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❍❡✐❣❤t ✢✉❝t✉❛t✐♦♥s t❤r♦✉❣❤ ❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s✳ ❱❛❞✐♠ ●♦r✐♥ ▼■❚ ✭❈❛♠❜r✐❞❣❡✮ ❛♥❞ ■■❚P ✭▼♦s❝♦✇✮ ▲❡❝t✉r❡ ✷ ❋❡❜r✉❛r② ✷✵✶✼

❖✈❡r✈✐❡✇ N ( ∂ i ) a ln( G ) 1 � • x 1 = ··· = x N =1 → c a � • ( ∂ i ) a ( ∂ j ) b ln( G ) P ( ℓ ) s ℓ ( x 1 , . . . , x N ) � ··· =1 → d a , b � G = � s ℓ (1 , . . . , 1) • [ � k � a =1 ∂ i a ] ln( G ) =1 → 0 ✱ |{ i a }| > 2 ℓ � 1 ■❢ ❛♥❞ ♦♥❧② ✐❢✿ • N p k → p ( k ) � k N � ℓ i • E p k p m − E p k E p m → cov ( k , m ) � p k = N • p k − E p k → ●❛✉ss✐❛♥s i =1 ❚♦❞❛②✿ ❲❤❛t ✐❢ ②♦✉ ❞♦ ◆❖❚ ❦♥♦✇ ❙✳●✳❋✳❄

❍❡①❛❣♦♥ ✐s ❛ ♣❛rt✐❝✉❧❛r ❝❛s❡✳ ❚r❛♣❡③♦✐❞s N C N ❧♦③❡♥❣❡s ♦♥ t❤❡ r✐❣❤t✿ ❛r❜✐tr❛r② ❞❡t❡r♠✐♥✐st✐❝ ♦r r❛♥❞♦♠✳ ❈♦♥❞✐t✐♦♥❛❧❧② ♦♥ t❤❡✐r ♣♦s✐t✐♦♥s✱ t❤❡ t✐❧✐♥❣ ✐s ✉♥✐❢♦r♠❧② r❛♥❞♦♠✳

❚r❛♣❡③♦✐❞s N C N ❧♦③❡♥❣❡s ♦♥ t❤❡ r✐❣❤t✿ ❛r❜✐tr❛r② ❞❡t❡r♠✐♥✐st✐❝ ♦r r❛♥❞♦♠✳ ❈♦♥❞✐t✐♦♥❛❧❧② ♦♥ t❤❡✐r ♣♦s✐t✐♦♥s✱ t❤❡ t✐❧✐♥❣ ✐s ✉♥✐❢♦r♠❧② r❛♥❞♦♠✳ ❍❡①❛❣♦♥ ✐s ❛ ♣❛rt✐❝✉❧❛r ❝❛s❡✳

❙●❋ ❝♦♠♣✉t❛t✐♦♥ N ❋✐①❡❞ ( t 1 > t 2 > . . . t L ) ✱ x 5 1 { x j x 4 i } ✱ 1 ≤ i ≤ j ≤ L ✖ 1 ❤♦r✐③♦♥t❛❧ ❧♦③❡♥❣❡s✳ x 3 1 x 5 2 x 2 x 4 ❚♦t❛❧ ♥✉♠❜❡r ♦❢ t✐❧✐♥❣s ❂ s t (1 L ) ✳ 1 2 x 1 x 3 x 5 1 2 3 x 2 x 4 2 3 x 3 3 x 5 4 x 4 4 x 5 L 5 ❙❝❤✉r ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ N ✕♣❛rt✐❝❧❡ ✈❡rt✐❝❛❧✳ = s t ( u 1 , . . . , u N , 1 L − N ) � s λ ( u 1 , . . . , u N ) � ( x N 1 , x N 2 , . . . , x N � N ) = λ P s t (1 L ) s λ (1 , . . . , 1) λ ❈♦♠❜✐♥❛t♦r✐❛❧ ❢♦r♠✉❧❛ ❢♦r ❙❝❤✉r ❢✉♥❝t✐♦♥s✳

❙●❋ ❝♦♠♣✉t❛t✐♦♥ N ❋✐①❡❞ ( t 1 > t 2 > . . . t L ) ✱ x 5 1 { x j x 4 i } ✱ 1 ≤ i ≤ j ≤ L ✖ 1 ❤♦r✐③♦♥t❛❧ ❧♦③❡♥❣❡s✳ x 3 1 x 5 2 x 2 x 4 1 2 x 1 x 3 x 5 1 2 3 ❙●❋ ♦❢ t❤❡ N ✕♣❛rt✐❝❧❡ ✈❡rt✐❝❛❧✳ x 2 x 4 2 3 x 3 3 x 5 4 x 4 s t ( u 1 , . . . , u N , 1 L − N ) 4 x 5 L 5 s t (1 L ) • ❆♣♣r♦❛❝❤ ✶✳ ❯s❡ ❛s②♠♣t♦t✐❝ ♦❢ ❙❝❤✉r ❢✉♥❝t✐♦♥s ❬●✳✲P❛♥♦✈❛✲✶✷❪ • ❆♣♣r♦❛❝❤ ✷✳ ❯s❡ t❤❛t ♦✉r t❤❡♦r❡♠ ✐s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✳

❙●❋ ♥♦✲❝♦♠♣✉t❛t✐♦♥ N x 5 1 x 4 1 s t ( u 1 , . . . , u N , 1 L − N ) x 3 1 s t (1 L ) x 5 2 x 2 x 4 1 2 x 1 x 3 x 5 1 2 3 x 2 x 4 2 3 x 3 3 x 5 4 ▲♦❣✕❞❡r✐✈❛t✐✈❡s ❞♦ ♥♦t x 4 4 x 5 L 5 ❞❡♣❡♥❞ ♦♥ N ✳ N ( ∂ i ) a ln( G ) 1 � • x 1 = ··· = x N =1 → c a � • ( ∂ i ) a ( ∂ j ) b ln( G ) P ( ℓ ) s ℓ ( x 1 , . . . , x N ) � � ··· =1 → d a , b G = � s ℓ (1 , . . . , 1) • [ � k � ℓ a =1 ∂ i a ] ln( G ) =1 → 0 ✱ |{ i a }| > 2 � 1 ■❢ ❛♥❞ ♦♥❧② ✐❢✿ • N p k → p ( k ) � k N � ℓ i • E p k p m − E p k E p m → cov ( k , m ) p k = � N • p k − E p k → ●❛✉ss✐❛♥s i =1

❙●❋ ♥♦✲❝♦♠♣✉t❛t✐♦♥ N x 5 1 x 4 1 s t ( u 1 , . . . , u N , 1 L − N ) x 3 s t (1 L ) 1 x 5 2 x 2 x 4 1 2 x 1 x 3 x 5 1 2 3 x 2 x 4 ▲♦❣✕❞❡r✐✈❛t✐✈❡s ❞♦ ♥♦t 2 3 x 3 3 ❞❡♣❡♥❞ ♦♥ N ✳ x 5 4 x 4 4 x 5 L 5 ❲❤❛t ❞❡♣❡♥❞s❄ N ( ∂ i ) a ln( G ) 1 � • x 1 = ··· = x N =1 → c a � • ( ∂ i ) a ( ∂ j ) b ln( G ) P ( ℓ ) s ℓ ( x 1 , . . . , x N ) � � ··· =1 → d a , b G = � s ℓ (1 , . . . , 1) • [ � k � ℓ a =1 ∂ i a ] ln( G ) =1 → 0 ✱ |{ i a }| > 2 � 1 ■❢ ❛♥❞ ♦♥❧② ✐❢✿ • N p k → p ( k ) � k N � ℓ i • E p k p m − E p k E p m → cov ( k , m ) p k = � N • p k − E p k → ●❛✉ss✐❛♥s i =1

❙●❋ ♥♦✲❝♦♠♣✉t❛t✐♦♥ N x 5 1 s t ( u 1 , . . . , u N , 1 L − N ) x 4 1 s t (1 L ) x 3 1 x 5 2 x 2 x 4 1 2 x 1 x 3 x 5 1 2 3 ▲♦❣✕❞❡r✐✈❛t✐✈❡s ❞♦ ♥♦t x 2 x 4 ❞❡♣❡♥❞ ♦♥ N ✳ 2 3 x 3 3 x 5 ❲❤❛t ❞❡♣❡♥❞s❄ 4 x 4 4 x 5 L 5 ❲❤❛t ✐❢ N = L ❄ N ( ∂ i ) a ln( G ) 1 � • x 1 = ··· = x N =1 → c a � • ( ∂ i ) a ( ∂ j ) b ln( G ) P ( ℓ ) s ℓ ( x 1 , . . . , x N ) � � ··· =1 → d a , b G = � s ℓ (1 , . . . , 1) • [ � k � ℓ a =1 ∂ i a ] ln( G ) =1 → 0 ✱ |{ i a }| > 2 � 1 ■❢ ❛♥❞ ♦♥❧② ✐❢✿ • N p k → p ( k ) � k N � ℓ i • E p k p m − E p k E p m → cov ( k , m ) p k = � N • p k − E p k → ●❛✉ss✐❛♥s i =1

❙●❋ ♥♦✲❝♦♠♣✉t❛t✐♦♥ N x 5 1 s t ( u 1 , . . . , u N , 1 L − N ) x 4 1 s t (1 L ) x 3 1 x 5 2 ▲♦❣✕❞❡r✐✈❛t✐✈❡s ❞♦ ♥♦t ❞❡♣❡♥❞ x 2 x 4 1 2 x 1 x 3 x 5 1 2 3 ♦♥ N ✳ x 2 x 4 2 3 x 3 3 ❈♦♥❝❧✉s✐♦♥✿ ▲❡✈❡❧ N ❣✐✈❡s x 5 4 x 4 ❝♦♠♣❧❡t❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t 4 x 5 L 5 ❡♥t✐r❡ ♣✐❝t✉r❡✳ N ( ∂ i ) a ln( G ) 1 � • x 1 = ··· = x N =1 → c a � • ( ∂ i ) a ( ∂ j ) b ln( G ) P ( ℓ ) s ℓ ( x 1 , . . . , x N ) � � ··· =1 → d a , b G = � s ℓ (1 , . . . , 1) • [ � k � ℓ a =1 ∂ i a ] ln( G ) =1 → 0 ✱ |{ i a }| > 2 � 1 ■❢ ❛♥❞ ♦♥❧② ✐❢✿ • N p k → p ( k ) � k N � ℓ i • E p k p m − E p k E p m → cov ( k , m ) p k = � N • p k − E p k → ●❛✉ss✐❛♥s i =1

●❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❝❛♥✬t ❝♦♠♣✉t❡ ❙●❋ ◆♦❜♦❞② ❦♥♦✇s ❙✳●✳❋✳ ♦r ❡✈❡♥ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ t✐❧✐♥❣s✦ ❙♣❡❝✐❛❧ t❤❛♥❦s t♦ ▲❡♦♥✐❞ P❡tr♦✈ ❢♦r s❛♠♣❧✐♥❣✦ ❈♦♥s✐❞❡r t❤❡ ❤❡①❛❣♦♥ ✇✐t❤ ❛ ❤♦❧❡ ♦❢ ✜①❡❞ ❤❡✐❣❤t✳

●❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❝❛♥✬t ❝♦♠♣✉t❡ ❙●❋ ◆♦❜♦❞② ❦♥♦✇s ❙✳●✳❋✳ ♦r ❡✈❡♥ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ t✐❧✐♥❣s✦ ❍♦✇❡✈❡r✱ ❚❤❡♦r❡♠✳ ❚❤❡ ❣❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ❝♦♥✈❡r❣❡ t♦ ❛ ●❛✉ss✐❛♥ ❋✐❡❧❞ ❛s t❤❡ ♠❡s❤ s✐③❡ ❣♦❡s t♦ 0 ✳ ❬❈♦♠❜✐♥❛t✐♦♥ ♦❢ ❇✉❢❡t♦✈✕●♦r✐♥✲✶✼ ❛♥❞ ❇♦r♦❞✐♥✕●♦r✐♥✕●✉✐♦♥♥❡t✲✶✻❪

●❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❝❛♥✬t ❝♦♠♣✉t❡ ❙●❋ ◆♦❜♦❞② ❦♥♦✇s ❙✳●✳❋✳ ♦r ❡✈❡♥ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ t✐❧✐♥❣s✦ ❍♦✇❡✈❡r✱ ❚❤❡♦r❡♠✳ ❚❤❡ ❣❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ❝♦♥✈❡r❣❡ t♦ ❛ ●❛✉ss✐❛♥ ❋✐❡❧❞ ❛s t❤❡ ♠❡s❤ s✐③❡ ❣♦❡s t♦ 0 ✳ ❍♦✇ ❞♦❡s ✐t ✇♦r❦❄ ❬❈♦♠❜✐♥❛t✐♦♥ ♦❢ ❇✉❢❡t♦✈✕●♦r✐♥✲✶✼ ❛♥❞ ❇♦r♦❞✐♥✕●♦r✐♥✕●✉✐♦♥♥❡t✲✶✻❪

●❧♦❜❛❧ ✢✉❝t✉❛t✐♦♥s ✐❢ ②♦✉ ❝❛♥✬t ❝♦♠♣✉t❡ ❙●❋ ◆♦❜♦❞② ❦♥♦✇s ❙✳●✳❋✳ ♦r ❡✈❡♥ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ t✐❧✐♥❣s✦ ❍♦✇❡✈❡r✱ ❖❜s❡r✈❛t✐♦♥✳ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❤♦r✐③♦♥t❛❧ ❧♦③❡♥❣❡s ❛❧♦♥❣ t❤❡ ✈❡rt✐❝❛❧ ❛①✐s ♦❢ t❤❡ ❤♦❧❡ ✭❛♥❞ ♦♥❧② ❛❧♦♥❣ ✐t✦✮ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞✳ ( a ) n = a ( a +1) · · · ( a + n − 1) N � � � ( ℓ i − ℓ j ) 2 � ( A + B + C +1 − t − ℓ i ) t − B ( ℓ i ) t − C ( H − ℓ i ) D ( H − ℓ i ) D i < j i =1