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  1. ■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢ ❚r❛❝②✲❲✐❞♦♠ ❧✐♠✐t ❢♦r t❤❡ ❧❛r❣❡st ❡✐❣❡♥✈❛❧✉❡ ♦❢ ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s ❲❛♥❣ ❩❤♦✉ ✶ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ✶ ✶ ❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s ❛♥❞ ❆♣♣❧✐❡❞ Pr♦❜❛❜✐❧✐t② ◆❛t✐♦♥❛❧ ❯♥✐✈❡rs✐t② ♦❢ ❙✐♥❣❛♣♦r❡ ❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s

  2. ■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢ ❖✉t❧✐♥❡ ✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ▼❛✐♥ r❡s✉❧ts ✸ ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢ ❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s

  3. ■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢ ■♥tr♦❞✉❝t✐♦♥ ❊✐❣❡♥✈❛❧✉❡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

  4. ■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢ ■♥tr♦❞✉❝t✐♦♥ ▲❡t y 1 , . . . , y N ❜❡ ✐✳✐✳❞✳ M ✲✈❛r✐❛t❡ r❛♥❞♦♠ ✈❡❝t♦rs ✇✐t❤ ♠❡❛♥ 0 ❛♥❞ ✐❞❡♥t✐t② ♣♦♣✉❧❛t✐♦♥ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✱ ✐✳❡✳ E y 1 y 1 = I ✳ ▲❡t Σ ❜❡ ❛ M × M s②♠♠❡tr✐❝ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ♠❛tr✐①✳ ■♥ ♠❛♥② st❛t✐st✐❝❛❧ ♣r♦❜❧❡♠s✱ ✇❡ ♦❜s❡r✈❡ ✐✳✐✳❞✳ ❞❛t❛ Σ 1 / 2 y 1 , . . . , Σ 1 / 2 y N ✳ ❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ♠❛❦❡ ✐♥❢❡r❡♥❝❡ ♦♥ t❤❡ ✉♥❦♥♦✇♥ ♠❛tr✐① Σ ❢r♦♠ t❤❡ s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① W = N − 1 � N i =1 Σ 1 / 2 y i y ∗ i Σ 1 / 2 ✇❤❡r❡ ∗ ✐s t❤❡ tr❛♥s♣♦s❡ ♦❢ ♠❛tr✐❝❡s✳ ▲❡t λ 1 ❜❡ t❤❡ ❧❛r❣❡st ❡✐❣❡♥✈❛❧✉❡ ♦❢ W ✳ ■t ✐s ❦♥♦✇♥ t❤❛t ✐❢ ❡❛❝❤ y i ❝♦♥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❡❞ λ 1 ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ ❚r❛❝②✲❲✐❞♦♠ ❞✐str✐❜✉t✐♦♥✳ ❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s

  5. ■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢ ■♥tr♦❞✉❝t✐♦♥ ◗✉❡st✐♦♥ ❲❤❛t ✐❢ t❤❡ ✐✳✐✳❞✳ ❝♦♠♣♦♥❡♥ts ❛ss✉♠♣t✐♦♥ ♦♥ y i ✐s ✈✐♦❧❛t❡❞✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❡❧❧✐♣t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❞❛t❛ x = ξ Σ 1 / 2 u ✇❤❡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❛ x 1 , . . . , x N ✇❤❡r❡ x i = ξ i Σ 1 / 2 u i ❢♦r i = 1 , . . . , N ❛r❡ ❡❧❧✐♣t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ r❛♥❞♦♠ ✈❡❝t♦rs✳ ❯♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s ♦❢ ξ i ✱ t❤❡ ♥♦r♠❛❧✐③❡❞ ❧❛r❣❡st ❡✐❣❡♥✈❛❧✉❡ ♦❢ t❤❡ s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① W = N − 1 � N i =1 x i x ∗ i st✐❧❧ ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ ❚r❛❝②✲❲✐❞♦♠ ❞✐str✐❜✉t✐♦♥✳ ❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s

  6. ■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢ ◆♦t❛t✐♦♥ ▲❡t X N , Y N ❜❡ t✇♦ s❡q✉❡♥❝❡s ♦❢ ♥♦♥♥❡❣❛t✐✈❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ❲❡ s❛② t❤❛t X N ✐s st♦❝❤❛st✐❝❛❧❧② ❞♦♠✐♥❛t❡❞ ❜② Y N ✱ ❞❡♥♦t❡❞ ❛s X N ≺ Y N ✐❢ ❢♦r ❛♥② ✭s♠❛❧❧✮ ε > 0 ❛♥❞ ✭❧❛r❣❡✮ D > 0 ✱ t❤❡r❡ ❡①✐sts N 0 ( ε, D ) s✉❝❤ t❤❛t P ( X N ≥ N ε Y N ) ≤ N − D ❢♦r ❛❧❧ N ≥ N 0 ( ε, D ) ✳ ■t ❝❛♥ ❜❡ ✐♥t✉✐t✐✈❡❧② ✐♥t❡r♣r❡t❡❞ ❛s ✏✇✐t❤ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✱ X N ✐s ♥♦ ❣r❡❛t❡r t❤❛♥ Y N ✉♣ t♦ ❛ s♠❛❧❧ ♠✉❧t✐♣❧❡ ❞❡♣❡♥❞✐♥❣ ♦♥ N ✑✳ ❲❛♥❣ ❩❤♦✉ ❥♦✐♥t❧② ✇✐t❤ ❏✉♥ ❲❡♥ ❚r❛❝②✲❲✐❞♦♠ ✐♥ ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s

  7. ■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ r❡s✉❧ts ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢ ●r❡❡♥ ❢✉♥❝t✐♦♥ ❛♥❞ ❙t✐❡❧t❥❡s tr❛♥s❢♦r♠ ❋♦r ❛♥② ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ρ ✱ ✐ts ❙t✐❡❧t❥❡s tr❛♥s❢♦r♠ ✐s ❞❡✜♥❡❞ ❛s � 1 ∀ z ∈ C + . m ρ ( z ) = x − z ρ (d x ) , ❚❤❡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 = ( x 1 , . . . , x N ) ❜❡ t❤❡ M × N ♠❛tr✐①✳ ❉❡♥♦t❡ W = N − 1 X ∗ X. ▲❡t λ 1 , . . . , λ N ❜❡ t❤❡ s❡t ♦❢ ❡✐❣❡♥✈❛❧✉❡s ♦❢ W ✳ ❖♥❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ❡♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ λ 1 , . . . , λ N ✱ ρ N ( x ) := N − 1 � N i =1 1 { λ i ≤ x } ✱ ✐s ✐♥ ❢❛❝t ❛ ✭r❛♥❞♦♠✮ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡✳ ❖✉r ❛♥❛❧②s✐s ✇✐❧❧ ❢♦❝✉s ♦♥ ✐ts ❙t✐❡❧t❥❡s tr❛♥s❢♦r♠ m ρ N ( 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

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