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❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ❏❛♥✉❛r② ✶✻✱ ✷✵✶✽

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❘❛♥❦✐♥❣ ✐♥❝♦♠❡ ❞✐str✐❜✉t✐♦♥ ❛♥❞ ▲♦r❡♥③ ❝✉r✈❡s✿ P❛rt✐❛❧ ❛♥❞ ❝♦♠♣❧❡t❡ ♦r❞❡r✐♥❣s

✭✐✮ P❛rt✐❛❧ ♦r❞❡r✐♥❣s✿ ❙t♦❝❤❛st✐❝ ❛♥❞ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡✱ ▲♦r❡♥③ ❞♦♠✐♥❛♥❝❡ ✭✐✐✮ ❈♦♠♣❧❡t❡ ♦r❞❡r✐♥❣s✿ ❛✳ ❙♦❝✐❛❧ ✇❡❧❢❛r❡ ❝r✐t❡r✐❛ ❜❛s❡❞ ♦♥ ❡①♣❡❝t❡❞ ✉t✐❧✐t② t❤❡♦r② ❜✳ ❘❛♥❦✲❞❡♣❡♥❞❡♥t s♦❝✐❛❧ ✇❡❧❢❛r❡ ❝r✐t❡r✐❛ ■♠♣♦rt❛♥t ✐ss✉❡ ✐♥ ❜♦t❤ ♣♦❧✐❝② ✇♦r❦✱ ❞❡s❝r✐♣t✐✈❡ ❛♥❛❧②s✐s ❛♥❞ ❝❛✉s❛❧ ✐♥❢❡r❡♥❝❡✿

✶ ❙t❛t✐st✐❝❛❧ ♦✣❝❡s ❛♥❞ ❣♦✈ ❛❣❡♥❝✐❡s ❝♦♠♣❛r❡ ❞✐str✐❜✉t✐♦♥

❢✉♥❝t✐♦♥s ❛♥❞ ▲♦r❡♥③ ❝✉r✈❡s ❛❝r♦ss ❝♦✉♥tr✐❡s✱ s✉❜❣r♦✉♣s ❛♥❞ t✐♠❡

✷ ❘❡s❡❛r❝❤ ❝♦♠♣❛r❡s ❞✐str✐❜✉t✐♦♥s ♦❢ ❡❛r♥✐♥❣s✱ ✐♥❝♦♠❡✱

❝♦♥s✉♠♣t✐♦♥ ❛♥❞ ✇❡❛❧t❤ t♦ ❡✈❛❧✉❛t❡ ❡❝♦♥♦♠✐❝ ♣♦❧✐❝✐❡s ❛♥❞ s♦❝✐❛❧ ✇❡❧❢❛r❡

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❚❤❡ ❝✉♠✉❧❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ✐♥✈❡rs❡

▲❡t F ❜❡ ❛ ♠❡♠❜❡r ♦❢ t❤❡ s❡t F ♦❢ ❝✉♠✉❧❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s ✇✐t❤ ♠❡❛♥ µF ❛♥❞ ❧❡❢t ✐♥✈❡rs❡ ❞❡✜♥❡❞ ❜② F −✶(t) = inf {x : F(x) ≥ t} ◆♦t❡ t❤❛t ❜♦t❤ ❞✐s❝r❡t❡ ❛♥❞ ❝♦♥t✐♥✉♦✉s ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s ❛r❡ ❛❧❧♦✇❡❞ ✐♥ F✱ ❛♥❞ t❤♦✉❣❤ t❤❡ ❢♦r♠❡r ✐s ✇❤❛t ✇❡ ❛❝t✉❛❧❧② ♦❜s❡r✈❡✱ t❤❡ ❧❛tt❡r ♦❢t❡♥ ❛❧❧♦✇s s✐♠♣❧❡r ❞❡r✐✈❛t✐♦♥ ♦❢ t❤❡♦r❡t✐❝❛❧ r❡s✉❧ts ❛♥❞ ✐s ❛ ✈❛❧✐❞ ❧❛r❣❡ s❛♠♣❧❡ ❛♣♣r♦①✐♠❛t✐♦♥✳ ❚❤✉s✱ ✐♥ ♠♦st ❝❛s❡s ❜❡❧♦✇ F ✇✐❧❧ ❜❡ ❛ss✉♠❡❞ t♦ ❜❡ ❛ ❝♦♥t✐♥✉♦✉s ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥✱ ❜✉t t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ ❛ ❞✐s❝r❡t❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ✇✐❧❧ ❜❡ ✉s❡❞ ✇❤❡r❡ ❛♣♣r♦♣r✐❛t❡✳ ❚♦ ✜① ✐❞❡❛s✱ ✇❡ ✇✐❧❧ r❡❢❡r t♦ F ❛s t❤❡ ✐♥❝♦♠❡ ❞✐str✐❜✉t✐♦♥✱ ❛❧t❤♦✉❣❤ ♦✉r ❢r❛♠❡✇♦r❦ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ❛♥② t②♣❡ ♦❢ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s✳ ■♥ ♦r❞❡r t♦ r❛♥❦ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ♦r❞❡r✐♥❣ r❡❧❛t✐♦♥

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❘❛♥❦✐♥❣ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s✿ ❊①❛♠♣❧❡s

❙✉♣♣♦s❡ ✇❡ ✇❛♥t t♦ r❛♥❦ t✇♦ ❞✐str✐❜✉t✐♦♥s✱ F✶ ❛♥❞ F✵ ❆ss✉♠❡ t❤❛t t❤❡ ♦r❞❡r✐♥❣ r❡❧❛t✐♦♥ s❛t✐s✜❡s ✜rst✲❞❡❣r❡❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡✱ ✐✳❡✳ F✶(x) ≤ F✵(x) for all x ∈ [✵,∞)⇔F −✶

✶ (t) ≥ F −✶ ✵ (t)

for all t ∈ [✵,✶]✳ ❈❛♥ ❜❡ ✉s❡❞ ❛s ❛ r❛♥❦✐♥❣ ❝r✐t❡r✐♦♥ ✇❤❡♥ ❞✐str✐❜t✐♦♥ ❞♦♥✬t ❝r♦ss✳ ❇✉t ❤♦✇ ❞♦ ✇❡ ❞❡❛❧ ✇✐t❤ ✐♥t❡rs❡❝t✐♥❣ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s ✭❋✐❣✉r❡s ✷ ❛♥❞ ✸✮❄ ❈♦♥✈❡♥t✐♦♥❛❧ ❛♣♣r♦❛❝❤ ✐♥ ❡♠♣✐r✐❝❛❧ ✇♦r❦✿ ❯s✐♥❣ s✉♠♠❛r② ♠❡❛s✉r❡s ❧✐❦❡ t❤❡ ♠❡❛♥✱ t❤❡ ♠❡❞✐❛♥ ❛♥❞ t❤❡ ✈❛r✐❛♥❝❡ ♦r ✇❡✐❣❤t❡❞ ♠❡❛♥s✳

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❙❡❝♦♥❞✲❞❡❣r❡❡ st♦❝❤❛st✐❝ ❛♥❞ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡

❉❡✜♥✐t✐♦♥ ❆ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ F✶ ✐s s❛✐❞ t♦ s❡❝♦♥❞✲❞❡❣r❡❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛t❡ ❛ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ F✵ ✐❢ ❛♥❞ ♦♥❧② ✐❢

y

• ✵

F✶(x)dx ≤

y

• ✵

F✵(x)dx for all y ∈ [✵,∞) ❛♥❞ t❤❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s str✐❝t❧② ❢♦r s♦♠❡ y ∈ (✵,∞)✳ ❆ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ F✶ ✐s s❛✐❞ t♦ s❡❝♦♥❞✲❞❡❣r❡❡ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛t❡ ❛ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ F✵ ✐❢ ❛♥❞ ♦♥❧② ✐❢

u

• ✵

F −✶

✶ (t)dt ≥ u

• ✵

F −✶

✵ (t)dt for all u ∈ [✵,✶]

❛♥❞ t❤❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s str✐❝t❧② ❢♦r s♦♠❡ u ∈ (✵,✶)✳

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❆s ✇❛s ❞❡♠♦♥str❛t❡❞ ❜② ❆t❦✐♥s♦♥ ✭✶✾✼✵✮✱ s❡❝♦♥❞✲❞❡❣r❡❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡ ✐s ❡q✉✐✈❛❧❡♥t t♦ s❡❝♦♥❞✲❞❡❣r❡❡ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡✱ ✇❤✐❝❤ ✐s ❝❛❧❧❡❞ ❣❡♥❡r❛❧✐③❡❞ ▲♦r❡♥③ ❞♦♠✐♥❛♥❝❡ ❜② ❙❤♦rr♦❝❦s ✭✶✾✽✸✮✳ ▼♦r❡♦✈❡r✱ ✉♥❞❡r t❤❡ r❡str✐❝t✐♦♥ ♦❢ ❡q✉❛❧ ♠❡❛♥ ✐♥❝♦♠❡s s❡❝♦♥❞ ❞❡❣r❡❡ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❝r✐t❡r✐♦♥ ♦❢ ♥♦♥✲✐♥t❡rs❡❝t✐♥❣ ▲♦r❡♥③ ❝✉r✈❡s✳

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❚❤✐r❞✲❞❡❣r❡❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡

❙✐♥❝❡ s✐t✉❛t✐♦♥s ✇❤❡r❡ s❡❝♦♥❞✲❞❡❣r❡❡ ❞♦♠✐♥❛♥❝❡ ❞♦❡s ♥♦t ♣r♦✈✐❞❡ ✉♥❛♠❜✐❣✉♦✉s r❛♥❦✐♥❣ ♦❢ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s ♠❛② ❛r✐s❡✱ ✐t ✇✐❧❧ ❜❡ ✉s❡❢✉❧ t♦ ✐♥tr♦❞✉❝❡ ✇❡❛❦❡r r❛♥❦✐♥❣ ❝r✐t❡r✐❛ t❤❛♥ s❡❝♦♥❞✲❞❡❣r❡❡ ❞♦♠✐♥❛♥❝❡✳ ❚♦ t❤✐s ❡♥❞ ✐t ❛♣♣❡❛rs ❛ttr❛❝t✐✈❡ t♦ ❝♦♥s✐❞❡r t❤✐r❞✲❞❡❣r❡❡ st♦❝❤❛st✐❝ ❛♥❞ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡✳ ❉❡✜♥✐t✐♦♥ ❆ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ F✶ ✐s s❛✐❞ t♦ t❤✐r❞✲❞❡❣r❡❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛t❡ ❛ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ F✵ ✐❢ ❛♥❞ ♦♥❧② ✐❢

z

• ✵

y

• ✵

F✶(x)dxdy ≤

z

• ✵

y

• ✵

F✵(x)dxdy for all z ∈ [✵,∞) ⇔

z

• ✵

(z −x)(F✶(x)−F✵(x))dx ≤ ✵for all z ∈ [✵,∞) ❛♥❞ t❤❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s str✐❝t❧② ❢♦r s♦♠❡ z ∈ (✵,∞)✳

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❚❤✐r❞✲❞❡❣r❡❡ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡

❉❡✜♥✐t✐♦♥ ❆ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ F✶ ✐s s❛✐❞ t♦ t❤✐r❞✲❞❡❣r❡❡ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛t❡ ❛ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ F✵ ✐❢ ❛♥❞ ♦♥❧② ✐❢

v

• ✵

u

• ✵

F −✶

✶ (t)dtdu ≥ v

• ✵

u

• ✵

F −✶

✵ (t)dtdu for all v ∈ [✵,✶] ⇔ v

• ✵

(v −t)

• F −✶

✶ (t)−F −✶ ✵ (t)

• dt ≤ ✵, for all v ∈ [✵,✶]

❛♥❞ t❤❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s str✐❝t❧② ❢♦r s♦♠❡ v ∈ (✵,✶)✳ ◆♦t❡ t❤❛t t❤✐r❞✲❞❡❣r❡❡ st♦❝❤❛st✐❝ ❛♥❞ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡ ❞♦ ♥♦t ❝♦✐♥❝✐❞❡✳

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❚r❛♥s❢❡r ♣r✐♥❝✐♣❧❡s ❛ss♦❝✐❛t❡❞ ✇✐t❤ s❡❝♦♥❞✲ ❛♥❞ t❤✐r❞✲❞❡❣r❡❡ ❞♦♠✐♥❛♥❝❡

❉❡✜♥✐t✐♦♥

✭❚❤❡ P✐❣♦✉✲❉❛❧t♦♥ ♣r✐♥❝✐♣❧❡ ♦❢ tr❛♥s❢❡rs✮✳ ❈♦♥s✐❞❡r ❛ ❞✐s❝r❡t❡ ✐♥❝♦♠❡ ❞✐str✐❜✉t✐♦♥ F ✳ ❆ tr❛♥s❢❡r δ > ✵ ❢r♦♠ ❛ ♣❡rs♦♥ ✇✐t❤ ✐♥❝♦♠❡ x +h ✭♦r F −✶(t)✮ t♦ ❛ ♣❡rs♦♥ ✇✐t❤ ✐♥❝♦♠❡ x ✭♦r F −✶(s)✮ ✐s s❛✐❞ t♦ r❡❞✉❝❡ ✐♥❡q✉❛❧✐t② ✐♥ F ✇❤❡♥ h > ✵ ✭s < t) ❛♥❞ t♦ r❛✐s❡ ✐♥❡q✉❛❧✐t② ✐♥ F ✇❤❡♥ h < ✵ (s > t)✳ ✭✐✮ ■❢ µF✶ = µF✵, t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ s❡❝♦♥❞✲❞❡❣r❡❡ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❡ P✐❣♦✉✲❉❛❧t♦♥ tr❛♥s❢❡r ♣r✐♥❝✐♣❧❡✳

❉❡✜♥✐t✐♦♥

✭❚❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ❞✐♠✐♥✐s❤✐♥❣ tr❛♥s❢❡rs✱ ❑♦❧♠✱✶✾✼✻✮✳ ❈♦♥s✐❞❡r ❛ ❞✐s❝r❡t❡ ✐♥❝♦♠❡ ❞✐str✐❜✉t✐♦♥ F ✳ ❆ tr❛♥s❢❡r δ > ✵ ❢r♦♠ ❛ ♣❡rs♦♥ ✇✐t❤ ✐♥❝♦♠❡ x +h✶ t♦ ❛ ♣❡rs♦♥ ✇✐t❤ ✐♥❝♦♠❡ x ✐s s❛✐❞ t♦ r❡❞✉❝❡ ✐♥❡q✉❛❧✐t② ✐♥ F ♠♦r❡ t❤❛♥ ❛ tr❛♥s❢❡r δ ❢r♦♠ ❛ ♣❡rs♦♥ ✇✐t❤ ✐♥❝♦♠❡ x +h✶ +h✷ t♦ ❛ ♣❡rs♦♥ ✇✐t❤ ✐♥❝♦♠❡ x +h✷✳ ✭✐✐✮ ■❢ µF✶ = µF✵, t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ t❤✐r❞✲❞❡❣r❡❡ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ❞✐♠✐♥✐s❤✐♥❣ tr❛♥s❢❡rs✳

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❘❛♥❦✲♣r❡s❡r✈✐♥❣ tr❛♥s❢❡rs

❉❡✜♥✐t✐♦♥ ✭❚❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ♣♦s✐t✐♦♥❛❧ tr❛♥s❢❡r s❡♥s✐t✐✈✐t②✱ ▼❡❤r❛♥✱ ✶✾✼✻✮✳ ❈♦♥s✐❞❡r ❛ ❞✐s❝r❡t❡ ✐♥❝♦♠❡ ❞✐str✐❜✉t✐♦♥ F ✳ ❆ r❛♥❦✲♣r❡s❡r✈✐♥❣ tr❛♥s❢❡r δ > ✵ ❢r♦♠ ❛ ♣❡rs♦♥ ✇✐t❤ ✐♥❝♦♠❡ F −✶(s +h) t♦ ❛ ♣❡rs♦♥ ✇✐t❤ ✐♥❝♦♠❡ F −✶(s) ✐s s❛✐❞ t♦ ❤❛✈❡ ❛ str♦♥❣❡r ❡q✉❛❧✐③✐♥❣ ❡✛❡❝t ♦♥ F t❤❛♥ ❛ tr❛♥s❢❡r δ > ✵ ❢r♦♠ ❛ ♣❡rs♦♥ ✇✐t❤ ✐♥❝♦♠❡ F −✶(t +h) t♦ ❛ ♣❡rs♦♥ ✇✐t❤ ✐♥❝♦♠❡ F −✶(t) ✇❤❡♥ s < t✳

✭✐✐✐✮ ■❢ µF✶ = µF✵, t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ t❤✐r❞✲❞❡❣r❡❡ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ✜rst✲❞❡❣r❡❡ ❞♦✇♥s✐❞❡ ♣♦s✐t✐♦♥❛❧ tr❛♥s❢❡r s❡♥s✐t✐✈✐t②✳

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

▲♦r❡♥③ ❞♦♠✐♥❛♥❝❡

❉❡✜♥✐t✐♦♥ ❆ ▲♦r❡♥③ ❝✉r✈❡ L✶ ✐s s❛✐❞ t♦ ✜rst✲❞❡❣r❡❡ ❞♦♠✐♥❛t❡ ❛ ▲♦r❡♥③ ❝✉r✈❡ L✵ ✐❢ L✶(u) ≥ L✵(u) for all u ∈ [✵,✶] ❛♥❞ t❤❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s str✐❝t❧② ❢♦r s♦♠❡ u ∈ [✵,✶]✳ ❋✐rst✲❞❡❣r❡❡ ▲♦r❡♥③ ❞♦♠✐♥❛♥❝❡ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❡ P✐❣♦✉✲❉❛❧t♦♥ tr❛♥s❢❡r ♣r✐♥❝✐♣❧❡✳ ❆ s♦❝✐❛❧ ♣❧❛♥♥❡r ✇❤♦ ♣r❡❢❡rs t❤❡ ❞♦♠✐♥❛t✐♥❣ ♦♥❡ ♦❢ ♥♦♥✲✐♥t❡rs❡❝t✐♥❣ ▲♦r❡♥③ ❝✉r✈❡s ❢❛✈♦rs tr❛♥s❢❡rs ♦❢ ✐♥❝♦♠❡s ✇❤✐❝❤ r❡❞✉❝❡ t❤❡ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ t❤❡ ✐♥❝♦♠❡ s❤❛r❡s ♦❢ t❤❡ ❞♦♥♦r ❛♥❞ t❤❡ r❡❝✐♣✐❡♥t✱ ❛♥❞ ✐s t❤❡r❡❢♦r❡ s❛✐❞ t♦ ❜❡ ✐♥❡q✉❛❧✐t② ❛✈❡rs❡✳

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❙❡❝♦♥❞✲❞❡❣r❡❡ ▲♦r❡♥③ ❞♦♠✐♥❛♥❝❡

❚♦ ❞❡❛❧ ✇✐t❤ s✐t✉❛t✐♦♥s ✇❤❡r❡ ▲♦r❡♥③ ❝✉r✈❡s ✐♥t❡rs❡❝t ❛ ✇❡❛❦❡r ♣r✐♥❝✐♣❧❡ t❤❛♥ ✜rst✲❞❡❣r❡❡ ▲♦r❡♥③ ❞♦♠✐♥❛♥❝❡ ✐s ❝❛❧❧❡❞ ❢♦r✳ ❚♦ t❤✐s ❡♥❞ ✐t ✐s ♥♦r♠❛❧ t♦ ❡♠♣❧♦② s❡❝♦♥❞✲❞❡❣r❡❡ ✉♣✇❛r❞ ▲♦r❡♥③ ❞♦♠✐♥❛♥❝❡ ❞❡✜♥❡❞ ❜② ❉❡✜♥✐t✐♦♥ ❆ ▲♦r❡♥③ ❝✉r✈❡ L✶ ✐s s❛✐❞ t♦ s❡❝♦♥❞✲❞❡❣r❡❡ ✉♣✇❛r❞ ❞♦♠✐♥❛t❡ ❛ ▲♦r❡♥③ ❝✉r✈❡ L✵ ✐❢

u

✵ L✶(t)dt ≥

u

✵ L✵(t)dt for all u ∈ [✵,✶]

❛♥❞ t❤❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s str✐❝t❧② ❢♦r s♦♠❡ u ∈ [✵,✶]✳

❙❡❝♦♥❞✲❞❡❣r❡❡ ✉♣✇❛r❞ ▲♦r❡♥③ ❞♦♠✐♥❛♥❝❡ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ✜rst✲❞❡❣r❡❡ ❞♦✇♥s✐❞❡ ♣♦s✐t✐♦♥❛❧ tr❛♥s❢❡r s❡♥s✐t✐✈✐t②✳ ❯♥❞❡r t❤❡ r❡str✐❝t✐♦♥ ♦❢ ❡q✉❛❧ ♠❡❛♥ ✐♥❝♦♠❡s t❤✐r❞✲❞❡❣r❡❡ ✭✉♣✇❛r❞✮ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❝r✐t❡r✐♦♥ ♦❢ s❡♦♥❞✲❞❡❣r❡❡ ✉♣✇❛r❞ ▲♦r❡♥③ ❞♦♠✐♥❛♥❝❡✳

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❍✐❣❤❡r ❞❡❣r❡❡s ♦❢ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡ ❛♥❞ ▲♦r❡♥③ ❞♦♠✐♥❛♥❝❡

❙✐♥❝❡ s✐t✉❛t✐♦♥s ✇❤❡r❡ s❡❝♦♥❞✲❞❡❣r❡❡ ✭✉♣✇❛r❞ ♦r ❞♦✇♥✇❛r❞✮ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❛♥❞ ▲♦r❡♥③ ❞♦♠✐♥❛♥❝❡ ❞♦ ♥♦t ♣r♦✈✐❞❡ ✉♥❛♠❜✐❣✉♦✉s r❛♥❦✐♥❣ ♦❢ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s ❛♥❞ ▲♦r❡♥③ ❝✉r✈❡s ♠❛② ❛r✐s❡✱ ✐t ✐s ✉s❡❢✉❧ t♦ ✐♥tr♦❞✉❝❡ ✇❡❛❦❡r ❞♦♠✐♥❛♥❝❡ ❝r✐t❡r✐❛ t❤❛♥ t❤✐r❞✲❞❡❣r❡❡ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡ ❛♥❞ s❡❝♦♥❞✲❞❡❣r❡❡ ▲♦r❡♥③ ❞♦♠✐♥❛♥❝❡✳ ❚♦ t❤✐s ❡♥❞ t✇♦ ❤✐❡r❛r❝❤✐❝❛❧ s❡q✉❡♥❝❡s ♦❢ ♥❡st❡❞ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ✭▲♦r❡♥③ ❞♦♠✐♥❛♥❝❡✮ ❝r✐t❡r✐❛ ♠✐❣❤t ❜❡ ✐♥tr♦❞✉❝❡❞❀ ♦♥❡ ❞❡♣❛rts ❢r♦♠ t❤✐r❞✲❞❡❣r❡❡ ✉♣✇❛r❞ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡ ✭s❡❝♦♥❞✲❞❡❣r❡❡ ✉♣✇❛r❞ ▲♦r❡♥③ ❞♦♠✐♥❛♥❝❡✮ ❛♥❞ t❤❡ ♦t❤❡r ❢r♦♠ t❤✐r❞✲❞❡❣r❡❡ ❞♦✇♥✇❛r❞ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡ ✭❞♦✇♥✇❛r❞ ▲♦r❡♥③ ❞♦♠✐♥❛♥❝❡✮✳ ▼♦r❡ ♦♥ t❤✐s ✐♥ ❆❛❜❡r❣❡ ✭✷✵✵✾✱ ❙❈❲✮✳

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❈♦♠♣❧❡t❡ ♦r❞❡r✐♥❣s✿ ❙♦❝✐❛❧ ✇❡❧❢❛r❡ ❝r✐t❡r✐❛ ❜❛s❡❞ ♦♥ ❡①♣❡❝t❡❞ ✉t✐❧✐t② t❤❡♦r②

❚❤❡ ♣r♦❜❧❡♠ ♦❢ r❛♥❦✐♥❣ ✐♥❝♦♠❡ ❞✐str✐❜✉t✐♦♥s ❢♦r♠❛❧❧② ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❤♦♦s✐♥❣ ❜❡t✇❡❡♥ ✉♥❝❡rt❛✐♥ ♣r♦s♣❡❝ts✳ ❚❤✐s r❡❧❛t✐♦♥s❤✐♣ ❤❛s ❜❡❡♥ ✉t✐❧✐③❡❞ ❜② ❡✳❣✳ ❑♦❧♠ ✭✶✾✻✾✮ ❛♥❞ ❆t❦✐♥s♦♥ ✭✶✾✼✵✮ t♦ ❝❤❛r❛❝t❡r✐③❡ t❤❡ ❝r✐t❡r✐♦♥ ♦❢ s❡❝♦♥❞ ♦r❞❡r ✭✉♣✇❛r❞✮ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡✳ ❆t❦✐♥s♦♥ r❡✐♥t❡r♣r❡t❡❞ t❤❡ st❛♥❞❛r❞ t❤❡♦r② ♦❢ ❝❤♦✐❝❡ ✉♥❞❡r ✉♥❝❡rt❛✐♥t② ❛♥❞ ❞❡♠♦♥str❛t❡❞ t❤❛t ✐♥❡q✉❛❧✐t② ❛✈❡rs✐♦♥ ❝❛♥ ✐♥ ❢❛❝t ❜❡ ✈✐❡✇❡❞ ❛s ❜❡✐♥❣ ❡q✉✐✈❛❧❡♥t t♦ r✐s❦ ❛✈❡rs✐♦♥✳ ❚❤✐s ✇❛s ♠♦t✐✈❛t❡❞ ❜② t❤❡ ❢❛❝t t❤❛t ✐♥ ❝❛s❡s ♦❢ ❡q✉❛❧ ♠❡❛♥ ✐♥❝♦♠❡s t❤❡ ❝r✐t❡r✐♦♥ ♦❢ ♥♦♥✲✐♥t❡rs❡❝t✐♥❣ ▲♦r❡♥③ ❝✉r✈❡s ✐s ❡q✉✐✈❛❧❡♥t t♦ s❡❝♦♥❞✲❞❡❣r❡❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t t❤❡ P✐❣♦✉✲❉❛❧t♦♥ tr❛♥s❢❡r ♣r✐♥❝✐♣❧❡ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ♠❡❛♥ ♣r❡s❡r✈✐♥❣ s♣r❡❛❞ ✐♥tr♦❞✉❝❡❞ ❜② ❘♦t❤s❝❤✐❧❞ ❛♥❞ ❙t✐❣❧✐t③ ✭✶✾✼✵✮✳ ❚♦ ❝❤♦♦s❡ ❜❡t✇❡❡♥ F✵ ❛♥❞ F✶ ✇❡ ❝❛♥ t❤❡♥ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝r✐t❡r✐♦♥

∞

✵ u(x)dF✶(x) ≥

∞

✵ u(x)dF✵(x)

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❆①✐♦♠❛t✐❝ ❥✉st✐✜❝❛t✐♦♥ ♦❢ t❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤

❆ss✉♠❡ t❤❛t t❤❡ ♣r❡❢❡r❡♥❝❡ r❡❧❛t✐♦♥ ♦❢ t❤❡ s♦❝✐❛❧ ♣❧❛♥♥❡r s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❛①✐♦♠s✿ ✭❖r❞❡r✮✳ ✐s ❛ tr❛♥s✐t✐✈❡ ❛♥❞ ❝♦♠♣❧❡t❡ ♦r❞❡r✐♥❣ ♦♥ F✳ ✭❈♦♥t✐♥✉✐t②✮✳ ❋♦r ❡❛❝❤ F ∈ F t❤❡ s❡ts {F ⋆ ∈ F : F F ⋆}❛♥❞ {F ⋆ ∈ F : F ⋆ F} ❛r❡ ❝❧♦s❡❞ ✭✇✳r✳t✳ L✶✲♥♦r♠✮✳ ✭❉♦♠✐♥❛♥❝❡✮✳ ▲❡t F✵,F✶ ∈ F✳ ■❢ F −✶

✶ (t) ≥ F −✶ ✵ (y) ❢♦r ❛❧❧ t ∈ [✵,✶] ❛♥❞ t❤❡

✐♥❡q✉❛❧✐t② ❤♦❧❞s str✐❝t❧② ❢♦r s♦♠❡ t∈ (✵,✶) t❤❡♥ F✶ F✵✳ ✭■♥❞❡♣❡♥❞❡♥❝❡✮✳ ▲❡t F✵✱ F✶ ❛♥❞ F✷ ❜❡ ♠❡♠❜❡rs ♦❢ F ❛♥❞ ❧❡t α ∈ [✵,✶]✳ ❚❤❡♥ F✶ F✵ ✐♠♣❧✐❡s (αF✶(x)+(✶−α)F✷(x)) (αF✵(x)+(✶−α)F✷(x))✳ ❱♦♥ ◆❡✉♠❛♥ ❛♥❞ ▼♦r❣❡♥st❡r♥ ✭✶✾✸✻✮ ♣r♦✈❡❞ t❤❛t ❛ ♣r❡❢❡r❡♥❝❡ r❡❧❛t✐♦♥ t❤❛t s❛t✐s✜❡s ❆①✐♦♠s ✶✲✹ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛♠✐❧② ♦❢ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥s

Eu(X) =

• u(x)dF(x)

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

▼❡❛s✉r❡s ♦❢ ✐♥❡q✉❛❧✐t② ❜❛s❡❞ ♦♥ t❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤

❆t❦✐♥s♦♥ ✭✶✾✼✵✮ ♣r♦♣♦s❡❞ t♦ ✉s❡ Iu(F) = ✶− u−✶(Eu(X)) µ as a measure of inequality✱ where u−✶(Eu(X)) is denoted the equally distributed equivalent income and µIu(F) is measure of the loss in social welfare due to inequality in the distribution F.

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❚❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ❞♦♠✐♥❛♥❝❡ ❝r✐t❡r✐❛ ❛♥❞ ♣r✐♠❛❧ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥s

❚❤❡♦r❡♠ ▲❡t F✶ ❛♥❞ F✵ ❜❡ ♠❡♠❜❡rs ♦❢ F✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❛r❡ ❡q✉✐✈❛❧❡♥t✳ ✭✐✮ F✶ s❡❝♦♥❞✲❞❡❣r❡❡ ✉♣✇❛r❞ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛t❡s F✵ ✭✐✐✮ EF✶u(X) > EF✵u(X) ❢♦r ❛❧❧ ✐♥❝r❡❛s✐♥❣ ❝♦♥❝❛✈❡ u ❚❤❡♦r❡♠ ▲❡t L✶ ❛♥❞ L✵ ❜❡ ♠❡♠❜❡rs ♦❢ L✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❛r❡ ❡q✉✐✈❛❧❡♥t✳ ✭✐✮ L✶ ✜rst✲❞❡❣r❡❡ ❞♦♠✐♥❛t❡s L✵ ✭✐✐✮ Iu(F✶) < Iu(F✵) ❢♦r ❛❧❧ ✐♥❝r❡❛s✐♥❣ ❝♦♥❝❛✈❡ u

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❈♦♠♣❧❡t❡ ♦r❞❡r✐♥❣s✿ ❚❤❡ ❢❛♠✐❧② ♦❢ r❛♥❦✲❞❡♣❡♥❞❡♥t s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥s

❚❤❡ ❣❡♥❡r❛❧ ❢❛♠✐❧② ♦❢ r❛♥❦✲❞❡♣❡♥❞❡♥t ♠❡❛s✉r❡s ♦❢ s♦❝✐❛❧ ✇❡❧❢❛r❡ ✐♥tr♦❞✉❝❡❞ ❜② ❨❛❛r✐ ✭✶✾✽✼✱✶✾✽✽✮ ✐s ❞❡✜♥❡❞ ❜② WP(F) =

✶

✵ P′(t)F −✶(t)dt,

❛♥❞ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s t❤❡ ❡q✉❛❧❧② ❞✐str✐❜✉t❡❞ ❡q✉✐✈❛❧❡♥t ✐♥❝♦♠❡✳ ❚❤❡ ✇❡✐❣❤t✐♥❣ ❢✉♥❝t✐♦♥ P′ ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ♣r❡❢❡r❡♥❝❡ ❢✉♥❝t✐♦♥ t❤❛t ✐s ❛ ♠❡♠❜❡r ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡ s❡t ♦❢ ♣r❡❢❡r❡♥❝❡ ❢✉♥❝t✐♦♥s✿ P = {P : P′(t) > ✵and P′′(t) < ✵ for all t ∈ (✵,✶), P(✵) = P′(✶) = ✵, P(✶) = ✶} WP ♣r❡s❡r✈❡s ✶st✲❞❡❣r❡❡ ❞♦♠✱ s✐♥❝❡ P′(t) > ✵✱ ❛♥❞

WP ♣r❡s❡r✈❡s ✷♥❞✲❞❡❣r❡❡ ❞♦♠ ✭❛♥❞ P✐❣♦✉✲❉❛❧t♦♥✮✱ s✐♥❝❡ P′′(t) < ✵

WP ≤ µF ✱ ❛♥❞ WP = µF ✐✛ F ✐s t❤❡ ❡❣❛❧✐t❛r✐❛♥ ❞✐str✐❜✉t✐♦♥

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❈♦♠♣❧❡t❡ ♦r❞❡r✐♥❣s✿ ❚❤❡ ❢❛♠✐❧② ♦❢ r❛♥❦✲❞❡♣❡♥❞❡♥t s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥s

❚❤❡ ❣❡♥❡r❛❧ ❢❛♠✐❧② ♦❢ r❛♥❦✲❞❡♣❡♥❞❡♥t ♠❡❛s✉r❡s ♦❢ s♦❝✐❛❧ ✇❡❧❢❛r❡ ✐♥tr♦❞✉❝❡❞ ❜② ❨❛❛r✐ ✭✶✾✽✼✱✶✾✽✽✮ ✐s ❞❡✜♥❡❞ ❜② WP(F) =

✶

✵ P′(t)F −✶(t)dt,

❛♥❞ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s t❤❡ ❡q✉❛❧❧② ❞✐str✐❜✉t❡❞ ❡q✉✐✈❛❧❡♥t ✐♥❝♦♠❡✳ ❚❤❡ ✇❡✐❣❤t✐♥❣ ❢✉♥❝t✐♦♥ P′ ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ♣r❡❢❡r❡♥❝❡ ❢✉♥❝t✐♦♥ t❤❛t ✐s ❛ ♠❡♠❜❡r ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡ s❡t ♦❢ ♣r❡❢❡r❡♥❝❡ ❢✉♥❝t✐♦♥s✿ P = {P : P′(t) > ✵and P′′(t) < ✵ for all t ∈ (✵,✶), P(✵) = P′(✶) = ✵, P(✶) = ✶} WP ♣r❡s❡r✈❡s ✶st✲❞❡❣r❡❡ ❞♦♠✱ s✐♥❝❡ P′(t) > ✵✱ ❛♥❞

WP ♣r❡s❡r✈❡s ✷♥❞✲❞❡❣r❡❡ ❞♦♠ ✭❛♥❞ P✐❣♦✉✲❉❛❧t♦♥✮✱ s✐♥❝❡ P′′(t) < ✵

WP ≤ µF ✱ ❛♥❞ WP = µF ✐✛ F ✐s t❤❡ ❡❣❛❧✐t❛r✐❛♥ ❞✐str✐❜✉t✐♦♥

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❉✉❛❧ ♠❡❛s✉r❡s ♦❢ ✐♥❡q✉❛❧✐t②

❙✐♥❝❡ WP(F) ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s t❤❡ ❡q✉❛❧❧② ❞✐str✐❜✉t❡❞ ❡q✉✐✈❛❧❡♥t ✐♥❝♦♠❡ t❤❡ ❞✉❛❧ ❢❛♠✐❧② ♦❢ ✐♥❡q✉❛❧✐t② ♠❡❛s✉r❡s ✐s ❞❡✜♥❡❞ ❜② JP(F) = ✶− WP(F) µ , ✇❤❡r❡ µ = EX =

xdF(X).

◆♦t❡ t❤❛t µJP(F) ✐s ❛ ♠❡❛s✉r❡ ♦❢ t❤❡ ❧♦ss ✐♥ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❞✉❡ t♦ ✐♥❡q✉❛❧✐t② ✐♥ t❤❡ ❞✐str✐❜✉t✐♦♥ F.

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❆①✐♦♠❛t✐❝ ❥✉st✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❆ss✉♠❡ t❤❛t t❤❡ ♣r❡❢❡r❡♥❝❡ r❡❧❛t✐♦♥ ♦❢ t❤❡ s♦❝✐❛❧ ♣❧❛♥♥❡r s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❛①✐♦♠s✿ ✭❖r❞❡r✮✳ ✐s ❛ tr❛♥s✐t✐✈❡ ❛♥❞ ❝♦♠♣❧❡t❡ ♦r❞❡r✐♥❣ ♦♥ F✳ ✭❈♦♥t✐♥✉✐t②✮✳ ❋♦r ❡❛❝❤ F ∈ F t❤❡ s❡ts {F ⋆ ∈ F : F F ⋆}❛♥❞ {F ⋆ ∈ F : F ⋆ F} ❛r❡ ❝❧♦s❡❞ ✭✇✳r✳t✳ L✶✲♥♦r♠✮✳ ✭❉♦♠✐♥❛♥❝❡✮✳ ▲❡t F✵,F✶ ∈ F✳ ■❢ F −✶

✶ (t) ≥ F −✶ ✵ (y) ❢♦r ❛❧❧ t ∈ [✵,✶] ❛♥❞ t❤❡

✐♥❡q✉❛❧✐t② ❤♦❧❞s str✐❝t❧② ❢♦r s♦♠❡ t∈ (✵,✶) t❤❡♥ F✶ F✵✳ ✭■♥❞❡♣❡♥❞❡♥❝❡✮✳ ▲❡t F✵✱ F✶ ❛♥❞ F✷ ❜❡ ♠❡♠❜❡rs ♦❢ F ❛♥❞ ❧❡t α ∈ [✵,✶]✳ ❚❤❡♥ F✶ F✵ ✐♠♣❧✐❡s

• αF −✶

✶ (t)+(✶−α)F −✶ ✷ (t)

−✶

• αF −✶

✵ (t)+(✶−α)F −✶ ✷ (t)

−✶ ✳ ❨❛❛r✐ ✭✶✾✸✻✮ ♣r♦✈❡❞ t❤❛t ❛ ♣r❡❢❡r❡♥❝❡ r❡❧❛t✐♦♥ t❤❛t s❛t✐s✜❡s ❆①✐♦♠s ✶✲✹ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛♠✐❧② ♦❢ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥s

WP(F) =

✶

✵ P′(t)F −✶(t)dt,

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

◆♦r♠❛t✐✈❡ ❥✉st✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❣❡♥❡r❛❧ ❢❛♠✐❧②

❚❤❡ ♥♦r♠❛t✐✈❡ ❥✉st✐✜❝❛t✐♦♥ ♦❢ WP ❝❛♥ ❜❡ ♠❛❞❡ ✐♥ t❡r♠s ♦❢ ❛ ✭❛✮ ❚❤❡♦r② ❢♦r r❛♥❦✐♥❣ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s✿ ❲✐t❤ ❜❛s✐❝ ♦r❞❡r✐♥❣ ❛♥❞ ❝♦♥t✐♥✉✐t② ❛ss✉♠♣t✐♦♥s✱ t❤❡ ❞✉❛❧ ✐♥❞❡♣❡♥❞❡♥❝❡ ❛①✐♦♠ ❝❤❛r❛❝t❡r✐③❡s WP ✭❨❛❛r✐✱ ✶✾✽✽✮ ✭❜✮ ❱❛❧✉❡ ❥✉❞❣❡♠❡♥t ♦❢ t❤❡ tr❛❞❡✲♦✛ ❜❡t✇❡❡♥ t❤❡ ♠❡❛♥ ❛♥❞ ✭✐♥✮❡q✉❛❧✐t② ✐♥ t❤❡ ❞✐str✐❜✉t✐♦♥s ✭❊❜❡rt✱ ✶✾✽✼❀ ❆❛❜❡r❣❡✱ ✷✵✵✶✮ WP = µF[✶−JP(F)] ✇❤❡r❡ µF ✐s t❤❡ ♠❡❛♥ ♦❢ F ❛♥❞ JP(F) ✐s t❤❡ ❢❛♠✐❧② ♦❢ r❛♥❦✲❞❡♣❡♥❞❡♥t ♠❡❛s✉r❡s ♦❢ ✐♥❡q✉❛❧✐t② ❛❣❣r❡❣❛t✐♥❣ t❤❡ P′✲✇❡✐❣❤t❡❞ ▲♦r❡♥③ ❝✉r✈❡ ♦❢ F

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❚❤❡ ●✐♥✐ s✉❜❢❛♠✐❧②

■❢ ✇❡ ❝❤♦♦s❡ P✶k(t) = ✶−(✶−t)k−✶, k > ✷ t❤❡♥ WP ✐s ❡q✉❛❧ t♦ t❤❡ ❡①t❡♥❞❡❞ ●✐♥✐ ❢❛♠✐❧② ♦❢ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥s ✭❉♦♥❛❧❞s♦♥ ❛♥❞ ❲❡②♠❛r❦✱ ✶✾✽✵✮ WGk = µ [✶−Gk(F)] =, k > ✷ ✇❤❡r❡ Gk(F) ✐s t❤❡ ❡①t❡♥❞❡❞ ●✐♥✐ ❢❛♠✐❧② ♦❢ ✐♥❡q✉❛❧✐t② ♠❡❛s✉r❡s G✸(F) ✐s t❤❡ ●✐♥✐ ❝♦❡✣❝✐❡♥t ❛♥❞ WG✷ = µ ◆♦t❡ t❤❛t {µ,WGi(F) : i = ✸,✹,...} ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡s t❤❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ F ✭❆❛❜❡r❣❡✱ ✷✵✵✵✮

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❚❤❡ ▲♦r❡♥③ s✉❜❢❛♠✐❧②

■❢ ✇❡ ✐♥st❡❛❞ ❝❤♦♦s❡ P✷k(t) = (k −✶)t −tk−✶ k −✷ , k > ✷ t❤❡♥ WP ✐s t❤❡ ▲♦r❡♥③ ❢❛♠✐❧② ♦❢ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥s ✭❆❛❜❡r❣❡✱ ✷✵✵✵✮ WDk = µ [✶−Dk(F)], k > ✷ ✇❤❡r❡ Dk(F) ✐s t❤❡ ▲♦r❡♥③ ❢❛♠✐❧② ♦❢ ✐♥❡q✉❛❧✐t② ♠❡❛s✉r❡s D✸(F) ✐s t❤❡ ●✐♥✐ ❝♦❡✣❝✐❡♥t ◆♦t❡ t❤❛t {µ,WDi(F) : i = ✸,✹,...} ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡s t❤❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ F ✭❆❛❜❡r❣❡✱ ✷✵✵✵✮

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❚❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ❞♦♠✐♥❛♥❝❡ ❝r✐t❡r✐❛ ❛♥❞ ❞✉❛❧ ❝r✐t❡r✐❛ ♦❢ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❛♥❞ ✐♥❡q✉❛❧✐t②

❚❤❡♦r❡♠ ▲❡t F✶ ❛♥❞ F✵ ❜❡ ♠❡♠❜❡rs ♦❢ F✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❛r❡ ❡q✉✐✈❛❧❡♥t✳ ✭✐✮ F✶ s❡❝♦♥❞✲❞❡❣r❡❡ ✉♣✇❛r❞ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛t❡s F✵ ✭✐✐✮

✶

✵ P′(t)F −✶ ✶ (t)dt >

✶

✵ P′(t)F −✶ ✵ (t)dt ❢♦r ❛❧❧ ✐♥❝r❡❛s✐♥❣ ❝♦♥❝❛✈❡

P (P′′(t) < ✵) ❚❤❡♦r❡♠ ▲❡t L✶ ❛♥❞ L✵ ❜❡ ♠❡♠❜❡rs ♦❢ L✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❛r❡ ❡q✉✐✈❛❧❡♥t✳ ✭✐✮ L✶ ✜rst✲❞❡❣r❡❡ ❞♦♠✐♥❛t❡s L✵ ✭✐✐✮ JP(F✶) < J(F✵) ❢♦r ❛❧❧ ✐♥❝r❡❛s✐♥❣ ❝♦♥❝❛✈❡ P

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

P❛rt✐❛❧ ❞✉❛❧ ♦r❞❡r✐♥❣ ✲ ❚❤✐r❞ ❞❡❣r❡❡ ✉♣✇❛r❞ ❞♦♠✐♥❛♥❝❡

◆♦t❡ t❤❛t s❡❝♦♥❞ ❞❡❣r❡❡ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡ ✐s ❞❡✜♥❡❞ ❜② Λ✷

F(u) ≡

u

✵ F −✶(t)dt,

u ∈ [✵,✶] ❚♦ ❞❡✜♥❡ t❤✐r❞ ❞❡❣r❡❡ ✉♣✇❛r❞ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡✱ ✇❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥ Λ✸

F(u) ≡

u

✵ Λ✷ F(t)dt =

u

✵ (u −t)F −✶(t)dt,

u ∈ [✵,✶] ❉❡✜♥✐t✐♦♥ ❆ ❞✐str✐❜✉t✐♦♥ F✶ ✐s s❛✐❞ t♦ t❤✐r❞ ❞❡❣r❡❡ ✉♣✇❛r❞ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛t❡ ❛ ❞✐str✐❜✉t✐♦♥ F✵ ✐❢ ❛♥❞ ♦♥❧② ✐❢ Λ✸

F✶(u) ≥ Λ✸ F✵(u) for all u ∈ [✵,✶]

❛♥❞ t❤❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s str✐❝t❧② ❢♦r s♦♠❡ u ∈ (✵,✶)✳

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❚r❛♥s❢❡r ♣r✐♥❝✐♣❧❡

∆sWP(δ,h): change in WP of a fixed progressive transfer δ from an individual with rank s +h to an individual with rank s. ∆✶

stWP(δ,h) ≡ ∆sWP(δ,h)−∆tWP(δ,h).

❉❡✜♥✐t✐♦♥ ✭❩♦❧✐✱ ✶✾✾✾❀ ❆❛❜❡r❣❡✱ ✷✵✵✵✱ ✷✵✵✾✮ WP s❛t✐s✜❡s t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ✜rst ❞❡❣r❡❡ ❞♦✇♥s✐❞❡ ♣♦s✐t✐♦♥❛❧ tr❛♥s❢❡r s❡♥s✐t✐✈✐t② ✭❉P❚❙✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ∆✶

stWP(δ,h) > ✵,

when s < t.

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡

❚❤❡ ♣r✐♠❛❧ ❛♣♣r♦❛❝❤ ❚❤❡ ❞✉❛❧ ❛♣♣r♦❛❝❤

❊q✉✐✈❛❧❡♥❝❡ r❡s✉❧t

▲❡t P✸ ❜❡ t❤❡ ❢❛♠✐❧② ♦❢ ♣r❡❢❡r❡♥❝❡ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❜②

P✸ =

• P ∈ P : P

′′′(t) > ✵,

• ❚❤❡♦r❡♠

▲❡t F✶ ❛♥❞ F✵ ❜❡ ♠❡♠❜❡rs ♦❢ F✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❛r❡ ❡q✉✐✈❛❧❡♥t✳ ✭✐✮ F✶ t❤✐r❞✲❞❡❣r❡❡ ✉♣✇❛r❞ ✐♥✈❡rs❡ st♦❝❤❛st✐❝ ❞♦♠✐♥❛t❡s F✵ ✭✐✐✮ WP(F✶) > WP(F✵) ❢♦r ❛❧❧ P ∈ P✸ ✭✐✐✐✮ WP(F✶) > WP(F✵) ❢♦r ❛❧❧ P ∈ P ✇❤❡r❡ WP s❛t✐s✜❡s ✜rst✲❞❡❣r❡❡ ❉P❚❙ ⇒ ✭✐✮ ❛♥❞ ✭✐✐✮✿ ❧❡❛st✲r❡str✐❝t✐✈❡ s❡t ♦❢ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥s t❤❛t ✉♥❛♠❜✐❣✉♦✉s❧② r❛♥❦ ✐♥ ❛❝❝♦r❞❛♥❝❡ ✇✐t❤ ✸✲❯■❉ ⇒ ✭✐✮ ❛♥❞ ✭✐✐✐✮✿ ♥♦r♠❛t✐✈❡ ❥✉st✐✜❝❛t✐♦♥ ❢♦r ✸✲❯■❉

❘♦❧❢ ❆❛❜❡r❣❡✱ ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱❙t❛t✐st✐❝s ◆♦r✇❛② ❊✲♠❛✐❧ ❛❞❞r❡ss✿ r♦❧❢✳❛❛❜❡r❣❡❅ss❜✳♥♦ ▼❡❛s✉r❡♠❡♥t ♦❢ ■♥❡q✉❛❧✐t② ❛♥❞ ❙♦❝✐❛❧ ❲❡❧❢❛r❡