ❝♦♥✈❡r❣❡♥❝❡ ♦❢ mρN(z) ❣✐✈❡♥ ❛♥② ✜①❡❞ z ∈ C+. ❚♦ ✐♥✈❡st✐❣❛t❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❧❛r❣❡st ❡✐❣❡♥✈❛❧✉❡ λ1✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ ❝♦♥s✐❞❡r t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ mρN(z) t♦❣❡t❤❡r ✇✐t❤ z ❛♣♣r♦❛❝❤✐♥❣ ❢r♦♠ C+ t♦ t❤❡ r❡❛❧ ❧✐♥❡ ❛t ❝❡rt❛✐♥ r❛t❡ ❛s N → ∞✳ ❚♦ ❜❡ s♣❡❝✐✜❝✱ t♦ ❝❛rr② ♦✉t t❤❡ ❛♥❛❧②s✐s✱ ✇❡ ❛ss✉♠❡ z = E + ıη s✉❝❤ t❤❛t N−1+τ ≤ η ≤ τ−1 ❢♦r s♦♠❡ ✜①❡❞ s♠❛❧❧ ❝♦♥st❛♥t τ > 0 ❛♥❞ E ✐s r❡str✐❝t❡❞ ✐♥t♦ ❛ s♠❛❧❧ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ λ+✳ ❍❡r❡ λ+ ✐s t❤❡ r✐❣❤t ❡♥❞♣♦✐♥t ♦❢ ρ✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
❆ss✉♠♣t✐♦♥s
✶ N, M → ∞ ✇✐t❤ M/N → φ > 0 ❛♥❞ t❤❡ s♣❡❝tr❛❧ ♥♦r♠ ||Σ||
♦❢ Σ ✐s ❜♦✉♥❞❡❞ ✉♥✐❢♦r♠❧② ❢♦r ❛❧❧ ❧❛r❣❡ M, N✳
✷ u1, . . . , uN ❛r❡ ✐✳✐✳❞✳ r❛♥❞♦♠ ✈❡❝t♦rs ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞
♦♥ t❤❡ M − 1 ✉♥✐t s♣❤❡r❡✳
✸ ξ1, . . . , ξN ❛r❡ ✐♥❞❡♣❡♥❞❡♥t r❡❛❧✲✈❛❧✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s
s✉❝❤ t❤❛t Eξ2
i = MN−1✱ E|ξi|p < ∞ ❢♦r ❛❧❧ p ∈ Z+ ❛♥❞
ξ2
i − Eξ2 i ≺ N−1/2 ✉♥✐❢♦r♠❧② ❢♦r ❛❧❧ i ∈ {1, . . . , M}✳
✹ ❙✉❜❝r✐t✐❝❛❧✐t② ❛ss✉♠♣t✐♦♥ ♦♥ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ Σ✳ ✭❚❤❡
❡✐❣❡♥✈❛❧✉❡s ♦❢ Σ s❛t✐s❢② t❤❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❧✐♠✐t✐♥❣ s❛♠♣❧❡ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ♥♦t s❡♣❛r❛t❡❞ ❢r♦♠ t❤❡ ❜✉❧❦✮
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
❆ss✉♠♣t✐♦♥ ✸ ❡①❝❧✉❞❡s s♦♠❡ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s✱ s✉❝❤ ❛s ♠✉❧t✐✈❛r✐❛t❡ st✉❞❡♥t✲t ❞✐str✐❜✉t✐♦♥s ❛♥❞ ♥♦r♠❛❧ s❝❛❧❡ ♠✐①t✉r❡s✳ ❇✉t t❤❡r❡ ❛r❡ st✐❧❧ ❛ ✇✐❞❡ r❛♥❣❡ ♦❢ ❞✐str✐❜✉t✐♦♥s s❛t✐s❢②✐♥❣ ❆ss✉♠♣t✐♦♥ ✸✱ ✐♥❝❧✉❞✐♥❣ f(x) ∼ |Σ|−1/2 exp(−x⊤Σ−1x/2), ♥♦r♠❛❧, f(x) ∼ |Σ|−1/2(1 − x⊤Σ−1x)β/2−1, P❡❛rs♦♥ t②♣❡ ■■, f(x) ∼ |Σ|−1/2(x⊤Σ−1x)k−1 exp(−β[x⊤Σ−1x]s), ❑♦t③✲t②♣❡. s❡❡ t❤❡ ❡①❛♠♣❧❡s ❛♥❞ ❚❛❜❧❡ ✶ ♦❢ ❍✉ ❡t ❛❧✳ ✭✷✵✶✾✮✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
❚❤❡♦r❡♠ ✭❙tr♦♥❣ ❧♦❝❛❧ ❧❛✇✮ ▲❡t δij ❜❡ t❤❡ ❑r♦♥❡❝❦❡r ❞❡❧t❛✳ ❉❡♥♦t❡ Λ(z) = maxi,j |Gij(z) − δijm(z)|✳ ❚❤❡♥ Λ(z) ≺
Nη + 1 Nη, ❛♥❞ |mρN(z) − m(z)| ≺ 1 Nη. ❚❤✐s t❤❡♦r❡♠ s❤♦✇s t❤❛t ❛s N, M → ∞ ✇✐t❤ η → 0 ❛t t❤❡ r❛t❡ N−1+τ✱ ✇✐t❤ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✱ mρN(z) ❛♥❞ m(z) ❛r❡ ❝❧♦s❡ t♦ ❡❛❝❤ ♦t❤❡r ✐♥ t❤❡ ♦r❞❡r N−τ✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
❚❤❡♦r❡♠ ✭❊✐❣❡♥✈❛❧✉❡ r✐❣✐❞✐t②✮ |λ1 − λ+| ≺ N−2/3, ✭✷✮ ❛♥❞ ❢♦r ❛♥② r❡❛❧ ♥✉♠❜❡rs E1, E2 ✐♥ ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ λ+
✭✸✮ ❚❤✐s t❤❡♦r❡♠ s❤♦✇s t❤❛t t❤❡ ❧❛r❣❡st s❛♠♣❧❡ ❡✐❣❡♥✈❛❧✉❡ λ1 ✐s ❝❧♦s❡ t♦ λ+ ✐♥ t❤❡ ♦r❞❡r ♦❢ N−2/3 ✇❤✐❝❤ s✉❣❣❡sts t❤❛t t❤❡ s❝❛❧❡r ♦❢ λ1 t♦ ♦❜t❛✐♥ t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ✐s N2/3✳ ❚❤✐s t❤❡♦r❡♠ ❛❧s♦ s❤♦✇s t❤❛t t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ❜❡t✇❡❡♥ ρN ❛♥❞ ρ ♦❢ ❛ s♠❛❧❧ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ λ+ ✐s ♦❢ ♦r❞❡r N−1 ✇✐t❤ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
❚❤❡♦r❡♠ ✭❊❞❣❡ ✉♥✐✈❡rs❛❧✐t②✮ ▲❡t ˜ X = (˜ x1, . . . ,˜ xN) s✉❝❤ t❤❛t ˜ x1, . . . ,˜ xN ❛r❡ ✐✳✐✳❞✳ M✲✈❛r✐❛t❡ ♥♦r♠❛❧ r❛♥❞♦♠ ✈❡❝t♦rs ✇✐t❤ ♠❡❛♥ ✵ ❛♥❞ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① Σ✳ ❉❡♥♦t❡ ˜ λ1 ❜❡ t❤❡ ❧❛r❣❡st ❡✐❣❡♥✈❛❧✉❡ ♦❢ t❤❡ s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ ˜ x1, . . . ,˜ xN✳ ❚❤❡♥ ❢♦r ❛♥② s ❛♥❞ ε, δ > 0✱ ❛s N ❧❛r❣❡ P(N2/3(˜ λ1 − λ+) ≤ s − N−ε) − N−δ ≤ P(N2/3(λ1 − λ+) ≤ s) ≤ P(N2/3(˜ λ1 − λ+) ≤ s + N−ε) + N−δ. ❚❤✐s t❤❡♦r❡♠ s❤♦✇s t❤❛t t❤❡ ❧✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ λ1 ✐s t❤❡ s❛♠❡ ❛s t❤❛t ♦❢ ˜ λ1 ✇❤✐❝❤ ✐s ❛❧r❡❛❞② ✇❡❧❧✲❦♥♦✇♥ t♦ ❜❡ t❤❡ ❚r❛❝②✲❲✐❞♦♠ ❞✐str✐❜✉t✐♦♥✳ ❚❤✉s ♦✉r ✜♥❛❧ r❡s✉❧t ❢♦❧❧♦✇s✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
❋✐rst ✇❡ ❡st❛❜❧✐s❤ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥ ❜♦✉♥❞s ♦❢ s♣❤❡r✐❝❛❧ ✉♥✐❢♦r♠ ❞✐str✐❜✉t❡❞ r❛♥❞♦♠ ✈❡❝t♦rs✳ ❙♣❡❝✐✜❝❛❧❧②✱ ❢♦r ❛♥② M × M ♠❛tr✐① A✱ |u∗Au − M−1TrA| ≺ M−1||A||F. ✭✹✮ ❖✉r st❛rt✐♥❣ ♣♦✐♥t ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s♦❧✈❡♥t ✐❞❡♥t✐t② ✭❢r♦♠ ❡❧❡♠❡♥t❛r② ❧✐♥❡❛r ❛❧❣❡❜r❛✮ 1 Gii(z) = −z − zx∗
i G(i)(z)xi,
∀i = 1, . . . , N. ✇❤❡r❡ G(i)(z) ✐s t❤❡ ●r❡❡♥ ❢✉♥❝t✐♦♥ ❢♦r♠❡❞ ❢r♦♠ x1, . . . , xN ❡①❝❧✉❞✐♥❣ xi✳ ◆❡①t✱ ❜② ❛♣♣❧②✐♥❣ ✭✹✮ ❛♥❞ ❛ ❜♦♦tstr❛♣♣✐♥❣ ♣r♦❝❡❞✉r❡s ✭s❡❡ ❡✳❣✳ ❇❛♦ ❡t ❛❧✳ ✭✷✵✶✸✮❀ ❑♥♦✇❧❡s ❛♥❞ ❨✐♥ ✭✷✵✶✼✮✮✱ ✇❡ ♦❜t❛✐♥ ❛ ✇❡❛❦❡r ✈❡rs✐♦♥ ♦❢ t❤❡ ❧♦❝❛❧ ❧❛✇ ✇❤✐❝❤ ✐s Λ(z) ≺ (Nη)−1/4.
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
❚❤❡♥✱ t❤❡ ❜♦✉♥❞ |mρN(z) − m(z)| ≺ (Nη)−1 ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ✇❡❛❦ ❧♦❝❛❧ ❧❛✇ ❛♥❞ t❤❡ ✢✉❝t✉❛t✐♦♥ ❛✈❡r❛❣✐♥❣ t❡❝❤♥✐q✉❡ ✭s❡❡ ❡✳❣✳ ❇❡♥❛②❝❤✲●❡♦r❣❡s ❛♥❞ ❑♥♦✇❧❡s ✭✷✵✶✻✮✮✳ ❚❤❡ ❦❡② ♠❡❝❤❛♥✐s♠ ❜❡❤✐♥❞ ✢✉❝t✉❛t✐♦♥ ❛✈❡r❛❣✐♥❣ ✐s t❤❛t s✐♠✐❧❛r t♦ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡r✱ t❤❡ ❛✈❡r❛❣❡ N−1 N
i=1{(Gii(z))−1 − Ei(Gii(z))−1} r❡❞✉❝❡s
t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❡❛❝❤ {(Gii(z))−1 − Ei(Gii(z))−1}✳ ❇✉t t❤❡ ♦r❞❡r ✐s (Nη)−1/2 ✐♥st❡❛❞ ♦❢ N−1/2✱ ✇❤❡r❡ Ei ✐s t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ❣✐✈❡♥ ❛❧❧ ξ1, . . . , ξN, u1, . . . , uN ❡①❝❡♣t ξi, ui✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
✭✷✮ ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ s❤♦✇✐♥❣ t❤❛t ✇✐t❤ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✱ t❤❡r❡ ✐s ♥♦ ❡✐❣❡♥✈❛❧✉❡ ✐♥ t❤❡ ✐♥t❡r✈❛❧ [λ+ + N−2/3+ε, C]✳ ❚❤❡ ♣r♦♦❢ ✐s ❜❛s❡❞ ♦♥ st❛♥❞❛r❞ ❛r❣✉♠❡♥ts ✐♥ ❧✐t❡r❛t✉r❡ ❛❜♦✉t ❧♦❝❛❧ ❧❛✇ ❛♥❞ ✭✸✮ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ❍❡❧✛❡r✲❙❥östr❛♥❞ ❛r❣✉♠❡♥ts ✭s❡❡ ❡✳❣✳ ❇❡♥❛②❝❤✲●❡♦r❣❡s ❛♥❞ ❑♥♦✇❧❡s ✭✷✵✶✻✮✮✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
■t ✐s ♥♦t ❞✐✣❝✉❧t t♦ ✈❡r✐❢② t❤❛t ❣✐✈❡♥ ❛♥ ✐♥t❡r✈❛❧ [E1, E2]✱ t❤❡ ♥✉♠❜❡r ♦❢ ❡✐❣❡♥✈❛❧✉❡s ♦❢ s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ q✉❛♥t✐t② E2
E1
NℑmρN(y + ıη)dy. ❚❤❡r❡❢♦r❡✱ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❡✈❛❧✉❛t✐♥❣ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t λ1 ❢❛❧❧s ✐♥ ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ λ+ ❝♦♥✈❡rts t♦ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❝♦♠♣❛r✐♥❣ t❤❡ ❙t✐❡❧t❥❡s tr❛♥s❢♦r♠ mρN(z) ✇✐t❤ m(z)✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❡❞❣❡ ✉♥✐✈❡rs❛❧✐t② r❡s✉❧t r❡❧✐❡s ♦♥ ❛ ●r❡❡♥ ❢✉♥❝t✐♦♥ ❝♦♠♣❛r✐s♦♥ t❤❡♦r❡♠ ✇❤✐❝❤ st❛t❡s t❤❛t ❢♦r ❛ ❢♦✉r t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s F s❛t✐s❢②✐♥❣ s♦♠❡ s❡t ♦❢ r❡❣✉❧❛r✐t② ❝♦♥❞✐t✐♦♥s✱ t❤❡ ❞✐✛❡r❡♥❝❡
E2
E1
NℑmρN(y + ıη)dy
E2
E1
Nℑ˜ mρN(y + ıη)dy
✐s ❜♦✉♥❞❡❞ ❜② ❛ ♥❡❣❛t✐✈❡ ♣♦✇❡r ♦❢ N✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢
❚❤❡ ❦❡② ✐❞❡❛ ❜❡❤✐♥❞ ✐ts ♣r♦♦❢ ✐s t❤❡ ▲✐♥❞❜❡r❣ t②♣❡ ❛r❣✉♠❡♥t ✇❤✐❝❤ r❡♣❧❛❝❡s ❛ ✇❤♦❧❡ ❝♦❧✉♠♥ xi ❜② ˜ xi ❡❛❝❤ t✐♠❡✳ ■♥ ❢❛❝t✱ ✐t ✐s ❡♥♦✉❣❤ t♦ r❡♣❧❛❝❡ t❤❡ r❛❞✐✉s ✈❛r✐❛❜❧❡ ξi✱ ❜② ˜ ξi i = 1, . . . , N ✇❤❡r❡ ˜ ξi ❛r❡ ✐✳✐✳❞✳ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❢♦❧❧♦✇✐♥❣ ❝❤✐✲sq✉❛r❡ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ M ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❛♥❞ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ✇✐t❤ u1, . . . , uN✳ ■t ❝❛♥ ❜❡ ✈❡r✐✜❡❞ t❤❛t ˜ ξiui ❢♦❧❧♦✇s st❛♥❞❛r❞ ♠✉❧t✐✈❛r✐❛t❡ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✳ ❙✐♥❝❡ t❤❡ ❧✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♥♦r♠❛❧✐③❡❞ ❧❛r❣❡st ❡✐❣❡♥✈❛❧✉❡ ❣✐✈❡♥ ♠✉❧t✐✈❛r✐❛t❡ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ ❞❛t❛ ❤❛s ❜❡❡♥ ❦♥♦✇♥ t♦ ❜❡ ❚r❛❝②✲❲✐❞♦♠ ✭❏♦❤♥st♦♥❡ ✭✷✵✵✶✮✮✱ ♦✉r ♣r♦❜❧❡♠ ❝♦♥✈❡rts t♦ ❜♦✉♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ✭✺✮ ❜❡t✇❡❡♥ ❣❡♥❡r❛❧ ❡❧❧✐♣t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❞❛t❛ ❛♥❞ ♠✉❧t✐✈❛r✐❛t❡ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ ❞❛t❛✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
❆♣♣❡♥❞✐① ❋♦r ❋✉rt❤❡r ❘❡❛❞✐♥❣
❇❛♦✱ ❩✳ ●✳✱ P❛♥✱ ●✳ ▼✳✱ ❛♥❞ ❩❤♦✉✱ ❲✳ ✭✷✵✶✸✮✳ ▲♦❝❛❧ ❞❡♥s✐t② ♦❢ t❤❡ s♣❡❝tr✉♠ ♦♥ t❤❡ ❡❞❣❡ ❢♦r s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ✇✐t❤ ❣❡♥❡r❛❧ ♣♦♣✉❧❛t✐♦♥✳ Pr❡♣r✐♥t✳ ❇❡♥❛②❝❤✲●❡♦r❣❡s✱ ❋✳ ❛♥❞ ❑♥♦✇❧❡s✱ ❆✳ ✭✷✵✶✻✮✳ ▲❡❝t✉r❡s ♦♥ t❤❡ ❧♦❝❛❧ s❡♠✐❝✐r❝❧❡ ❧❛✇ ❢♦r ❲✐❣♥❡r ♠❛tr✐❝❡s✳ ❆r❳✐✈ ❡✲♣r✐♥ts✳ ❍✉✱ ❏✳✱ ▲✐✱ ❲✳✱ ▲✐✉✱ ❩✳✱ ❛♥❞ ❩❤♦✉✱ ❲✳ ✭✷✵✶✾✮✳ ❍✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥ t♦ s♣❤❡r✐❝❛❧ t❡st✳ ❆♥♥✳ ❙t❛t✐st✳✱ ✹✼✭✶✮✿✺✷✼✕✺✺✺✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
❆♣♣❡♥❞✐① ❋♦r ❋✉rt❤❡r ❘❡❛❞✐♥❣
❏♦❤♥st♦♥❡✱ ■✳ ▼✳ ✭✷✵✵✶✮✳ ❖♥ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❧❛r❣❡st ❡✐❣❡♥✈❛❧✉❡ ✐♥ ♣r✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥ts ❛♥❛❧②s✐s✳ ❆♥♥✳ ❙t❛t✐st✳✱ ✷✾✿✷✾✺✕✸✷✼✳ ❑♥♦✇❧❡s✱ ❆✳ ❛♥❞ ❨✐♥✱ ❏✳ ✭✷✵✶✼✮✳ ❆♥✐s♦tr♦♣✐❝ ❧♦❝❛❧ ❧❛✇s ❢♦r r❛♥❞♦♠ ♠❛tr✐❝❡s✳ Pr♦❜❛❜✳ ❚❤❡♦r② ❘❡❧❛t❡❞ ❋✐❡❧❞s✱ ✶✻✾✭✶✮✿✷✺✼✕✸✺✷✳
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s
❆♣♣❡♥❞✐① ❋♦r ❋✉rt❤❡r ❘❡❛❞✐♥❣
❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s