❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❚❊❈◆❖▲Ó●■❈❆❙ ✭❈❈❊❚✮ P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊❙❚❆❚❮❙❚■❈❆
▲✉✐s ❊r♥❡st♦ ❇✉❡♥♦ ❙❛❧❛s❛r
❊❧✐♠✐♥❛çã♦ ❞❡ ♣❛râ♠❡tr♦s ♣❡rt✉r❜❛❞♦r❡s ❡♠
✉♠ ♠♦❞❡❧♦ ❞❡ ❝❛♣t✉r❛✲r❡❝❛♣t✉r❛
❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛✲ tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r✲ ❧♦s ✲ ❉❊s✴❯❋❙❈❛r✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ❊st❛✲ tíst✐❝❛✳
Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária/UFSCar
S161ep
Salasar, Luis Ernesto Bueno.
Eliminação de parâmetros perturbadores em um modelo de captura-recaptura / Luis Ernesto Bueno Salasar. -- São Carlos : UFSCar, 2012.
67 f.
Tese (Doutorado) -- Universidade Federal de São Carlos, 2011.
1. Estatística. 2. Estimativas de máxima verossimilhança. 3. Tamanho populacional. 4. População fechada. I. Título.
❘❡s✉♠♦
❖ ♣r♦❝❡ss♦ ❞❡ ❝❛♣t✉r❛✲r❡❝❛♣t✉r❛✱ ❛♠♣❧❛♠❡♥t❡ ✉t✐❧✐③❛❞♦ ♥❛ ❡st✐♠❛çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ ♣♦♣✉❧❛çã♦ ❞❡ ❛♥✐♠❛✐s✱ é t❛♠❜é♠ ❛♣❧✐❝❛❞♦ ❛ ♦✉tr❛s ár❡❛s ❞♦ ❝♦♥❤❡✲ ❝✐♠❡♥t♦ ❝♦♠♦ ❊♣✐❞❡♠✐♦❧♦❣✐❛✱ ▲✐♥❣✉íst✐❝❛✱ ❈♦♥✜❛❜✐❧✐❞❛❞❡ ❞❡ ❙♦❢t✇❛r❡✱ ❊❝♦❧♦❣✐❛✱ ❡♥tr❡ ♦✉tr❛s✳ ❯♠❛ ❞❛s ♣r✐♠❡✐r❛s ❛♣❧✐❝❛çõ❡s ❞❡st❡ ♠ét♦❞♦ ❢♦✐ ❢❡✐t❛ ♣♦r ▲❛♣❧❛❝❡ ❡♠ 1783✱ ❝♦♠
♦ ♦❜❥❡t✐✈♦ ❞❡ ❡st✐♠❛r ♦ ♥ú♠❡r♦ ❞❡ ❤❛❜✐t❛♥t❡s ❞❛ ❋r❛♥ç❛✳ P♦st❡r✐♦r♠❡♥t❡✱ ❈❛r❧ ●✳ ❏✳ P❡t❡rs❡♥ ❡♠ 1889 ❡ ▲✐♥❝♦❧♥ ❡♠ 1930 ✉t✐❧✐③❛r❛♠ ♦ ♠❡s♠♦ ❡st✐♠❛❞♦r ♥♦ ❝♦♥t❡①t♦ ❞❡ ♣♦✲
❆❜str❛❝t
❚❤❡ ❝❛♣t✉r❡✲r❡❝❛♣t✉r❡ ♣r♦❝❡ss✱ ❧❛r❣❡❧② ✉s❡❞ ✐♥ t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❡❧❡✲ ♠❡♥ts ♦❢ ❛♥✐♠❛❧ ♣♦♣✉❧❛t✐♦♥✱ ✐s ❛❧s♦ ❛♣♣❧✐❡❞ t♦ ♦t❤❡r ❜r❛♥❝❤❡s ♦❢ ❦♥♦✇❧❡❞❣❡ ❧✐❦❡ ❊♣✐❞❡♠✐♦✲ ❧♦❣②✱ ▲✐♥❣✉✐st✐❝s✱ ❙♦❢t✇❛r❡ r❡❧✐❛❜✐❧✐t②✱ ❊❝♦❧♦❣②✱ ❛♠♦♥❣ ♦t❤❡rs✳ ❖♥❡ ♦❢ t❤❡ ✜rst ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤✐s ♠❡t❤♦❞ ✇❛s ❞♦♥❡ ❜② ▲❛♣❧❛❝❡ ✐♥ 1783✱ ✇✐t❤ ❛✐♠ ❛t ❡st✐♠❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥❤❛✲
❜✐t❛♥ts ♦❢ ❋r❛♥❝❡✳ ▲❛t❡r✱ ❈❛r❧ ●✳ ❏✳ P❡t❡rs❡♥ ✐♥ 1889 ❛♥❞ ▲✐♥❝♦❧♥ ✐♥ 1930 ❛♣♣❧✐❡❞ t❤❡
s❛♠❡ ❡st✐♠❛t♦r ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❛♥✐♠❛❧ ♣♦♣✉❧❛t✐♦♥s✳ ❚❤✐s ❡st✐♠❛t♦r ❤❛s ❜❡✐♥❣ ❦♥♦✇♥ ✐♥ ❧✐t❡r❛t✉r❡ ❛s ✏▲✐♥❝♦❧♥✲P❡t❡rs❡♥✑ ❡st✐♠❛t♦r✳ ■♥ t❤❡ ♠✐❞✲t✇❡♥t✐❡t❤ ❝❡♥t✉r② s❡✈❡r❛❧ r❡s❡❛r✲ ❝❤❡rs ❞❡❞✐❝❛t❡❞ t❤❡♠s❡❧✈❡s t♦ t❤❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢ st❛t✐st✐❝❛❧ ♠♦❞❡❧s ❛♣♣r♦♣r✐❛t❡❞ ❢♦r t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ ♣♦♣✉❧❛t✐♦♥ s✐③❡✱ ✇❤✐❝❤ ❝❛✉s❡❞ ❛ s✉❜st❛♥t✐❛❧ ✐♥❝r❡❛s❡ ✐♥ t❤❡ ❛♠♦✉♥t ♦❢ t❤❡♦✲ r❡t✐❝❛❧ ❛♥❞ ❛♣♣❧✐❡❞ ✇♦r❦s ♦♥ t❤❡ s✉❜❥❡❝t✳ ❚❤❡ ❝❛♣t✉r❡✲r❡❝❛♣t✉r❡ ♠♦❞❡❧s ❛r❡ ❝♦♥str✉❝t❡❞ ✉♥❞❡r ❝❡rt❛✐♥ ❛ss✉♠♣t✐♦♥s r❡❧❛t✐♥❣ t♦ t❤❡ ♣♦♣✉❧❛t✐♦♥✱ t❤❡ s❛♠♣❧✐♥❣ ♣r♦❝❡❞✉r❡ ❛♥❞ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ ❝♦♥❞✐t✐♦♥s✳ ❚❤❡ ♠❛✐♥ ❛ss✉♠♣t✐♦♥ t❤❛t ❞✐st✐♥❣✉✐s❤❡s ♠♦❞❡❧s ❝♦♥❝❡r♥s t❤❡ ❝❤❛♥❣❡ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥❞✐✈✐❞✉❛❧s ✐♥ t❤❡ ♣♦♣✉❧❛t✐♦♥ ❞✉r✐♥❣ t❤❡ ♣❡r✐♦❞ ♦❢ t❤❡ ❡①♣❡r✐✲ ♠❡♥t✳ ▼♦❞❡❧s t❤❛t ❛❧❧♦✇ ❢♦r ❜✐rt❤s✱ ❞❡❛t❤s ♦r ♠✐❣r❛t✐♦♥ ❛r❡ ❝❛❧❧❡❞ ♦♣❡♥ ♣♦♣✉❧❛t✐♦♥ ♠♦❞❡❧s✱ ✇❤✐❧❡ ♠♦❞❡❧s t❤❛t ❞♦❡s ♥♦t ❛❧❧♦✇ ❢♦r t❤❡s❡ ❡✈❡♥ts t♦ ♦❝❝✉r ❛r❡ ❝❛❧❧❡❞ ❝❧♦s❡❞ ♣♦♣✉❧❛t✐♦♥ ♠♦❞❡❧s✳ ■♥ t❤✐s ✇♦r❦✱ t❤❡ ❣♦❛❧ ✐s t♦ ❝❤❛r❛❝t❡r✐③❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥s ♦❜t❛✐♥❡❞ ❜② ❛♣♣❧②✐♥❣ ♠❡t❤♦❞s ♦❢ ❡❧✐♠✐♥❛t✐♦♥ ♦❢ ♥✉✐ss❛♥❝❡ ♣❛r❛♠❡t❡rs ✐♥ t❤❡ ❝❛s❡ ♦❢ ❝❧♦s❡❞ ♣♦♣✉❧❛t✐♦♥ ♠♦❞❡❧s✳ ❇❛s❡❞ ♦♥ t❤❡s❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥s✱ ✇❡ ❞✐s❝✉ss ♠❡t❤♦❞s ❢♦r ♣♦✐♥t ❛♥❞ ✐♥t❡r✈❛❧ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ s✐③❡✳ ❚❤❡ ❡st✐♠❛t✐♦♥ ♠❡t❤♦❞s ❛r❡ ✐❧❧✉str❛t❡❞ ♦♥ ❛ r❡❛❧ ❞❛t❛✲s❡t ❛♥❞ t❤❡✐r ❢r❡q✉❡♥t✐st ♣r♦♣❡rt✐❡s ❛r❡ ❛♥❛❧✐s❡❞ ✈✐❛ ▼♦♥t❡ ❈❛r❧♦ s✐♠✉❧❛t✐♦♥✳
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ♣✳ ✻
✷ ▼♦❞❡❧♦ ❇✐♥♦♠✐❛❧ ♣✳ ✽
✷✳✶ ▼♦❞❡❧♦ ❊st❛tíst✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✽ ✷✳✷ ❋✉♥çõ❡s ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✵ ✷✳✷✳✶ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ P❡r✜❧❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✵ ✷✳✷✳✷ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ❈♦♥❞✐❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✶ ✷✳✷✳✸ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ■♥t❡❣r❛❞❛ ❯♥✐❢♦r♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✷ ✷✳✷✳✹ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ■♥t❡❣r❛❞❛ ❞❡ ❏❡✛r❡②s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✷
✸ ❊st✐♠❛çã♦ ❞❡ ▼á①✐♠❛ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ❡ ■♥t❡r✈❛❧❛r ♣✳ ✶✹ ✸✳✶ ❊st✐♠❛çã♦ ❞❡ ▼á①✐♠❛ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✹ ✸✳✶✳✶ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ P❡r✜❧❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✹ ✸✳✶✳✷ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ❈♦♥❞✐❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✷ ✸✳✶✳✸ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ■♥t❡❣r❛❞❛ ❯♥✐❢♦r♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✵ ✸✳✶✳✹ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ■♥t❡❣r❛❞❛ ❞❡ ❏❡✛r❡②s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✼ ✸✳✶✳✺ ❊①❡♠♣❧♦ ■❧✉str❛t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✷ ✸✳✷ ❊st✐♠❛çã♦ ■♥t❡r✈❛❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✹ ✸✳✸ ❊①❡♠♣❧♦ ◆✉♠ér✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✼
✹ ❊st✉❞♦ ❞❡ ❙✐♠✉❧❛çã♦ ♣✳ ✺✵
✺ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ♣✳ ✻✸
✻
✶ ■♥tr♦❞✉çã♦
❖ ♣r♦❝❡ss♦ ❞❡ ❛♠♦str❛❣❡♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❝❛♣t✉r❛✲r❡❝❛♣t✉r❛ é ❢r❡q✉❡♥t❡♠❡♥t❡ ✉s❛❞♦ ♥❛ ❡st✐♠❛çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ✐♥❞✐✈í❞✉♦s ❡♠ ✉♠❛ ❞❛❞❛ r❡❣✐ã♦✳ ❊st❡ ♣r♦❝❡ss♦ ❝♦♥s✐st❡ ♥❛ s❡❧❡çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✜①♦ ♦✉ ❛❧❡❛tór✐♦ ❞❡ ✐♥❞✐✈í❞✉♦s ❡♠ ❞✐❢❡r❡♥t❡s é♣♦❝❛s ♦✉ ♦❝❛s✐õ❡s ❞❡ ❛♠♦str❛❣❡♠✳ ◆❛ ♣r✐♠❡✐r❛ é♣♦❝❛ ❞❡ ❛♠♦str❛❣❡♠✱ ♦s ✐♥❞✐✈í❞✉♦s s❡❧❡❝✐♦♥❛❞♦s sã♦ t♦❞♦s ♠❛r❝❛❞♦s✱ ❝♦♥t❛❞♦s ❡ ❞❡✈♦❧✈✐❞♦s à ♣♦♣✉❧❛çã♦✳ ❆♣ós ✉♠ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ q✉❡ ♣❡r♠✐t❛ ❛♦s ✐♥❞✐✈í❞✉♦s ♠❛r❝❛❞♦s s❡ ♠✐st✉r❛r❡♠ ❛♦s ♥ã♦ ♠❛r❝❛❞♦s ♥❛ ♣♦♣✉❧❛çã♦✱ ✉♠❛ s❡❣✉♥❞❛ ❛♠♦str❛ é s❡❧❡❝✐♦♥❛❞❛ ♥❛ q✉❛❧ ❝♦♥t❛✲s❡ ♦ ♥ú♠❡r♦ ❞❡ ✐♥❞✐✈í❞✉♦s ❝♦♠ ❡ s❡♠ ♠❛r❝❛s✱ ♠❛r❝❛✲ s❡ ♦s q✉❡ ❛✐♥❞❛ ♥ã♦ ♣♦ss✉❡♠ ♠❛r❝❛ ❡ t♦❞♦s ♦s ✐♥❞✐✈í❞✉♦s sã♦ ❞❡✈♦❧✈✐❞♦s à ♣♦♣✉❧❛çã♦✳ ❖ ♣r♦❝❡❞✐♠❡♥t♦ é r❡♣❡t✐❞♦ ❛té s❡ ❛t✐♥❣✐r ✉♠ ♥ú♠❡r♦ ✜①❛❞♦ ❞❡ é♣♦❝❛s ❞❡ ❛♠♦str❛❣❡♠✳
❯♠❛ ❞❛s ♣r✐♠❡✐r❛s ❛♣❧✐❝❛çõ❡s ❞♦ ♠ét♦❞♦ ❞❡ ❝❛♣t✉r❛✲r❡❝❛♣t✉r❛ ❢♦✐ ❢❡✐t❛ ♣♦r ▲❛♣❧❛❝❡ ✭✶✼✽✸✮ ♣❛r❛ ❡st✐♠❛r ♦ t❛♠❛♥❤♦ ❞❛ ♣♦♣✉❧❛çã♦ ❞❛ ❋r❛♥ç❛✳ ▼❛✐s t❛r❞❡✱ P❡t❡rs❡♥ ✭✶✽✾✻✮ ❛♣❧✐❝♦✉ ♦ ♠ét♦❞♦ ♣❛r❛ ❡st✉❞❛r ♦ ✢✉①♦ ♠✐❣r❛tór✐♦ ❞❡ ♣❡✐①❡s ♥♦ ♠❛r ❇á❧t✐❝♦ ❡✱ ✐♥❞❡♣❡♥✲ ❞❡♥t❡♠❡♥t❡✱ ▲✐♥❝♦❧♥ ✭✶✾✸✵✮ ❛♣❧✐❝♦✉ ♦ ♠❡s♠♦ ♠ét♦❞♦ ♥❛ ❡st✐♠❛çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ♣❛t♦s s❡❧✈❛❣❡♥s ♥❛ ❆♠ér✐❝❛ ❞♦ ◆♦rt❡✳ ❖ ♠ét♦❞♦ ✉t✐❧③❛❞♦ ♣♦r ❡st❡s ♣❡sq✉✐s❛❞♦r❡s ❜❛s❡✐❛✲s❡ ❡♠ ✉♠ ❡①♣❡r✐♠❡♥t♦ ❝♦♠ ❞✉❛s é♣♦❝❛s ❞❡ ❛♠♦str❛❣❡♠ ❡ ♦ ❡st✐♠❛❞♦r ♣♦♥t✉❛❧ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❡st✐♠❛r ♦ t❛♠❛♥❤♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ✜❝♦✉ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❡st✐♠❛❞♦r ❞❡ ✏▲✐♥❝♦❧♥✲P❡t❡rs❡♥✑✳ ❊st❡ ❡st✐♠❛❞♦r é ♦❜t✐❞♦ ❛♦ s❡ ✐❣✉❛❧❛r ❛ ♣r♦♣♦rçã♦ ❞❡ ✐♥❞✐✈í❞✉♦s ♠❛r❝❛❞♦s ♥❛ s❡❣✉♥❞❛ ❛♠♦str❛ ❝♦♠ ❛ ♣r♦♣♦rçã♦ ❞❡ ✐♥❞✐✈í❞✉♦s ♠❛r❝❛❞♦s ♥❛ ♣♦♣✉❧❛çã♦ ✐♠❡❞✐❛t❛♠❡♥t❡ ❛♥t❡s ❞❛ s❡❣✉♥❞❛ s❡❧❡çã♦✳
❆ ♣❛rt✐r ❞❛ ❞é❝❛❞❛ ❞❡1950❢♦r❛♠ ♣✉❜❧✐❝❛❞♦s ✈ár✐♦s ❛rt✐❣♦s ❝✐❡♥tí✜❝♦s r❡❧❡✈❛♥t❡s s♦❜r❡
♦ t❡♠❛ ❡♥tr❡ ❡❧❡s ❈❤❛♣♠❛♥ ✭✶✾✺✹✮✱ ❉❛rr♦❝❤ ✭✶✾✺✽✮✱ ❉❛rr♦❝❤ ✭✶✾✺✾✮✱ ❙❡❜❡r ✭✶✾✻✺✮✱ ❏♦❧❧② ✭✶✾✻✺✮ ❡ ❈♦r♠❛❝❦ ✭✶✾✻✽✮✳ ❆t✉❛❧♠❡♥t❡✱ ♦s ♠ét♦❞♦s ❞❡ ❝❛♣t✉r❛✲r❡❝❛♣t✉r❛ t❡♠ ❛♣❧✐❝❛çõ❡s ♥❛s ♠❛✐s ❞✐❢❡r❡♥t❡s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ t❛✐s ❝♦♠♦ ❈♦♥✜❛❜✐❧✐❞❛❞❡ ❞❡ ❙♦❢t✇❛r❡ ✭◆❆❨❆❑✱
✶✾✽✽❀ ❇❆❙❯❀ ❊❇❘❆❍■▼■✱ ✷✵✵✶✮✱ ❊♣✐❞❡♠✐♦❧♦❣✐❛ ✭❙❊❇❊❘❀ ❍❯❆❑❆❯❀ ❙■▼▼❖◆❙✱ ✷✵✵✵❀ ▲❊❊ ❡t ❛❧✳✱ ✷✵✵✶❀ ❈❍❆❖ ❡t ❛❧✳✱ ✷✵✵✶❀ ▲❊❊✱ ✷✵✵✷✮✱ ▲✐♥❣✉íst✐❝❛ ✭❇❖❊◆❉❊❘❀ ❘■◆❖❖❨ ❑❆◆✱ ✶✾✽✼❀ ❚❍■❙❚❊❉❀ ❊❋❘❖◆✱ ✶✾✽✼✮ ❡♥tr❡ ♦✉tr❛s✳ ◆❛ ❧✐t❡r❛t✉r❛ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✈ár✐❛s r❡✈✐sõ❡s ❞♦s
✼
✶✾✽✻❀ ❙❊❇❊❘✱ ✶✾✾✷❀ ❙❈❍❲❆❘❩❀ ❙❊❇❊❘✱ ✶✾✾✾❀ ❲❍■❚❊❀ ●❆❘❘❖❚✱ ✶✾✾✵❀ P❖▲▲❖❈❑✱ ✶✾✾✶❀ P❖▲▲❖❈❑✱ ✷✵✵✵❀ ❈❍❆❖✱ ✷✵✵✶❀ ❆▼❙❚❘❯P❀ ▼❈❉❖◆❆▲❉❀ ▼❆◆▲❨✱ ✷✵✵✸✮✳
❖s ♠♦❞❡❧♦s ❡st❛tíst✐❝♦s ♣❛r❛ ❡st✐♠❛çã♦ ❞♦ t❛♠❛♥❤♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❛ ♣❛rt✐r ❞♦ ♣r♦❝❡ss♦ ❞❡ ❝❛♣t✉r❛✲r❡❝❛♣t✉r❛ ❞✐st✐♥❣✉❡♠✲s❡ ❡♥tr❡ s✐✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♣❡❧❛s s✉♣♦s✐çõ❡s ❛❞♦t❛❞❛s✳ ❆ s✉♣♦s✐çã♦ ❜ás✐❝❛ q✉❡ ❞✐❢❡r❡♥❝✐❛ ♦s ♠♦❞❡❧♦s é ❛ ❞❡ q✉❡ ❛ ♣♦♣✉❧❛çã♦ é ❢❡❝❤❛❞❛✱ ✐st♦ é✱ ♦ ♥ú♠❡r♦ ❞❡ ✐♥❞✐✈í❞✉♦s ♥ã♦ s❡ ❛❧t❡r❛ ❞✉r❛♥t❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❛♠♦str❛❣❡♠✱ ♥ã♦ ❤❛✈❡♥❞♦ ♣♦rt❛♥t♦ ♠✐❣r❛çã♦✱ ♠♦rt❡s ♦✉ ♥❛s❝✐♠❡♥t♦s✳ ◗✉❛♥❞♦ ❡st❛ s✉♣♦s✐çã♦ ♥ã♦ é s❛t✐s❢❡✐t❛✱ ❞✐✲ ③❡♠♦s q✉❡ ❛ ♣♦♣✉❧❛çã♦ é ❛❜❡rt❛✳ ❖✉tr❛s s✉♣♦s✐çõ❡s ✉s✉❛❧♠❡♥t❡ ❛❞♦t❛❞❛s ♣❡❧♦s ♠♦❞❡❧♦s ♠❛✐s r❡str✐t✐✈♦s sã♦✿ ✭✐✮ ♦s ✐♥❞✐✈í❞✉♦s ♥ã♦ ♣❡r❞❡♠ s✉❛s ♠❛r❝❛s ❞✉r❛♥t❡ ♦ ❡①♣❡r✐♠❡♥t♦❀ ✭✐✐✮ t♦❞❛s ❛s ♠❛r❝❛s sã♦ ♦❜s❡r✈❛❞❛s ❡ r❡❣✐str❛❞❛s ❝♦rr❡t❛♠❡♥t❡❀ ✭✐✐✐✮ ♦s ✐♥❞✐✈í❞✉♦s t❡♠ ❛ ♠❡s♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ s❡r❡♠ ❝❛♣t✉r❛❞♦s ❡♠ q✉❛❧q✉❡r é♣♦❝❛ ❞❡ ❛♠♦str❛❣❡♠✱ ✐st♦ é✱ ♦ ♣r♦❝❡ss♦ ❞❡ ❝❛♣t✉r❛ ❡ ♠❛r❝❛çã♦ ♥ã♦ ❛❧t❡r❛ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❝❛♣t✉r❛✳ ❖t✐s ❡t ❛❧✳ ✭✶✾✼✽✮ ❞✐s❝✉t❡8♠♦❞❡❧♦s ♣❛r❛ ♣♦♣✉❧❛çã♦ ❢❡❝❤❛❞❛ ❡♠ q✉❡ ❛ s✉♣♦s✐çã♦ ✭✐✐✐✮ é ✢❡①✐❜✐❧✐③❛❞❛✱ ❝♦♠❜✐✲
♥❛♥❞♦ três t✐♣♦s ❞❡ ✈❛r✐❛çã♦ ♣❛r❛ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❝❛♣t✉r❛✿ ✭✶✮ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❝❛♣t✉r❛ ✈❛r✐❛♠ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ é♣♦❝❛ ❞❡ ❛♠♦str❛❣❡♠❀ ✭✷✮ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❝❛♣t✉r❛ ❞❡ ✉♠ ✐♥❞✐✈í❞✉♦ s❡ ❛❧t❡r❛ q✉❛♥❞♦ ❡st❡ ✐♥❞✐✈í❞✉♦ ❥á ❢♦✐ ❝❛♣t✉r❛❞♦❀ ✭✸✮ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❝❛♣t✉r❛♠ ✈❛r✐❛♠ ❞❡ ✐♥❞✐✈í❞✉♦ ♣❛r❛ ✐♥❞✐✈í❞✉♦✳
P❛r❛ ♠✉✐t♦s ♠♦❞❡❧♦s ❞❡ ❝❛♣t✉r❛✲r❡❝❛♣t✉r❛✱ ♦ ✐♥t❡r❡ss❡ ♣r✐♥❝✐♣❛❧ r❡s✐❞❡ ♥❛ ❡st✐♠❛çã♦ ❞♦ t❛♠❛♥❤♦ ♣♦♣✉❧❛❝✐♦♥❛❧✳ ❆ss✐♠✱ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❝❛♣t✉r❛ t♦r♥❛♠✲s❡ ♣❛râ♠❡tr♦s ♣❡rt✉r❜❛❞♦r❡s ♦✉ ♥✉✐s❛♥❝❡✱ ♠❛s q✉❡ sã♦ ✐♠♣r❡s❝✐♥❞í✈❡✐s ♥❛ ❝♦♥str✉çã♦ ❞♦s ♠♦❞❡❧♦s ♣r♦✲ ❜❛❜✐❧íst✐❝♦s✳ ◆❡st❡ ❝♦♥t❡①t♦✱ ♦ ❡st✉❞♦ ❞❡ ❢✉♥çõ❡s ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ q✉❡ ❞❡♣❡♥❞❛♠ ❛♣❡♥❛s ❞♦ ♣❛râ♠❡tr♦ ❞❡ ✐♥t❡r❡ss❡ t♦r♥❛✲s❡ ✐♠♣♦rt❛♥t❡✳ ❖ ✉s♦ ❞❡ ♠ét♦❞♦s ♣❛r❛ ❡❧✐♠✐♥❛çã♦ ❞❡ ♣❛✲ râ♠❡tr♦s ♣❡rt✉r❜❛❞♦r❡s sã♦ ❛♠♣❧❛♠❡♥t❡ ❞✐s❝✉t✐❞♦s ♥❛ ❧✐t❡r❛t✉r❛ ❡st❛tíst✐❝❛ ✭❈❖❳✱ ✶✾✼✺❀ ❇❊❘●❊❘❀ ▲■❙❊❖❀ ❲❖▲P❊❘❚✱ ✶✾✾✾❀❇❆❙❯✱ ✶✾✼✼✮✳
✽
✷ ▼♦❞❡❧♦ ❇✐♥♦♠✐❛❧
◆❡st❡ ❝❛♣ít✉❧♦ ❞✐s❝✉t✐♠♦s ❝♦♠♦ ♦s ❞✐❢❡r❡♥t❡s ♠ét♦❞♦s ❞❡ ❡❧✐♠✐♥❛çã♦ ❞❡ ♣❛râ♠❡tr♦s ♣❡rt✉r❜❛❞♦r❡s ✭♥✉✐s❛♥❝❡✮ ❝♦♥s✐❞❡r❛❞♦s ♥♦ ❈❛♣ít✉❧♦ 1 ♣♦❞❡♠ s❡r ✉s❛❞♦s ♥❛ ♦❜t❡♥çã♦ ❞❡
❡st✐♠❛t✐✈❛s ♣♦♥t✉❛✐s ❡ ✐♥t❡r✈❛❧❛r❡s ♣❛r❛ ♦ t❛♠❛♥❤♦ ♣♦♣✉❧❛❝✐♦♥❛❧✱ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠ ♠♦❞❡❧♦ ❞❡ ❝❛♣t✉r❛✲r❡❝❛♣t✉r❛ ❝♦♠ ♠❛r❝❛çã♦ ♣❛r❛ ✉♠❛ ♣♦♣✉❧❛çã♦ ❢❡❝❤❛❞❛✳
✷✳✶ ▼♦❞❡❧♦ ❊st❛tíst✐❝♦
❈♦♥s✐❞❡r❡♠♦s ✉♠ ♣r♦❝❡ss♦ ❞❡ ❝❛♣t✉r❛✲r❡❝❛♣t✉r❛ ❡♠ q✉❡ ♦s ✐♥❞✐✈í❞✉♦s sã♦ s❡❧❡❝✐♦♥❛✲ ❞♦s ❞❛ ♣♦♣✉❧❛çã♦ ❡♠ k ♦❝❛s✐õ❡s ❞❡ ❛♠♦str❛❣❡♠✱ k ≥ 2✳ ❊♠ ❝❛❞❛ ♦❝❛s✐ã♦ ❝♦♥t❛✲s❡ ♦
♥ú♠❡r♦ ❞❡ ✐♥❞✐✈í❞✉♦s r❡❝❛♣t✉r❛❞♦s ✭❥á ♣♦ss✉❡♠ ♠❛r❝❛✮ ❡ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ✐♥❞✐✈í❞✉♦s✱ ❡❢❡t✉❛✲s❡ ❛ ♠❛r❝❛çã♦ ❞♦s ✐♥❞✐✈í❞✉♦s s❡♠ ♠❛r❝❛ ❡ t♦❞♦s ♦s ✐♥❞✐✈í❞✉♦s sã♦ ❞❡✈♦❧✈✐❞♦s à ♣♦♣✉❧❛çã♦✳ ❯♠❛ ♣ró①✐♠❛ ❝❛♣t✉r❛ é r❡❛❧✐③❛❞❛ ❛♣ós ♣❡r♠✐t✐r q✉❡ ♦s ✐♥❞✐✈í❞✉♦s ♠❛r❝❛❞♦s ❡ ♥ã♦ ♠❛r❝❛❞♦s s❡ ❞✐str✐❜✉❛♠ ♥❛ ♣♦♣✉❧❛çã♦✳
❉❡♥♦t❡♠♦s ♣♦r N ♦ ♥ú♠❡r♦ ❞❡ ✐♥❞✐✈í❞✉♦s ♥❛ ♣♦♣✉❧❛çã♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡✱ ❡♠ ❝❛❞❛
♦❝❛s✐ã♦✱ ✉♠ ✐♥❞✐✈í❞✉♦ s❡❥❛ ❝❛♣t✉r❛❞♦ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❡ ❝♦♠ ❛ ♠❡s♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞♦s ❞❡♠❛✐s ❡ q✉❡ ❛s ♦❝❛s✐õ❡s ❞❡ ❝❛♣t✉r❛ s❡❥❛♠ ✐♥❞❡♣❡♥❞❡♥t❡s ❡♥tr❡ s✐✱ ✐st♦ é✱ ♦ ❢❛t♦ ❞❡ ✉♠ ✐♥❞✐✈í❞✉♦ s❡r ❝❛♣t✉r❛❞♦ ♦✉ ♥ã♦ ❡♠ ✉♠❛ ♦❝❛s✐ã♦ ♥ã♦ ✐♥✢✉❡♥❝✐❛ ♥❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❝❛♣t✉r❛ ❡♠ ♦✉tr❛ ♦❝❛s✐ã♦✳
◆❡st❡ ❝♦♥t❡①t♦✱ ❞❡♥♦t❡♠♦s ♣♦r
• pi ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ✐♥❞✐✈í❞✉♦ s❡r ❝❛♣t✉r❛❞♦ ♥❛ i✲és✐♠❛ ♦❝❛s✐ã♦❀
• ni ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ✐♥❞✐✈í❞✉♦s ❝❛♣t✉r❛❞♦s ♥❛i✲és✐♠❛ ♦❝❛s✐ã♦❀
• mi ♦ ♥ú♠❡r♦ ❞❡ ✐♥❞✐✈í❞✉♦s r❡❝❛♣t✉r❛❞♦s ♥❛i✲és✐♠❛ ♦❝❛s✐ã♦ ✭m1 = 0✮❀ • Mi =
iP−1
j=1
(nj−mj)♦ ♥ú♠❡r♦ ❞❡ ✐♥❞✐✈í❞✉♦s ❞✐st✐♥t♦s ♠❛r❝❛❞♦s ♣r❡s❡♥t❡s ♥❛ ♣♦♣✉❧❛çã♦
✾
• r = Mk+1 =
k
P
j=1
(nj −mj) ♦ ♥ú♠❡r♦ ❞❡ ✐♥❞✐✈í❞✉♦s ❞✐st✐♥t♦s ❝❛♣t✉r❛❞♦s ❞✉r❛♥t❡ ♦
♣r♦❝❡ss♦❀
♣❛r❛ i= 1, . . . , k✳
P♦rt❛♥t♦✱ ❞❛❞♦ ♦ ✈❡t♦r ❞❡ ♣❛râ♠❡tr♦s θ= (N,p)✱ ♦♥❞❡N é ♦ ♣❛râ♠❡tr♦ ❞❡ ✐♥t❡r❡ss❡
❡ p = (p1, . . . , pk) ♦ ✈❡t♦r ❞❡ ♣❛râ♠❡tr♦s ♥✉✐s❛♥❝❡✱ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞♦s
❞❛❞♦s ❛♠♦str❛✐s n= (n1, . . . , nk) ❡ m= (m1, . . . , mk)é ❞❛❞❛ ♣♦r
Pθ
n,m =Pθ
n1, m1 Pθ
n2, m2|n1, m1 · · ·Pθ
nk, mk|n1, m1;n2, m2;. . .;nk−1, mk−1
=
k
Y
i=1
N −Mi
ni−mi
pni−mi
i (1−pi)N−Mi−ni+mi
Mi
mi
pmi
i (1−pi)Mi−mi
=
k
Y
i=1
Mi
mi
×
k
Y
i=1
N−Mi
ni−mi
pni
i (1−pi)N−ni. ✭✷✳✶✮
❈♦♠♦ Mi+1 =Mi+ni−mi ♣❛r❛ i= 1, . . . , k✱ ❡♥tã♦ k
Y
i=1
N −Mi
ni−mi
=
k
Y
i=1
(N −Mi)!
(ni−mi)!(N −Mi −ni+mi)!
=
k
Y
i=1
1 (ni−mi)!
k
Y
i=1
(N −Mi)!
(N −Mi+1)!
=
Yk
i=1
1 (ni−mi)!
× N!
(N −r)!. ✭✷✳✷✮
❙✉❜st✐t✉✐♥❞♦ ❛ r❡❧❛çã♦ (2.2)❡♠ (2.1)✱ s❡❣✉❡ q✉❡ ❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞❡ θ = (N,p) é ❞❛❞❛ ♣♦r
L(N,p) =Pθ{n,m}
=
Yk
i=1
Mi!
mi!(Mi−mi)!(ni−mi)!
× N!
(N −r)!
k
Y
i=1
pni
i (1−pi)N−ni, ✭✷✳✸✮
N ≥r✱ 0< pi <1✱ i= 1, . . . , k✳
P♦rt❛♥t♦✱ ♦ ♥ú❝❧❡♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ é ❞❛❞♦ ♣♦r
K(N,p) = N! (N −r)!
k
Y
i=1
pni
✶✵
❡ s❡✉ ❧♦❣❛r✐t♠♦ ❞❛❞♦ ♣♦r
log K(N,p)= log N!−log (N −r)!+
k
X
i=1
nilog(pi) + k
X
i=1
(N −ni) log(1−pi),
✭✷✳✺✮
N ≥r✱ 0< pi <1✱ i= 1, . . . , k✳
❙❛❧✐❡♥t❛♠♦s q✉❡ ♦ ♥ú❝❧❡♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✭✷✳✹✮ ❞❡♣❡♥❞❡ ❞♦s ❞❛❞♦s ❛♠♦s✲ tr❛✐s s♦♠❡♥t❡ ❛tr❛✈és ❞❡n ❡ r✱ ✐st♦ é✱ (n, r) é ✉♠❛ ❡st❛tíst✐❝❛ s✉✜❝✐❡♥t❡ ♣❛r❛(N, p)✳
✷✳✷ ❋✉♥çõ❡s ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛
◆❡st❛ s❡çã♦ ✈❛♠♦s ❛♣❧✐❝❛r ♦s ♠ét♦❞♦s ❞❡ ❡❧✐♠✐♥❛çã♦ ❞❡ ♣❛râ♠❡tr♦s ♣❡rt✉r❜❛❞♦r❡s ❞✐s✲ ❝✉t✐❞♦s ♥♦ ❈❛♣ít✉❧♦1à ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✭✷✳✸✮ ❡ ♦❜t❡r ❢✉♥çõ❡s ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛
q✉❡ ❞❡♣❡♥❞❡♠ ❡①❝❧✉s✐✈❛♠❡♥t❡ ❞♦ ♣❛râ♠❡tr♦ ❞❡ ✐♥t❡r❡ss❡ N✳
✷✳✷✳✶ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ P❡r✜❧❛❞❛
❆ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❡r✜❧❛❞❛ ❞❡N é ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❛ ❡①♣r❡ssã♦ ❞❛ ❢✉♥çã♦ ❞❡
✈❡r♦ss✐❧❤❛♥ç❛ ✭✷✳✸✮ s✉❜st✐t✉✐♥❞♦✲s❡ ♦ ✈❡t♦r ❞❡ ♣❛râ♠❡tr♦sp♣♦r s✉❛ ❡st✐♠❛t✐✈❛ ❞❡ ♠á①✐♠❛
✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❛r❛ ❝❛❞❛N ✜①❛❞♦✳ ❊♥tã♦✱ ❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❡r✜❧❛❞❛ é ♦❜t✐❞❛
❝♦♠♦
LP(N) = sup
p∈[0,1]k
L(N,p)
=L N,pb(N)
∝K N,pb(N),
♦♥❞❡pb(N) é ♦ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❞❡ L(N,p)♣❛r❛ ❝❛❞❛ N ≥r✱ N ✜①❛❞♦✳
❆ ♣❛rt✐r ❞❡ ✭✷✳✺✮ s❡❣✉❡ q✉❡
∂log K(N, p) pi
= ni pi
−N −ni
1−pi
= 0, i= 1, . . . , k,
♦ q✉❡ ✐♠♣❧✐❝❛
b
p(N) =
n1
N,· · · , nk
N
, N ≥r. ✭✷✳✻✮
✶✶
é ❞❛❞♦ ♣♦r
KP(N) = N!
(N −r)!
k
Y
i=1
(N −ni)N−ni
NN , N ≥r. ✭✷✳✼✮
✷✳✷✳✷ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ❈♦♥❞✐❝✐♦♥❛❧
❆ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧ é ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❛ ❢❛t♦r❛çã♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✭✷✳✸✮ ❝♦♠♦
L(N,p) =LC(N)×L′(N, p),
♦♥❞❡LC(N)é ❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦rr❡s♣♦♥❞❡♥t❡ à ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s
❝♦♥❞✐❝✐♦♥❛❧ ❞❡ m = (m1, m2, . . . , mk)✱ ❞❛❞♦ n = (n1, n2, . . . , nk)✱ ❡ L′(N,p) ❛ ❢✉♥çã♦ ❞❡
✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦rr❡s♣♦♥❞❡♥t❡ à ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡(n1, n2, . . . , nk)❞❛❞♦s
N ❡ p✳
❊♥tã♦✱ s❡❣✉❡ q✉❡ L′(N,p) é ❞❛❞❛ ♣♦r
L′(N,p) = Pθ{n1, . . . , nk}
=
k
Y
i=1
N ni
pni
i (1−pi)N−ni,
N ≥max{n1, n2, . . . , nk}✱0< pi <1✱ i= 1, . . . , k ❡ s❡❣✉❡ ❞❡ ✭✷✳✸✮ q✉❡ L′(N) é ❞❛❞❛ ♣♦r
L′(N) =P
θ{m1, . . . , mk|n1, . . . , nk}
= Pθ{m1, n1;. . .;mk, nk} Pθ{n1, . . . , nk}
=
k
Q
i=1
Mi!
mi!(Mi−mi)!(ni−mi)!
× N!
(N−r)!
k
Q
i=1
pni
i (1−pi)N−ni
k
Q
i=1
N ni
pni
i (1−pi)N−ni
=
Yk
i=1
Mi!ni!
mi!(Mi−mi)!(ni−mi)!
× N!
(N −r)!
k
Y
i=1
(N −ni)!
N! , ✭✷✳✽✮
N ≥r✳
P♦rt❛♥t♦✱ s❡❣✉❡ ❞❡ ✭✷✳✽✮ q✉❡ ♦ ♥ú❝❧❡♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧ é ❞❛❞♦ ♣♦r
KC(N) = N!
(N −r)!
k
Y
i=1
(N −ni)!
✶✷
✷✳✷✳✸ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ■♥t❡❣r❛❞❛ ❯♥✐❢♦r♠❡
❈♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❡❧✐♠✐♥❛r ♦ ♣❛râ♠❡tr♦ ♥✉✐s❛♥❝❡ p✱ s✉♣♦♥❤❛♠♦s q✉❡✱ ❞❛❞♦ N✱ p1, . . . , pk s❡❥❛♠ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♠ ♠❡s♠❛ ❞✐str✐❜✉✐çã♦ ❯♥✐❢♦r♠❡
(0,1)✳ ❆ss✐♠✱ ❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✐♥t❡❣r❛❞❛ ✉♥✐❢♦r♠❡ é ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❛ ✐♥✲
t❡❣r❛çã♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✭✷✳✸✮✱ ✐st♦ é✱
LU(N) = Z
(0,1)k
L(N,p)dp
∝ N!
(N −r)!
k
Y
i=1 1
Z
0
pni
i (1−pi)N−nidpi
= N!
(N −r)!
k
Y
i=1
Γ(ni+ 1)Γ(N −ni+ 1)
Γ(N + 2)
∝ N!
(N −r)!
k
Y
i=1
(N −ni)!
(N + 1)!, N ≥r.
P♦rt❛♥t♦✱ ♦ ♥ú❝❧❡♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✐♥t❡❣r❛❞❛ ✉♥✐❢♦r♠❡ é ❞❛❞♦ ♣♦r
KU(N) = N!
(N −r)!
k
Y
i=1
(N −ni)!
(N + 1)!, N ≥r. ✭✷✳✶✵✮
✷✳✷✳✹ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ■♥t❡❣r❛❞❛ ❞❡ ❏❡✛r❡②s
◆♦✈❛♠❡♥t❡✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ❛ ❡❧✐♠✐♥❛çã♦ ❞♦ ♣❛râ♠❡tr♦ ♥✉✐s❛♥❝❡ p ✈✐❛ ✐♥t❡❣r❛çã♦✳
❙✉♣♦♥❤❛♠♦s q✉❡✱ ❞❛❞♦N✱ps❡❥❛ ✉♠ ✈❡t♦r ❛❧❡❛tór✐♦ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❞❡ ❏❡✛r❡②sπJ(p|N)✳
❆ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❏❡✛r❡②s é ❞❡✜♥✐❞❛ ❝♦♠♦ s❡♥❞♦ ♣r♦♣♦r❝✐♦♥❛❧ ❛ r❛✐③ q✉❛❞r❛❞❛ ❞♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ❡s♣❡r❛❞❛ ❞❡ ❋✐s❤❡r IN(p) ✭s✉♣♦♥❞♦
N ❝♦♥❤❡❝✐❞♦✮✱ ✐st♦ é✱
IN(p) = ❊
−∂logL(N,p)
∂pi∂pj
!
k×k
∝ ❊
− ∂logK(N,p)
∂pi∂pj
!
k×k
✶✸
❆ ♣❛rt✐r ❞❡ ✭✷✳✺✮✱ t❡♠♦s q✉❡
∂logK(N,p) ∂pi∂pj
=
0, s❡ i6=j,
−nj+ 2njpj−N p2j
p2
j(1−pj)2
, s❡ ✐ ❂ ❥, 1≤i, j ≤k,
♦ q✉❡ ✐♠♣❧✐❝❛
❊
−∂logL(N,p)
∂pi∂pj
=
0, s❡i6=j,
N pi(1−pi)
, s❡ ✐ ❂ ❥, 1≤i, j ≤k.
P♦rt❛♥t♦✱ ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❏❡✛r❡②s é ❞❛❞❛ ♣♦r
πJ(p|N)∝
detIN(p)
1
2
∝
k
Y
i=1
1 p1i/2(1−pi)1/2
. ✭✷✳✶✶✮
P♦rt❛♥t♦✱ ❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✐♥t❡❣r❛❞❛ ❞❡ ❏❡✛r❡②s é ❞❛❞❛ ♣♦r
LJ(N) = Z
(0,1)k
L(N,p)πJ(p|N)dp
∝ N!
(N −r)!
k
Y
i=1 1
Z
0
pni−1/2
i (1−pi)N−ni−1/2dpi
= N!
(N −r)!
k
Y
i=1
Γ(ni + 1/2)Γ(N −ni + 1/2)
Γ(N + 1) ,
N ≥r✳
▲♦❣♦✱ ♦ ♥ú❝❧❡♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✐♥t❡❣r❛❞❛ ❞❡ ❏❡✛r❡②s é ❞❛❞♦ ♣♦r
KJ(N) = N!
(N −r)!
k
Y
i=1
Γ(N −ni+ 1/2)
✶✹
✸ ❊st✐♠❛çã♦ ❞❡ ▼á①✐♠❛
❱❡r♦ss✐♠✐❧❤❛♥ç❛ ❡ ■♥t❡r✈❛❧❛r
◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ♠ét♦❞♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞❛s ❡st✐♠❛t✐✈❛s ❞❡ ♠á①✐♠❛ ✈❡✲ r♦ss✐♠✐❧❤❛♥ç❛ ❡ ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❡ ✐♥t❡r✈❛❧♦s ❞❡ ❝♦♥✜❛♥ç❛ ♣❛r❛ ♦ t❛♠❛♥❤♦ ♣♦♣✉❧❛✲ ❝✐♦♥❛❧✱ ❜❛s❡❛❞♦s ♥❛s ❢✉♥çõ❡s ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❛♣r❡s❡♥t❛❞❛s ♥♦ ❈❛♣ít✉❧♦ ✷✳ ◆♦ q✉❡ s❡❣✉❡ ✈❛♠♦s s✉♣♦r q✉❡ ♦ ♣r♦❝❡ss♦ ❝♦♥s✐st❛ ❡♠ ♣❡❧♦ ♠❡♥♦s ❞✉❛s ♦❝❛s✐õ❡s ❞❡ ❝❛♣t✉r❛ ✭k ≥ 2✮ ❡ q✉❡✱ ❡♠ ❝❛❞❛ ♦❝❛s✐ã♦✱ ❤❛❥❛ ♣❡❧♦ ♠❡♥♦s ✉♠ ✐♥❞✐✈í❞✉♦ ❝❛♣t✉r❛❞♦✳ ❉❡♥♦t❡♠♦s
♣♦r m = max{n1, . . . , nk} ♦ ♥ú♠❡r♦ ♠á①✐♠♦ ❞❡ ✐♥❞✐✈í❞✉♦s ❝❛♣t✉r❛❞♦s ❡♠ ✉♠❛ ♠❡s♠❛
♦❝❛s✐ã♦ ❡ n = n1 +· · ·+nk ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ❝❛♣t✉r❛s r❡❛❧✐③❛❞❛s✳ ❖❜s❡r✈❡♠♦s q✉❡ ♦
♥ú♠❡r♦ ❞❡ ✐♥❞✐✈í❞✉♦s ❞✐st✐♥t♦s ❝❛♣t✉r❛❞♦sr é t❛❧ q✉❡ m≤r ≤n✳
✸✳✶ ❊st✐♠❛çã♦ ❞❡ ▼á①✐♠❛ ❱❡r♦ss✐♠✐❧❤❛♥ç❛
◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛♠♦s r❡s✉❧t❛❞♦s q✉❡ ❝❛r❛❝t❡r✐③❛♠ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s ❢✉♥✲ çõ❡s ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞♦ ❈❛♣ít✉❧♦ ✷✱ ❛❧é♠ ❞❡ r❡s✉❧t❛❞♦s ❛ r❡s♣❡✐t♦ ❞❛ ♦❜t❡♥çã♦ ❞❛s ❡st✐♠❛t✐✈❛s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳
✸✳✶✳✶ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ P❡r✜❧❛❞❛
❉❡✈✐❞♦ ❛♦ ❝❛rát❡r ❞✐s❝r❡t♦ ❞♦ ♣❛râ♠❡tr♦ ❞❡ ✐♥t❡r❡ss❡ N✱ s❡ ❡①✐st✐r ✉♠ ♣♦♥t♦ ❞❡ ♠á✲
①✐♠♦✱ (N ,b pb)✱ ❞♦ ♥ú❝❧❡♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✭✷✳✹✮✱ ❡♥tã♦ Nb ♠❛①✐♠✐③❛ ♦ ♥ú❝❧❡♦
❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❡r✜❧❛❞❛ ✭✷✳✼✮✳
❉❡ ❢❛t♦✱ s✉♣♦♥❤❛♠♦s q✉❡ (N ,b bp) ♠❛①✐♠✐③❡ ♦ ♥ú❝❧❡♦ ✭✷✳✹✮ ❡ q✉❡✱ ♣❛r❛ ❝❛❞❛ N ✜①❛❞♦✱ N ≥ r✱ ❡①✐st❛ ✉♠ ú♥✐❝♦ ♣♦♥t♦✱ bp(N)✱ q✉❡ ♠❛①✐♠✐③❡ ❛ ❢✉♥çã♦ hN(p) = K(N,p)✱ p ∈
[0,1]k✳ ❊♥tã♦✱ ♦s ♣♦ssí✈❡✐s ❝❛♥❞✐❞❛t♦s ❛ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❞❡ K(N,p) sã♦ ♦s ♣♦♥t♦s ❞♦
❝♦♥❥✉♥t♦{(N,pb(N)) :N ≥r}✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡Nb é ✉♠ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❞♦ ♥ú❝❧❡♦ ❞❛
✶✺
P♦rt❛♥t♦✱ ❛ ❡st✐♠❛çã♦ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞♦ ♣❛râ♠❡tr♦ ❞❡ ✐♥t❡r❡ss❡ N ❛
♣❛rt✐r ❞♦ ♥ú❝❧❡♦ ✭✷✳✹✮ é ❡q✉✐✈❛❧❡♥t❡ à ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞♦ ♥ú❝❧❡♦ ✭✷✳✼✮✳
Pr♦♣♦s✐çã♦ ✸✳✶✳ ❖ ♥ú❝❧❡♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❡r✜❧❛❞❛✱KP(N)✱ ❝♦♥✈❡r❣❡ ♣❛r❛
0 s❡ m≤r < n ❡ ❝♦♥✈❡r❣❡ ♣❛r❛ exp{−n} s❡ r =n✱ q✉❛♥❞♦ N −→ ∞✳
Pr♦✈❛✿
❆ ♣❛rt✐r ❞❡ ✭✷✳✼✮ s❡❣✉❡ q✉❡
KP(N) = N!
(N −r)!
k
Y
i=1
(N −ni)N−ni
NN
=N(N −1)· · ·(N −r+ 1)
k
Y
i=1
(N −ni)N−ni
NN−niNni
= N
r
Nn
1− 1
N
· · ·
1− r−1
N
Yk
i=1
1− ni
N
N
1− ni
N
−ni
.
❈♦♠♦
lim
N→∞
1− x
N
= 1, ♣❛r❛ t♦❞♦ x r❡❛❧ ✭✸✳✶✮
❡
lim
N→∞
1− x
N
N
= exp{−x}, ♣❛r❛ t♦❞♦x >0, ✭✸✳✷✮
❡♥tã♦
lim
N→∞K P
(N) = (
0, s❡ r < n, exp{−n}, s❡ r=n.
P❛r❛ ♣r♦✈❛r ♦s ♣ró①✐♠♦s r❡s✉❧t❛❞♦s✱ ✈❛♠♦s ✉t✐❧✐③❛r ♦ ▲❡♠❛ ❛ s❡❣✉✐r✳ ▲❡♠❛ ✸✳✶✳ P❛r❛ q✉❛❧q✉❡r y >0 ❡ t∈1,2, . . . ✱ t❡♠♦s
✭❆✮
log(y+t)−log(y)>
t−1
X
i=0
✶✻
✭❇✮
log(y+t)−log(y)< 1 2(y+t) +
t
X
i=1
1 y+t−i −
1 2y
= 1
2(y+t) +
t−1
X
i=1
1 y+t−i +
1 2y.
Pr♦✈❛✿
❈♦♥s✐❞❡r❡♠♦s y >0✱ ✜①❛❞♦✳
✭❆✮
P❛r❛ ♣r♦✈❛r ❡st❛ ❛✜r♠❛çã♦ ❜❛st❛ ♥♦t❛r q✉❡
log(y+t)−log(y) =
y+t
Z
y
1 xdx
=
y+1
Z
y
1
xdx+. . .+
y+t
Z
y+t−1
1 xdx
> 1
y+ 1 +· · ·+ 1 y+t.
✭❇✮
Pr✐♠❡✐r❛♠❡♥t❡✱ ✈❛♠♦s ♠♦str❛r ❛ ❛✜r♠❛çã♦ ♣❛r❛ t = 1✱ ✐st♦ é✱
log(y+ 1)−log(y)< 1 2
1 y+ 1 +
1 y
.
◆❡st❡ s❡♥t✐❞♦✱ ❝♦♥s✐❞❡r❡♠♦s ❛ ❢✉♥çã♦ f(x) = 1/x ❞❡✜♥✐❞❛ ♣❛r❛ x > 0✳ ❈♦♠♦ f é
❝♦♥✈❡①❛✱ ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ✉♥❡ ♦s ♣♦♥t♦s y,1/y❡ y+ 1,1/(y+ 1) ✜❝❛ ❛❝✐♠❛ ❞♦
❣rá✜❝♦ ❞❡ f ♥♦ ✐♥t❡r✈❛❧♦ (y, y+ 1)✱ ✐st♦ é✱ 1
y+ξ < 1 y +
1 y+ 1 −
1 y
✶✼
❊♥tã♦✱
log(y+ 1)−log(y) =
1
Z
0
1 y+ξdξ
<
Z 1
0
1 y +
1 y+ 1 −
1 y
ξ
dξ
= 1 2
1 y+ 1 +
1 y
✭✸✳✸✮ ❡ ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡t≥2✱ ♦❜s❡r✈❡♠♦s q✉❡
log(y+t)−log(y) =
t
X
i=1
log(y+t+ 1−i)−log(y+t−i)
.
▲♦❣♦✱ s❡❣✉❡ ❞❡ ✭✸✳✸✮ q✉❡
log(y+t)−log(y)<
t
X
i=1
1 2
1
y+t−i+ 1 + 1 y+t−i
= 1
2(y+t)+
t−1
X
i=1
1 y+t−i+
1 2y
= 1
2(y+t)+
t
X
i=1
1 y+t−i−
1 2y.
P❛r❛ ♣r♦✈❛r ♦s r❡s✉❧t❛❞♦s q✉❡ s❡❣✉❡♠✱ ❝♦♥s✐❞❡r❛♠♦s ♦ ❧♦❣❛r✐t♠♦ ❞♦ ♥ú❝❧❡♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❡r✜❧❛❞❛ ✭✷✳✼✮✱ q✉❡ é ❞❛❞♦ ♣♦r
log(KP(N)) =
r−1
X
i=0
log(N −i) +
k
X
i=1
(N−ni) log(N −ni)−Nlog(N)
, N ≥r.
❆ ❡①t❡♥sã♦ ❞❡st❛ ❢✉♥çã♦ ♣❛r❛ ✈❛❧♦r❡s r❡❛✐s é ❞❛❞❛ ♣♦r
f(x) =
r−1
X
i=0
log(x−i) +
k
X
i=1
(x−ni) log(x−ni)−xlog(x)
, x≥r, ✭✸✳✹✮
♦♥❞❡ s✉♣♦♠♦s0 log(0) = 0✳
❖❜s❡r✈❡♠♦s q✉❡ ❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ❡♠ ✭✸✳✹✮ é ❝♦♥tí♥✉❛ ❡ ❞❡r✐✈á✈❡❧ ♥♦ ✐♥t❡r✈❛❧♦(r,∞)✱
s❡♥❞♦lim
x↓rf(x) =f(r)✱ ✐st♦ é✱f é ❝♦♥tí♥✉❛ à ❞✐r❡✐t❛ ❡♠ r✳ P♦rt❛♥t♦✱ ♦ ❡st✉❞♦ ❞♦ ❝♦♠♣♦r✲
✶✽
❞♦ ❡st✉❞♦ ❞♦ s✐♥❛❧ ❞❛ ❞❡r✐✈❛❞❛ ♣r✐♠❡✐r❛ ❞❡f✱ ❝✉❥❛ ❡①♣r❡ssã♦ é ❞❛❞❛ ♣♦r
f′(x) =
r−1
X
i=0
1 x−i +
k
X
i=1
log(x−ni)−log(x)
, x > r. ✭✸✳✺✮
Pr♦♣♦s✐çã♦ ✸✳✷✳ ❙❡ r = m✱ ❡♥tã♦ ❛ ❢✉♥çã♦ KP é ❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡✱ ✐st♦ é✱
KP(N + 1) < KP(N)✱ ♣❛r❛ t♦❞♦ N ≥ r✳ P♦rt❛♥t♦✱ ❛ ❡st✐♠❛t✐✈❛ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐✲
❧❤❛♥ç❛ ♣❡r✜❧❛❞❛ é NbP =m✳
Pr♦✈❛✿
❙✉♣♦♥❤❛♠♦s✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ q✉❡ m = n1✳ P❛r❛ ♠♦str❛r q✉❡ KP é
❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡ é s✉✜❝✐❡♥t❡ ♠♦str❛r✱ ✉s❛♥❞♦ ❛ ❡①♣r❡ssã♦ ✭✸✳✺✮ ❝♦♠r =n1✱ q✉❡ f′(x)<0♣❛r❛ t♦❞♦ x > n1✱ ♦✉ s❡❥❛✱
f′(x) =
nX1−1
i=0
1 x−i +
k
X
i=1
log(x−ni)−log(x)
<0, ♣❛r❛ x > n1. ✭✸✳✻✮
❉♦ ▲❡♠❛ ✸✳✶ ✭❆✮✱ s❡❣✉❡ q✉❡
nX1−1
i=0
1
x−i <log(x)−log(x−n1),
♦ q✉❡ ♣r♦✈❛ ♦ r❡s✉❧t❛❞♦✳
Pr♦♣♦s✐çã♦ ✸✳✸✳ ❙❡ r = n✱ ❡♥tã♦ ❛ ❢✉♥çã♦ KP é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡✱ ✐st♦ é✱
KP(N + 1) > KP(N) ♣❛r❛ t♦❞♦ N ≥ r✳ P♦rt❛♥t♦✱ ❛ ❡st✐♠❛t✐✈❛ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐✲
♠✐❧❤❛♥ç❛ ♣❡r✜❧❛❞❛ é NbP = +∞✳
Pr♦✈❛✿
P❛r❛ ♠♦str❛r q✉❡KP é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ❜❛st❛ ♠♦str❛r q✉❡✱ ✉s❛♥❞♦ ❛ ❡①♣r❡ssã♦
✭✸✳✺✮ ❝♦♠ r=n✱ f′(x)>0♣❛r❛ t♦❞♦ x > n✱ ♦✉ s❡❥❛✱
f′(x) =
n−1
X
i=0
1 x−i+
k
X
i=1
log(x−ni)−log(x)
>0, x > n,
♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ ❜❛st❛ ♠♦str❛r q✉❡✱ ♣❛r❛k ≥2✱ ✈❛❧❡
k
X
i=1
log(x)−log(x−ni)
<
n1+n2X+···+nk−1
i=0
1
✶✾
♣❛r❛ q✉❛✐sq✉❡r n1, n2, . . . , nk ∈N∗ ={1,2, . . .} ❡ q✉❛❧q✉❡r r❡❛❧ x > n1+n2+· · ·+nk✳ ❆
r❡❧❛çã♦ ✭✸✳✼✮ s❡rá ♣r♦✈❛❞❛ ♣♦r ✐♥❞✉çã♦ ✜♥✐t❛ s♦❜r❡k✳
Pr✐♠❡✐r❛♠❡♥t❡✱ ♦❜s❡r✈❡♠♦s ♣❡❧♦ ▲❡♠❛ ✸✳✶ ✭❇✮ q✉❡
log(x)−log(x−ni)<
1 2x +
ni
X
j=1
1 x−j −
1
2(x−ni) ✭✸✳✽✮
= 1
2x +
nXi−1
j=1
1 x−j +
1 2(x−ni)
, ✭✸✳✾✮
♣❛r❛ x > ni✱i= 1,2, . . . , k✳
❙✉♣♦♥❞♦ k = 2 ❡✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ q✉❡n1 ≥n2✱ s❡❣✉❡ ❞❡ ✭3.8✮ ❡ ✭3.9✮ q✉❡ 2
X
i=1
log(x)−log(x−ni)
< 1
2x +
n1
X
j=1
1 x−j −
1 2(x−n1)
+ 1
2x +
nX2−1
j=1
1 x−j +
1 2(x−n2)
= 1
x +
n1
X
j=1
1 x−j +
nX2−1
j=1
1 x−j +
1 2
1 x−n2
− 1
x−n1
< 1 x +
n1
X
j=1
1 x−j +
nX2−1
j=1
1 x−n1−j
=
n1+Xn2−1
j=0
1 x−j,
x > n1+n2✱ ♦ q✉❡ r❡s✉❧t❛ ✭3.7✮ ❝♦♠ k= 2✳
❆❣♦r❛✱ s✉♣♦♥❤❛♠♦s ♣♦r ✐♥❞✉çã♦ q✉❡ ❛ ❛✜r♠❛çã♦ ✭✸✳✼✮ ✈❛❧❤❛ ♣❛r❛ k =k′✱k′ ≥2✳ ❙♦❜
❡st❛ ❤✐♣ót❡s❡✱ ♣r♦✈❛♠♦s q✉❡ ❛ ❛✜r♠❛çã♦ ✭✸✳✼✮ ✈❛❧❡ ♣❛r❛k =k′ + 1✳
❙❡❥❛♠ n1, . . . , nk′, nk′+1 ♣❡rt❡♥❝❡♥t❡s ❛ N∗ ={1,2, . . . ,} ❡ x > n1+· · ·+nk′ +nk′+1✳ ❈♦♠♦x > n1+· · ·+nk′✱ s❡❣✉❡ ❞❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ q✉❡
k′ X
i=1
log(x)−log(x−ni)
<
n1+n2+X···+nk′−1
j=0
✷✵
❡ ♣❡❧♦ ▲❡♠❛ ✸✳✶ ✭❇✮ s❡❣✉❡ q✉❡
k′+1 X
i=1
log(x)−log(x−ni)
=
k′ X
i=1
log(x)−log(x−ni)
+ log x−log x−nk′+1
<
n1+n2+X···+nk′−1
j=0
1 x−j +
nXk′+1
j=1
1 x−j +
1 2
1 x −
1 x−nk′+1
<
n1+n2+X···+nk′−1
j=0
1 x−j +
nXk′+1
j=1
1
x−n1− · · · −nk′ + 1
−j
=
n1+n2+X···+nk′+1−1
j=0
1 x−j,
♦ q✉❡ ✐♠♣❧✐❝❛ ✭✸✳✼✮ ♣❛r❛k =k′+ 1✳
Pr♦♣♦s✐çã♦ ✸✳✹✳ P❛r❛ m < r < n✱ ❞❡✜♥❛♠♦s
x0 = sup
x∈(0,1/r] :
r−1
X
j=1
j
1/x−j < n−r
.
✭❛✮ ❙❡x0 = 1/r✱ ❡♥tã♦ NbP =r❀
✭❜✮ ❙❡ 0 < x0 < 1/r✱ ❡♥tã♦ NbP é ♦ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❞❡ KP(N) r❡str✐t♦ ❛♦ ❝♦♥❥✉♥t♦ {r, r+ 1, . . . ,[1/x0] + 1}✳
Pr♦✈❛✿
❖❜s❡r✈❡♠♦s q✉❡ KP(N)✱ ❞❛❞❛ ❡♠ ✭✷✳✼✮✱ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦
KP(N) = 1
Nn−r r−1
Y
i=1
1− i
N
Yk
i=1
1− ni
N
N−ni
, N ≥r.
❈♦♥s✐❞❡r❡♠♦s ❛ ❢✉♥çã♦ f ❞❡✜♥✐❞❛ ♣♦r
✷✶
♦♥❞❡ ❛s ❢✉♥çõ❡s gr ❡h sã♦ ❞❡✜♥✐❞❛s ♣♦r
gr(x) = xn−r r−1
Y
i=1
(1−ix), 0< x≤1/r,
h(x) =
k
Y
i=1
(1−nix)1/x−ni, 0< x≤1/r,
♦ q✉❡ ✐♠♣❧✐❝❛
f
1 N
=gr
1 N
h
1 N
=KP(N), N ≥r.
❆ ❢✉♥çã♦ gr é ❝♦♥tí♥✉❛✱ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈❛ ♥♦ ✐♥t❡r✈❛❧♦(0,1/r)✱limx↓0gr(x) = 0 ❡
❡①✐st❡ ✉♠ ♣♦♥t♦x0✱ 0< x0 ≤1/r✱ t❛❧ q✉❡ gr′(x)>0 s❡0< x < x0✳ ❈♦♠ ❡❢❡✐t♦✱
g′
r(x) = (n−r)xn−r−1 r−1
Y
i=1
(1−ix) +xn−r r−1
X
j=1
(−j)Y
i6=j
(1−ix)
=xn−r−1
(n−r)
r−1
Y
i=1
(1−ix) +x
r−1
X
j=1
(−j)Y
i6=j
(1−ix)
, 0< x <1/r,
♦ q✉❡ ✐♠♣❧✐❝❛
g′
r(x)>0⇐⇒x r−1
X
j=1
jY
i6=j
(1−ix)<(n−r)
r−1
Y
i=1
(1−ix)⇐⇒
r−1
X
j=1
j
1/x−j < n−r.
❖❜s❡r✈❡♠♦s q✉❡ ❛ ❢✉♥çã♦ q(x) =
rP−1
j=1
j
1/x−j é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ❡♠ (0,1/r] ❡ limx↓0q(x) = 0✳ ❊♥tã♦✱ gr′(x)>0✱ s❡ ❡ s♦♠❡♥t❡✱ 0< x < x0✱ ♦♥❞❡
x0 = sup
x∈(0,1/r] :
r−1
X
j=1
j
1/x−j < n−r
.
◆♦t❡♠♦s q✉❡x0 = 1/r s❡ ❡ s♦♠❡♥t❡ s❡ q(1/r)≤n−r✳
P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ ❢✉♥çã♦ h é ❝♦♥tí♥✉❛✱ ♣♦s✐t✐✈❛ ❡ ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ♥♦ ✐♥t❡r✈❛❧♦ (0,1/r]❡limx↓0h(x) =e−n✳ P❛r❛ ♠♦str❛r q✉❡h(x)é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ❜❛st❛ ♠♦str❛r q✉❡log(h(x))é ❝r❡s❝❡♥t❡ ❡♠ (0,1/r]✳ ◆❡st❡ s❡♥t✐❞♦✱ ♣❛r❛ 0< x <1/r
log(h(x)) =
k
X
i=1
1/x−ni
✷✷
♦ q✉❡ ✐♠♣❧✐❝❛
log(h(x))
′
=
k
X
i=1
− 1
x2 log(1−nix)−
1 x −ni
ni
1−nix
=− 1
x2
Xk
i=1
log(1−nix) +nx
>− 1
x2
Xk
i=1
(−nix) + k
X
i=1
nix
= 0,
♦♥❞❡ ❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ s❡❣✉❡ ❞❡ e−y >1−y ♣❛r❛ y r❡❛❧✳
P❡❧❛s ❝♦♥s✐❞❡r❛çõ❡s ❢❡✐t❛s ❛❝✐♠❛✱ ❝♦♥❝❧✉í♠♦s q✉❡ f(x) é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ♥♦
✐♥t❡r✈❛❧♦ (0, x0)✱ ♣♦✐s f′(x) = gr′(x)h(x) +gr(x)h′(x)>0✱ ♣❛r❛ 0< x < x0✱ ♦ q✉❡ ✐♠♣❧✐❝❛
KP(N + 1) =f
1 N + 1
< f
1 N
=KP(N), ♣❛r❛ N ≥1/x0.
▲♦❣♦✱ s❡ x0 = 1/r✱ ❡♥tã♦ NbP = r✱ ♦ q✉❡ ♣r♦✈❛ ♦ ✐t❡♠ ✭❛✮✳ ❙❡ 0 < x0 < 1/r✱ ❡♥tã♦
b
NP é ♦ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❞❡KP(N)r❡str✐t♦ ❛♦ ❝♦♥❥✉♥t♦ {r, r+ 1, . . . ,[1/x
0] + 1}✱ ♦ q✉❡
♣r♦✈❛ ♦ ✐t❡♠ ✭❜✮✳
✸✳✶✳✷ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ❈♦♥❞✐❝✐♦♥❛❧
Pr♦♣♦s✐çã♦ ✸✳✺✳ ❖ ♥ú❝❧❡♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧✱ KC(N)✱ ❝♦♥✈❡r❣❡
♣❛r❛0 s❡ r < n ❡ ❝♦♥✈❡r❣❡ ♣❛r❛ 1 s❡ r =n✱ q✉❛♥❞♦ N −→ ∞✳
Pr♦✈❛✿
❆ ♣❛rt✐r ❞❡ ✭✷✳✾✮ s❡❣✉❡ q✉❡
KC(N) = N(N −1)· · ·(N −r+ 1)
k
Y
j=1
1
N(N−1)· · ·(N −nj + 1)
= N
r
Nn
1− 1
N
1− 2
N
· · ·
1−r−1
N
Yk
i=1
1 1− 1
N
1− 2
N
· · · 1− ni−1
N
,
♣❛r❛ N ≥r✳
▲♦❣♦✱ ❞♦ ❢❛t♦
lim
N→∞
Nr
Nn =
(
0, s❡ r < n, 1, s❡ r=n,
✷✸
◆♦ q✉❡ s❡❣✉❡✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ❛ r❛③ã♦
KC(N + 1)
KC(N) =
(N + 1) (N + 1−r)
k
Y
i=1
N + 1−ni
N+ 1
=
1− r
N + 1
−1 Yk
i=1
1− ni
N + 1
, N ≥r. ✭✸✳✶✵✮
Pr♦♣♦s✐çã♦ ✸✳✻✳ ❙❡ r = m✱ ❡♥tã♦ ❛ ❢✉♥çã♦ KC é ❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡✱ ✐st♦ é✱
KC(N + 1) < KC(N)✱ ♣❛r❛ t♦❞♦ N ≥ r✳ P♦rt❛♥t♦✱ ❛ ❡st✐♠❛t✐✈❛ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐✲
♠✐❧❤❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧ é NbC =m✳
Pr♦✈❛✿
❙✉♣♦♥❤❛♠♦s✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ q✉❡ m = n1✳ ❊♥tã♦✱ s✉❜st✐t✉✐♥❞♦ r ♣♦r n1 ❡♠ ✭✸✳✶✵✮✱ s❡❣✉❡ q✉❡
KC(N+ 1)
KC(N) =
k
Y
i=2
1− ni
N + 1
<1, ♣❛r❛ t♦❞♦ N ≥r,
♦ q✉❡ ✐♠♣❧✐❝❛ r❡s✉❧t❛❞♦✳
P❛r❛ ♣r♦✈❛r ♦ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ ✉t✐❧✐③❛♠♦s ♦ s❡❣✉✐♥t❡ ❧❡♠❛✳
▲❡♠❛ ✸✳✷✳ P❛r❛ t♦❞♦ ✐♥t❡✐r♦ k ≥ 2 ❡ q✉❛✐sq✉❡r r❡❛✐s x1, . . . , xk t❛✐s q✉❡ 0 < xi < 1✱
1≤i≤k✱ t❡♠♦s
k
Q
i=1
(1−xi)>1− k
P
i=1
xi
Pr♦✈❛✿
P❛r❛ ♣r♦✈❛r ❡st❡ r❡s✉❧t❛❞♦ ✉t✐❧✐③❛♠♦s ✐♥❞✉çã♦ ✜♥✐t❛ s♦❜r❡ k ≥2✳
P❛r❛ k = 2✱ t❡♠♦s q✉❡
(1−x1)(1−x2) = 1−x1−x2+x1x2
>1−x1−x2,
✷✹
❙✉♣♦♥❤❛♠♦s ❛❣♦r❛ q✉❡ ❛ t❡s❡ ✈❛❧❤❛ ♣❛r❛ k >2✱ ✐st♦ é✱
k
Y
i=1
(1−xi)>1− k
X
i=1
xi.
❊♥tã♦✱
k+1
Y
i=1
(1−xi) = (1−xk+1)
k
Y
i=1
(1−xi)
>(1−xk+1)
1−
k
X
i=1
xi
= 1−
k
X
i=1
xi−xk+1+xk+1
k
X
i=1
xi
>1−
k+1
X
i=1
xi,
♦ q✉❡ ♣r♦✈❛ ♦ r❡s✉❧t❛❞♦✳
Pr♦♣♦s✐çã♦ ✸✳✼✳ ❙❡ r = n✱ ❡♥tã♦ ❛ ❢✉♥çã♦ KC é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡✱ ✐st♦ é✱
KC(N + 1) > KC(N) ♣❛r❛ t♦❞♦ N ≥ r✳ P♦rt❛♥t♦✱ ❛ ❡st✐♠❛t✐✈❛ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐✲
❧❤❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧ é NbC =∞✳
Pr♦✈❛✿
❆ ♣❛rt✐r ❞♦ ▲❡♠❛ ✸✳✷ t❡♠♦s q✉❡
k
Y
i=1
1− ni
N + 1
>1− n
N + 1, N ≥r,
♦ q✉❡ ♣❡r♠✐t❡ ❝♦♥❝❧✉✐r
KC(N + 1)
KC(N) >1, N ≥r.
P❛r❛ ❞❡♠♦♥str❛r ♦ ♣ró①✐♠♦ r❡s✉❧t❛❞♦✱ ✈❛♠♦s ❞❡✜♥✐r ❛ ❢✉♥çã♦
fr(x) = (1−rx)−1 k
Y
i=1
✷✺
❖❜s❡r✈❡♠♦s q✉❡✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ✭✸✳✶✵✮✱ fr s❛t✐s❢❛③ ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦
fr
1 N + 1
= K
C(N + 1)
KC(N) , N ≥r. ✭✸✳✶✷✮
P❛r❛ ❡st✉❞❛r ♦ ❝r❡s❝✐♠❡♥t♦ ✭❞❡❝r❡s❝✐♠❡♥t♦✮ ❞❡KC ♥❡❝❡ss✐t❛♠♦s ❞❡t❡r♠✐♥❛r ♦s ✈❛❧♦r❡s
❞❡N ♣❛r❛ ♦s q✉❛✐s ♦❝♦rr❡ KC(N + 1) > KC(N)✱ KC(N + 1) =KC(N) ♦✉ KC(N + 1) <
KC(N)✳ ❆ ♣❛rt✐r ❞❡ ✭✸✳✶✷✮✱ s❡❣✉❡ q✉❡ ♣❛r❛ r❡❛❧✐③❛r ❡st❡ ❡st✉❞♦ é s✉✜❝✐❡♥t❡ ❞❡t❡r♠✐♥❛r
♣❛r❛ q✉❛✐s ✈❛❧♦r❡s ❞❡ x ♥♦ ✐♥t❡r✈❛❧♦ (0,1/r) ♦❝♦rr❡ fr(x) < 1✱ fr(x) = 1 ♦✉ fr(x) > 1✳
◆♦ ❡♥t❛♥t♦✱ ❛♥t❡s ❞❡ ❛♥❛❧✐s❛r fr✱ s❡rá út✐❧ ❝♦♥s✐❞❡r❛r ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❛ r❡s♣❡✐t♦ ❞❡
❢✉♥çõ❡s ❝ô♥❝❛✈❛s✳
▲❡♠❛ ✸✳✸✳ ❙❡❥❛ f :R −→R ✭R é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✮ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳ ❙❡f ❢♦r ❝ô♥❝❛✈❛ ♥♦ ✐♥t❡r✈❛❧♦[a, b]✱ f(a)>0 ❡ f(b)<0✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♣♦♥t♦x0✱ a < x0 < b t❛❧ q✉❡ f(x0) = 0✱ f(x) > 0 s❡ a ≤ x < x0 ❡ f(x) < 0 s❡ x0 < x ≤ b✳ ❙♦❜ ❡st❛s ❝♦♥❞✐çõ❡s✱x0 é ú♥✐❝♦✳
Pr♦✈❛✿
P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❢✉♥çã♦ f✱ s❡❣✉❡ q✉❡ ❡①✐st❡ x0✱ a < x0 < b✱ t❛❧ q✉❡
f(x0) = 0.
P❡❧❛ ❝♦♥❝❛✈✐❞❛❞❡ ❞❡ f ♥♦ ✐♥t❡r✈❛❧♦ [a, b]✱ t❡♠♦s q✉❡ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f ❡♥tr❡
q✉❛✐sq✉❡ry1 ❡y2✱ a≤y1 < y2 ≤b✱ ♥ã♦ t❡♠ ♥❡♥❤✉♠ ♣♦♥t♦ ❧♦❝❛❧✐③❛❞♦ ❛❜❛✐①♦ ❞♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ✉♥❡ ♦s ♣♦♥t♦s y1, f(y1)
❡ y2, f(y2)
✱ ♦✉ s❡❥❛✱
f y1+α(y2−y1)
≥f(y1) +α
f(y2)−f(y1)
, ✭✸✳✶✸✮
♣❛r❛ t♦❞♦0< α <1✳
▲♦❣♦✱ t♦♠❛♥❞♦ y1 =a ❡y2 =x0 ❡♠ ✭✸✳✶✸✮✱ t❡♠♦s q✉❡
f a+α(x0−a)
≥f(a)−αf(a)
= (1−α)f(a)
>0, ♣❛r❛ t♦❞♦ 0< α <1,
♦ q✉❡ ✐♠♣❧✐❝❛
✷✻
P❛r❛ x0 < x < b ❞❡✈❡♠♦s t❡r f(x) < 0✳ ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛♠♦s q✉❡ f(x) ≥ 0 ♣❛r❛ ❛❧❣✉♠x0 < x < b✳ ❊♥tã♦✱ s❡❣✉❡ ❞❡ ✭✸✳✶✸✮ ❝♦♠ y1 =a ❡ y2 =x q✉❡
f a+α(x−a)≥f(a) +α[f(x)−f(a)]
= (1−α)f(a) +αf(x)
>0,
♣❛r❛ t♦❞♦0< α <1 ❡ ♣❛r❛ α= x0−a
/ x−a t❡♠♦s f x0
>0✱ ♦ q✉❡ é ❛❜s✉r❞♦✳
▲❡♠❛ ✸✳✹✳ P❛r❛ m < r < n✱ ❡①✐st❡ x0✱ 0 < x0 < 1/r✱ t❛❧ q✉❡ fr(x0) = 1✱ fr(x) < 1 s❡
0< x < x0 ❡ fr(x)>1 s❡ x0 < x <1/r✳
Pr♦✈❛✿
❖❜s❡r✈❡♠♦s q✉❡✱ ❞❡✜♥✐♥❞♦
g(x) =
k
Q
i=1
(1−nix), x∈R ❡
hr(x) = 1−rx, x∈R,
t❡♠♦s
fr(x) =
g(x) hr(x)
, 0≤x <1/r.
❆s ❞❡r✐✈❛❞❛s ❞❡ ♣r✐♠❡✐r❛ ❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❞❡ g sã♦ ❞❛❞❛s ♣♦r
g′(x) =
k
X
i=1
(−ni) k
Y
j=1
j6=i
(1−njx),
g′′(x) =
k
X
i=1
(−ni) k
X
j=1
j6=i
(−nj) k
Y
s=1
s6=i,j
(1−nsx)
=
k
X
i=1
k
X
j=1
j6=i
ninj k
Y
s=1
s6=i,j
(1−nsx)
❡ ❝♦♠♦1−nix >0 ♣❛r❛ 0≤x≤1/r✱ i= 1, . . . , n✱ t❡♠♦s
g′(x)<0, ♣❛r❛ t♦❞♦ 0≤x≤1/r,
✷✼
P♦rt❛♥t♦✱ ❛s ❢✉♥çõ❡s g✱ g′ ❡ g′′ sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❡ g t❛♠❜é♠ é ❡str✐t❛♠❡♥t❡
❞❡❝r❡s❝❡♥t❡ ❡ ❝♦♥✈❡①❛ ♥♦ ✐♥t❡r✈❛❧♦ [0,1/r]✱ ❝♦♠ g(0) = 1 ❡ g(1/r) > 0✳ P♦r ♦✉tr♦ ❧❛❞♦✱
❛ ❢✉♥çã♦ hr é ❧✐♥❡❛r ❡ ❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡ ♥♦ ✐♥t❡r✈❛❧♦ [0,1/r] ❝♦♠ hr(0) = 1 ❡
hr(1/r) = 0✳ ❆ ❋✐❣✉r❛ ✶ ✐❧✉str❛ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s ❢✉♥çõ❡sg ❡hr ♥♦ ✐♥t❡r✈❛❧♦ [0,1/r]
❡✱ ❛ ♣❛rt✐r ❞❛ ♦❜s❡r✈❛çã♦ ❞❡st❡ ❣rá✜❝♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ❞❡✈❡ ❡①✐st✐r ✉♠ ú♥✐❝♦ ♣♦♥t♦
x0✱ 0 < x0 < 1/r t❛❧ q✉❡ g(x0) = hr(x0)✱ g(x) < hr(x) s❡ 0 < x < x0 ❡ g(x) > hr(x) s❡
x0 < x <1/r✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❛ t❡s❡✳ ❯♠❛ ❥✉st✐✜❝❛t✐✈❛ ❢♦r♠❛❧ ♣❛r❛ ❡st❛s ❝♦♥❝❧✉sõ❡s é ❞❛❞❛ ❛ s❡❣✉✐r✳
❋✐❣✉r❛ ✶✳ ●rá✜❝♦s ❞❛s ❢✉♥çõ❡s g ❡hr ♥♦ ✐♥t❡r✈❛❧♦ [0,1/r]✳
❉❡✜♥❛♠♦s ❛ ❞✐❢❡r❡♥ç❛ dr(x) = hr(x)−g(x)✱ x ∈ R✳ ❊♥tã♦✱ ❛s ❢✉♥çõ❡s dr✱ d′r ❡ d′′r
sã♦ ❝♦♥tí♥✉❛s ♥❛ r❡t❛ R✳ ❆❧é♠ ❞✐ss♦✱ dr é ✉♠❛ ❢✉♥çã♦ ❝ô♥❝❛✈❛ ♥♦ ✐♥t❡r✈❛❧♦ [0,1/r]✱ ♣♦✐s
d′′
r(x) = h′′r(x)−gr′′(x) = −gr′′(x)<0♣❛r❛ t♦❞♦ 0≤x≤1/r✳
◆♦t❡♠♦s t❛♠❜é♠ q✉❡
d′r(0) =h′r(0)−g′(0) =−r+
k
X
i=1
ni >0,
♦ q✉❡ ✐♠♣❧✐❝❛✱ ♣❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ d′
✷✽
t♦❞♦0≤x≤δ✳ ❈♦♠♦dr(0) = 0✱ s❡❣✉❡ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ q✉❡
dr(x)>0, ♣❛r❛ t♦❞♦0< x≤δ. ✭✸✳✶✹✮
❖❜s❡r✈❡♠♦s t❛♠❜é♠ q✉❡ dr(δ) > 0 ❡ dr(1/r) = hr(1/r)−g(1/r) = −g(1/r) < 0✳
▲♦❣♦✱ ♣❡❧♦ ▲❡♠❛ ✸✳✸ ❛♣❧✐❝❛❞♦ à ❢✉♥çã♦ dr ♥♦ ✐♥t❡r✈❛❧♦ [δ,1/r] ❡ ♣♦r ✭✸✳✶✹✮✱ s❡❣✉❡ q✉❡
❡①✐st❡ 0< x0 <1/r t❛❧ q✉❡
dr(x) >0, s❡0< x < x0,
dr(x) = 0, s❡x=x0,
dr(x) <0, s❡x0 < x≤1/r,
♦ q✉❡ ✐♠♣❧✐❝❛
g(x)< hr(x), s❡0< x < x0,
g(x) =hr(x), s❡x=x0,
g(x)> hr(x), s❡x0 < x < 1/r,
❡ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡✳
P♦rt❛♥t♦✱ s❡❣✉❡ ❞♦ ▲❡♠❛ ✸✳✹ q✉❡✱ ♣❛r❛ m < r < n✱ ♣♦❞❡♠♦s ❞❡✜♥✐r δr ❝♦♠♦ s❡♥❞♦ ♦
♠❡♥♦r ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦t t❛❧ q✉❡
fr
1 r+t
≤1, ✭✸✳✶✺✮
♦✉ s❡❥❛✱
δr = min
t∈N∗ :
k
Y
i=1
(r+t−ni)≤t(r+t)k−1
.
❆ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r ❝❛r❛❝t❡r✐③❛ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❢✉♥çã♦ KC q✉❛♥t♦ ❛♦ ❝r❡s❝✐✲
♠❡♥t♦ ✭❞❡❝r❡s❝✐♠❡♥t♦✮ ❡ ❢♦r♥❡❝❡ ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ❛ ❡st✐♠❛t✐✈❛ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐✲ ❧❤❛♥ç❛✱ ♥♦ ❝❛s♦ ❡♠ q✉❡m < r < n✳
Pr♦♣♦s✐çã♦ ✸✳✽✳ P❛r❛ m < r < n✱ t❡♠♦s q✉❡
✭❛✮ s❡ Qk
i=1
(r+δr−ni)< δr(r+δr)k−1✱ ❡♥tã♦ KC(N + 1) < KC(N) ♣❛r❛ N ≥r+δr−1
❡ KC(N + 1) > KC(N) ♣❛r❛ r ≤N < r+δ
r−1✳ P♦rt❛♥t♦✱ ❛ ❡st✐♠❛t✐✈❛ ❞❡ ♠á①✐♠❛
✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧ é ú♥✐❝❛ ❡ ❞❛❞❛ ♣♦rNbC =r+δ
r−1❀
✭❜✮ s❡ Qk
i=1
(r +δr −ni) = δr(r +δr)k−1✱ ❡♥tã♦ KC(N + 1) < KC(N) ♣❛r❛ N ≥ r +δr✱
KC(N + 1)> KC(N)♣❛r❛ r ≤N < r+δ
✷✾
❝❛s♦✱ ❤á ❞✉❛s ❡st✐♠❛t✐✈❛s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧ NbC
1 =r+δr−1 ❡
b NC
2 =Nb1C+ 1✳ Pr♦✈❛✿
✭❛✮
❙❡ Qk
i=1
(r+δr−ni) < δr(r+δr)k−1 t❡♠♦s✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ fr✱ ❞❛❞❛ ❡♠ ✭✸✳✶✶✮ q✉❡
fr 1/(r+δr)
<1✳ P♦rt❛♥t♦✱ s❡❣✉❡ ❞♦ ▲❡♠❛ ✸✳✹ ❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ δr ❞❛❞❛ ❡♠ ✭✸✳✶✺✮ q✉❡
fr
1 r+t
<1, s❡t ≥δr,
fr
1 r+t
>1, s❡1≤t < δr,
♦ q✉❡ ✐♠♣❧✐❝❛✱ ♣❡❧❛ r❡❧❛çã♦ ✭✸✳✶✷✮ q✉❡
KC(N + 1)
KC(N) <1, s❡N ≥r+δr−1,
KC(N + 1)
KC(N) >1, s❡r ≤N < r+δr−1,
♦ q✉❡ ♣r♦✈❛ ♦ r❡s✉❧t❛❞♦✳
✭❜✮
❙❡ Qk
i=1
(r+δr−ni) = δr(r+δr)k−1 t❡♠♦s✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ fr ❞❛❞❛ ❡♠ ✭✸✳✶✶✮✱ q✉❡
fr(1/(r+δr)) = 1✳ ▲♦❣♦✱ ♣❡❧❛ r❡❧❛çã♦ ✭✸✳✶✷✮ s❡❣✉❡
KC(r+δ
r−1) =KC(r+δr).
❡ ♣❡❧♦ ▲❡♠❛ ✭✸✳✹✮ ❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡δr✱ ❞❛❞❛ ❡♠ ✭✸✳✶✺✮✱ s❡❣✉❡
fr
1 r+t
<1, s❡ t≥δr+ 1,
fr
1 r+t
>1, s❡ 1≤t < δr,
♦ q✉❡ ✐♠♣❧✐❝❛✱ ♣❡❧❛ r❡❧❛çã♦ ✭✸✳✶✷✮✱ q✉❡
KC(N + 1)
KC(N) <1, s❡ N ≥r+δr,
KC(N + 1)
✸✵
♦✉ s❡❥❛✱ r+δr−1 ❡r+δr ♠❛①✐♠✐③❛♠ KC✳
✸✳✶✳✸ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ■♥t❡❣r❛❞❛ ❯♥✐❢♦r♠❡
Pr♦♣♦s✐çã♦ ✸✳✾✳ ❖ ♥ú❝❧❡♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✐♥t❡❣r❛❞❛ ✉♥✐❢♦r♠❡✱ KU(N)✱
❝♦♥✈❡r❣❡ ♣❛r❛0 q✉❛♥❞♦ N −→ ∞✳
Pr♦✈❛✿
❆ ♣❛rt✐r ❞❡ ✭✷✳✶✵✮ t❡♠♦s
KU(N) = N! (N −r)!
k
Y
i=1
(N −ni)!
(N + 1)!
=N(N −1)· · ·(N −r+ 1)
k
Y
i=1
1
(N + 1)N(N −1)· · ·(N −ni+ 1)
= N
r
(N + 1)k
1− 1
N
· · ·
1− r−1
N
Yk
i=1
1 Nni 1− 1
N
· · · 1− ni−1
N
= N
r
Nn(N+ 1)k
1− 1
N
· · ·
1−r−1
N
Yk
i=1
1 1− 1
N
· · · 1− ni−1
N
.
▲♦❣♦✱ ❞❡ ✭✸✳✶✮ ❡ ❞♦ ❢❛t♦
lim
N→∞
Nr
Nn(N+ 1)k = 0
s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳
◆♦ q✉❡ s❡❣✉❡ ❝♦♥s✐❞❡r❡♠♦s ❛ r❛③ã♦
KU(N + 1)
KU(N) =
(N + 1) (N + 1−r)(N + 2)k
k
Y
i=1
(N + 1−ni)
=
N + 1 N + 1−r
N + 1 N + 2
k Yk
i=1
N + 1−ni
N + 1
=
1− r
N + 1
−1
1 + 1
N + 1
−k Yk
i=1
1− ni
N + 1
, N ≥r. ✭✸✳✶✻✮
Pr♦♣♦s✐çã♦ ✸✳✶✵✳ ❙❡ r=m✱ ❡♥tã♦ KU é ❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡✱ ✐st♦ é✱ KU(N+ 1)<
KU(N) ♣❛r❛ t♦❞♦ N ≥ r✳ P♦rt❛♥t♦✱ ❛ ❡st✐♠❛t✐✈❛ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✐♥t❡❣r❛❞❛
