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  1. ❆❞❞✐t✐♦♥ ✐♥ t❤❡ ❏❛❝♦❜✐❛♥ ♦❢ ♥♦♥✲❤②♣❡r❡❧❧✐♣t✐❝ ❣❡♥✉s ✸ ❝✉r✈❡s ❛♥❞ r❛t✐♦♥❛❧✐t② ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♣♦✐♥ts ♦❢ ❛ ❧✐♥❡ ✇✐t❤ ❛ ♣❧❛♥❡ q✉❛rt✐❝ ❙té♣❤❛♥❡ ❋❧♦♥✱ ❘♦❣❡r ❖②♦♥♦✱ ❈❤r✐st♦♣❤❡ ❘✐t③❡♥t❤❛❧❡r ❈✳ ❘✐t③❡♥t❤❛❧❡r✱ ❈✳◆✳❘✳❙✳ ■♥st✐t✉t ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❞❡ ▲✉♠✐♥② ▲✉♠✐♥② ❈❛s❡ ✾✸✵✱ ❋✶✸✷✽✽ ▼❛rs❡✐❧❧❡ ❈❊❉❊❳ ✾ ❡✲♠❛✐❧ ✿ r✐t③❡♥t❤❅✐♠❧✳✉♥✐✈✲♠rs✳❢r ✇❡❜ ✿ ❤tt♣ ✿✴✴✐♠❧✳✉♥✐✈✲♠rs✳❢r✴ ∼ r✐t③❡♥t❤✴ ❙té♣❤❛♥❡ ❋❧♦♥✱ ❘♦❣❡r ❖②♦♥♦✱ ❈❤r✐st♦♣❤❡ ❘✐t③❡♥t❤❛❧❡r ✭✮ ❆❞❞✐t✐♦♥ ✐♥ t❤❡ ❏❛❝♦❜✐❛♥ ♦❢ ♥♦♥✲❤②♣❡r❡❧❧✐♣t✐❝ ❣❡♥✉s ✸ ❝✉r✈❡s ❛♥❞ r❛t✐♦♥❛❧✐t② ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♣♦✐♥ts ♦❢ ❛ ❧✐♥❡ ✇✐t❤ ❛ ♣❧❛♥❡ q✉❛rt✐❝ ✶ ✴ ✶✶

  2. ●❡♥❡r❛❧ s❡tt✐♥❣ ◆♦♥✲❤②♣❡r❡❧❧✐♣t✐❝ ❣❡♥✉s ✸ ❝✉r✈❡s ❂ s♠♦♦t❤ ♣❧❛♥❡ q✉❛rt✐❝s✳ ❈❤♦✐❝❡ ♦❢ ❛ ❣♦♦❞ ❞✐✈✐s♦r ❛t ✐♥✜♥✐t② � ❝♦♥❞✐t✐♦♥ ( ∗ ) ✿ ❚❤❡r❡ ✐s ❛ r❛t✐♦♥❛❧ ❧✐♥❡ ❧ ∞ ✇❤✐❝❤ ❝r♦ss❡s t❤❡ q✉❛rt✐❝ ❈ ✐♥ ❢♦✉r ❦ ✲♣♦✐♥ts P ∞ ✶ , P ∞ ✷ , P ∞ ✸ , P ∞ ✹ ✳ ▲❡t ❉ ∞ = P ∞ ✶ + P ∞ ✷ + P ∞ ✸ ✳ ❇❡❝❛✉s❡ ❙②♠ ✸ ❈ → ❏❛❝ ( ❈ ) ✱ ❉ + �→ ❉ + − ❉ ∞ ✐s s✉r❥❡❝t✐✈❡✱ ❛ ❡❧❡♠❡♥t ❉ ∈ ❏❛❝ ( ❈ ) ✐s ❛ s✉♠ ♦❢ t❤r❡❡ ♣♦✐♥ts ❝❛❧❧❡❞ ❉ + ✳ ◗✉❡st✐♦♥ ✿ ❤♦✇ ❞♦ ✇❡ ❝♦♠♣✉t❡ ✭❡✣❝✐❡♥t❧②✮ ❉ ✶ + ❉ ✷ ✐♥ t❡r♠s ♦❢ ❉ + ✶ , ❉ + ✷ ❄ ❙té♣❤❛♥❡ ❋❧♦♥✱ ❘♦❣❡r ❖②♦♥♦✱ ❈❤r✐st♦♣❤❡ ❘✐t③❡♥t❤❛❧❡r ✭✮ ❆❞❞✐t✐♦♥ ✐♥ t❤❡ ❏❛❝♦❜✐❛♥ ♦❢ ♥♦♥✲❤②♣❡r❡❧❧✐♣t✐❝ ❣❡♥✉s ✸ ❝✉r✈❡s ❛♥❞ r❛t✐♦♥❛❧✐t② ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♣♦✐♥ts ♦❢ ❛ ❧✐♥❡ ✇✐t❤ ❛ ♣❧❛♥❡ q✉❛rt✐❝ ✷ ✴ ✶✶

  3. ❆ ❣❡♦♠❡tr✐❝ ❛❞❞✐t✐♦♥ ❛❧❣♦r✐t❤♠ ▲❡t ❉ ✶ , ❉ ✷ ∈ ❏❛❝ ( ❈ )( ❦ ) ✳ ❚❤❡♥ ❉ ✶ + ❉ ✷ ✐s ❡q✉✐✈❛❧❡♥t t♦ ❛ ❞✐✈✐s♦r ❉ = ❉ + − ❉ ∞ ✱ ✇❤❡r❡ t❤❡ ♣♦✐♥ts ✐♥ t❤❡ s✉♣♣♦rt ♦❢ ❉ + ❛r❡ ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧❣♦r✐t❤♠ ✿ ✶ ❚❛❦❡ ❛ ❝✉❜✐❝ ❊ ✇❤✐❝❤ ❣♦❡s ✭✇✐t❤ ♠✉❧t✐♣❧✐❝✐t②✮ t❤r♦✉❣❤ t❤❡ s✉♣♣♦rt ♦❢ ❉ + ✶ , ❉ + ✷ ❛♥❞ P ∞ ✶ , P ∞ ✷ , P ∞ ✹ ✳ ❚❤✐s ❝✉❜✐❝ ❛❧s♦ ❝r♦ss❡s ❈ ✐♥ t❤❡ r❡s✐❞✉❛❧ ❡✛❡❝t✐✈❡ ❞✐✈✐s♦r ❉ ✸ ✳ ✷ ❚❛❦❡ ❛ ❝♦♥✐❝ ◗ ✇❤✐❝❤ ❣♦❡s t❤r♦✉❣❤ t❤❡ s✉♣♣♦rt ♦❢ ❉ ✸ ❛♥❞ P ∞ ✶ , P ∞ ✷ ✳ ❚❤✐s ❝♦♥✐❝ ❛❧s♦ ❝r♦ss❡s ❈ ✐♥ t❤❡ r❡s✐❞✉❛❧ ❡✛❡❝t✐✈❡ ❞✐✈✐s♦r ❉ + ✳ ❲❤② ❄ ❇❡❝❛✉s❡ ( ❧ ∞ · ❈ ) ∼ κ ✱ ( ◗ · ❈ ) ∼ ✷ κ ❛♥❞ ( ❊ · ❈ ) ∼ ✸ κ ✳ ❙té♣❤❛♥❡ ❋❧♦♥✱ ❘♦❣❡r ❖②♦♥♦✱ ❈❤r✐st♦♣❤❡ ❘✐t③❡♥t❤❛❧❡r ✭✮ ❆❞❞✐t✐♦♥ ✐♥ t❤❡ ❏❛❝♦❜✐❛♥ ♦❢ ♥♦♥✲❤②♣❡r❡❧❧✐♣t✐❝ ❣❡♥✉s ✸ ❝✉r✈❡s ❛♥❞ r❛t✐♦♥❛❧✐t② ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♣♦✐♥ts ♦❢ ❛ ❧✐♥❡ ✇✐t❤ ❛ ♣❧❛♥❡ q✉❛rt✐❝ ✸ ✴ ✶✶

  4. ❆ ❝❤♦r❞ ❝♦♥str✉❝t✐♦♥ ❙té♣❤❛♥❡ ❋❧♦♥✱ ❘♦❣❡r ❖②♦♥♦✱ ❈❤r✐st♦♣❤❡ ❘✐t③❡♥t❤❛❧❡r ✭✮ ❆❞❞✐t✐♦♥ ✐♥ t❤❡ ❏❛❝♦❜✐❛♥ ♦❢ ♥♦♥✲❤②♣❡r❡❧❧✐♣t✐❝ ❣❡♥✉s ✸ ❝✉r✈❡s ❛♥❞ r❛t✐♦♥❛❧✐t② ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♣♦✐♥ts ♦❢ ❛ ❧✐♥❡ ✇✐t❤ ❛ ♣❧❛♥❡ q✉❛rt✐❝ ✹ ✴ ✶✶

  5. ✶ ❞❡❣ ❤ ✶ ✷ ❛♥❞ ❞❡❣ ❤ ✷ ✸ ✐❢ P ✶ P ✷ ✭t❛♥❣❡♥t ❝❛s❡✮ ❀ ✷ ❞❡❣ ❤ ✶ ✶ ❛♥❞ ❞❡❣ ❤ ✷ ✸ ✐❢ P ✶ P ✷ P ✹ ✭✢❡① ❝❛s❡✮✳ ■❢ ② ✸ ❝❤❛r ❦ ✸ ✇❡ ❝❛♥ ✇r✐t❡ ❈ ❤ ✷ ① ② ❢ ✹ ① ✳ ✸ ❞❡❣ ❤ ✶ ✶ ❛♥❞ ❞❡❣ ❤ ✷ ✷ ✐❢ P ✶ P ✷ P ✸ P ✹ ✭❤②♣❡r✢❡① ❝❛s❡✮✳ ❚❤❡s❡ ❝✉r✈❡s ❛r❡ t❤❡ ❈ ✸ ✹ ✲❝✉r✈❡s✳ ✹ ■❢ ❝❤❛r ❦ ② ✸ ✸✱ ❈ ❢ ✹ ① ✭P✐❝❛r❞ ❝✉r✈❡s✮ ✐✐❢ P ✶ ✐s ❛ r❛t✐♦♥❛❧ ●❛❧♦✐s ♣♦✐♥t✳ ✿ t❤❡ ♠♦r❡ s♣❡❝✐❛❧✱ t❤❡ ❜❡tt❡r✳ ❙♣❡❝✐❛❧ ❢♦r♠s ♦❢ t❤❡ ❝✉r✈❡ ❈ ❛❞♠✐ts ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠ ❈ : ② ✸ + ❤ ✶ ( ① ) ② ✷ + ❤ ✷ ( ① ) ② = ❢ ✹ ( ① ) ✇✐t❤ ❞❡❣ ( ❢ ✹ ) ≤ ✹ ❛♥❞ ❙té♣❤❛♥❡ ❋❧♦♥✱ ❘♦❣❡r ❖②♦♥♦✱ ❈❤r✐st♦♣❤❡ ❘✐t③❡♥t❤❛❧❡r ✭✮ ❆❞❞✐t✐♦♥ ✐♥ t❤❡ ❏❛❝♦❜✐❛♥ ♦❢ ♥♦♥✲❤②♣❡r❡❧❧✐♣t✐❝ ❣❡♥✉s ✸ ❝✉r✈❡s ❛♥❞ r❛t✐♦♥❛❧✐t② ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♣♦✐♥ts ♦❢ ❛ ❧✐♥❡ ✇✐t❤ ❛ ♣❧❛♥❡ q✉❛rt✐❝ ✺ ✴ ✶✶

  6. ✷ ❞❡❣ ❤ ✶ ✶ ❛♥❞ ❞❡❣ ❤ ✷ ✸ ✐❢ P ✶ P ✷ P ✹ ✭✢❡① ❝❛s❡✮✳ ■❢ ② ✸ ❝❤❛r ❦ ✸ ✇❡ ❝❛♥ ✇r✐t❡ ❈ ❤ ✷ ① ② ❢ ✹ ① ✳ ✸ ❞❡❣ ❤ ✶ ✶ ❛♥❞ ❞❡❣ ❤ ✷ ✷ ✐❢ P ✶ P ✷ P ✸ P ✹ ✭❤②♣❡r✢❡① ❝❛s❡✮✳ ❚❤❡s❡ ❝✉r✈❡s ❛r❡ t❤❡ ❈ ✸ ✹ ✲❝✉r✈❡s✳ ✹ ■❢ ❝❤❛r ❦ ② ✸ ✸✱ ❈ ❢ ✹ ① ✭P✐❝❛r❞ ❝✉r✈❡s✮ ✐✐❢ P ✶ ✐s ❛ r❛t✐♦♥❛❧ ●❛❧♦✐s ♣♦✐♥t✳ ✿ t❤❡ ♠♦r❡ s♣❡❝✐❛❧✱ t❤❡ ❜❡tt❡r✳ ❙♣❡❝✐❛❧ ❢♦r♠s ♦❢ t❤❡ ❝✉r✈❡ ❈ ❛❞♠✐ts ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠ ❈ : ② ✸ + ❤ ✶ ( ① ) ② ✷ + ❤ ✷ ( ① ) ② = ❢ ✹ ( ① ) ✇✐t❤ ❞❡❣ ( ❢ ✹ ) ≤ ✹ ❛♥❞ ✶ ❞❡❣ ( ❤ ✶ ) ≤ ✷ ❛♥❞ ❞❡❣ ( ❤ ✷ ) ≤ ✸ ✐❢ P ∞ ✶ = P ∞ ✭t❛♥❣❡♥t ❝❛s❡✮ ❀ ✷ ❙té♣❤❛♥❡ ❋❧♦♥✱ ❘♦❣❡r ❖②♦♥♦✱ ❈❤r✐st♦♣❤❡ ❘✐t③❡♥t❤❛❧❡r ✭✮ ❆❞❞✐t✐♦♥ ✐♥ t❤❡ ❏❛❝♦❜✐❛♥ ♦❢ ♥♦♥✲❤②♣❡r❡❧❧✐♣t✐❝ ❣❡♥✉s ✸ ❝✉r✈❡s ❛♥❞ r❛t✐♦♥❛❧✐t② ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♣♦✐♥ts ♦❢ ❛ ❧✐♥❡ ✇✐t❤ ❛ ♣❧❛♥❡ q✉❛rt✐❝ ✺ ✴ ✶✶

  7. ✸ ❞❡❣ ❤ ✶ ✶ ❛♥❞ ❞❡❣ ❤ ✷ ✷ ✐❢ P ✶ P ✷ P ✸ P ✹ ✭❤②♣❡r✢❡① ❝❛s❡✮✳ ❚❤❡s❡ ❝✉r✈❡s ❛r❡ t❤❡ ❈ ✸ ✹ ✲❝✉r✈❡s✳ ✹ ■❢ ❝❤❛r ❦ ② ✸ ✸✱ ❈ ❢ ✹ ① ✭P✐❝❛r❞ ❝✉r✈❡s✮ ✐✐❢ P ✶ ✐s ❛ r❛t✐♦♥❛❧ ●❛❧♦✐s ♣♦✐♥t✳ ✿ t❤❡ ♠♦r❡ s♣❡❝✐❛❧✱ t❤❡ ❜❡tt❡r✳ ❙♣❡❝✐❛❧ ❢♦r♠s ♦❢ t❤❡ ❝✉r✈❡ ❈ ❛❞♠✐ts ❛♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠ ❈ : ② ✸ + ❤ ✶ ( ① ) ② ✷ + ❤ ✷ ( ① ) ② = ❢ ✹ ( ① ) ✇✐t❤ ❞❡❣ ( ❢ ✹ ) ≤ ✹ ❛♥❞ ✶ ❞❡❣ ( ❤ ✶ ) ≤ ✷ ❛♥❞ ❞❡❣ ( ❤ ✷ ) ≤ ✸ ✐❢ P ∞ ✶ = P ∞ ✭t❛♥❣❡♥t ❝❛s❡✮ ❀ ✷ ✷ ❞❡❣ ( ❤ ✶ ) ≤ ✶ ❛♥❞ ❞❡❣ ( ❤ ✷ ) ≤ ✸ ✐❢ P ∞ ✶ = P ∞ ✷ = P ∞ ✭✢❡① ❝❛s❡✮✳ ■❢ ✹ ❝❤❛r ( ❦ ) � = ✸ ✇❡ ❝❛♥ ✇r✐t❡ ❈ : ② ✸ + ❤ ✷ ( ① ) ② = ❢ ✹ ( ① ) ✳ ❙té♣❤❛♥❡ ❋❧♦♥✱ ❘♦❣❡r ❖②♦♥♦✱ ❈❤r✐st♦♣❤❡ ❘✐t③❡♥t❤❛❧❡r ✭✮ ❆❞❞✐t✐♦♥ ✐♥ t❤❡ ❏❛❝♦❜✐❛♥ ♦❢ ♥♦♥✲❤②♣❡r❡❧❧✐♣t✐❝ ❣❡♥✉s ✸ ❝✉r✈❡s ❛♥❞ r❛t✐♦♥❛❧✐t② ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♣♦✐♥ts ♦❢ ❛ ❧✐♥❡ ✇✐t❤ ❛ ♣❧❛♥❡ q✉❛rt✐❝ ✺ ✴ ✶✶

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