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❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ■❍P✱ ❏❛♥✉❛r② ✷✽✱ ✷✵✶✻

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❚❛❧❦ ❜❛s❡❞ ♦♥ r❡❝❡♥t ✇♦r❦s ✇✐t❤ ✲ ❙t❡♣❤❛♥❡ ▲❡ ❇♦r❣♥❡ ✭❘❡♥♥❡s✮✱ ❋❧♦r❡♥t ▼❛❧r✐❡✉ ✭❚♦✉rs✮ ✫ P✐❡rr❡✲❆♥❞ré ❩✐tt ✭▼❛r♥❡ ❧❛ ❱❛❧❧é❡✮ ✭❆♥♥❛❧❡s ❞❡ ❧✬■❍P✱ ✷✵✶✺✮✳ ✲ ❋r✐t③ ❈♦❧♦♥✐✉s ✫ ❘❛❧♣❤ ▲❡tt❛✉ ✭❆✉❣s❜✉r❣✮ ✭❲♦r❦ ✐♥ Pr♦❣r❡ss✮✳ ✲ ❈❧❛✉❞❡ ▲♦❜r② ✭◆✐❝❡✮ ✭❆r①✐✈ ♣r❡♣r✐♥t ❖❝t✳ ✷✵✶✺✮✳

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s ■◆❚❘❖❉❯❈❚■❖◆✱ ●❖❆▲❙ ❛♥❞ ❊❳❆▼P▲❊❙

■♥tr♦❞✉❝t✐♦♥✱ ●♦❛❧s ❛♥❞ ❊①❛♠♣❧❡s

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s P❉▼Ps (❩t) ❛♥❞ ( ˜ ❩♥)

■♥tr♦❞✉❝t✐♦♥

• P❉▼Ps ❛r❡ ▼❛r❦♦✈ Pr♦❝❡ss❡s ❣✐✈❡♥ ❜② ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝s

❜❡t✇❡❡♥ r❛♥❞♦♠ s✇✐t❝❤✐♥❣ ❧❛r❣❡ ❧✐t❡r❛t✉r❡ ♦♥ t❤❡ s✉❜❥❡❝t ✫ ♥✉♠❡r♦✉s t②♣❡s ♦❢ P❉▼Ps ✉s❡❞ ✐♥ ❛ ✈❛r✐❡t② ♦❢ ✜❡❧❞s ✭♠♦❧❡❝✉❧❛r ❜✐♦❧♦❣②✱ ❝♦♠♠✉♥✐❝❛t✐♦♥ ♥❡t✇♦r❦s✱ ✳✳✳✮ ❍❡r❡ ✇❡ r❡str✐❝t ❛tt❡♥t✐♦♥ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ s♣❡❝✐✜❝ ❝❧❛ss✿

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s P❉▼Ps (❩t) ❛♥❞ ( ˜ ❩♥)

■♥tr♦❞✉❝t✐♦♥

• P❉▼Ps ❛r❡ ▼❛r❦♦✈ Pr♦❝❡ss❡s ❣✐✈❡♥ ❜② ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝s

❜❡t✇❡❡♥ r❛♥❞♦♠ s✇✐t❝❤✐♥❣

• ∃ ❧❛r❣❡ ❧✐t❡r❛t✉r❡ ♦♥ t❤❡ s✉❜❥❡❝t ✫ ♥✉♠❡r♦✉s t②♣❡s ♦❢ P❉▼Ps

✉s❡❞ ✐♥ ❛ ✈❛r✐❡t② ♦❢ ✜❡❧❞s ✭♠♦❧❡❝✉❧❛r ❜✐♦❧♦❣②✱ ❝♦♠♠✉♥✐❝❛t✐♦♥ ♥❡t✇♦r❦s✱ ✳✳✳✮ ❍❡r❡ ✇❡ r❡str✐❝t ❛tt❡♥t✐♦♥ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ s♣❡❝✐✜❝ ❝❧❛ss✿

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s P❉▼Ps (❩t) ❛♥❞ ( ˜ ❩♥)

■♥tr♦❞✉❝t✐♦♥

• P❉▼Ps ❛r❡ ▼❛r❦♦✈ Pr♦❝❡ss❡s ❣✐✈❡♥ ❜② ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝s

❜❡t✇❡❡♥ r❛♥❞♦♠ s✇✐t❝❤✐♥❣

• ∃ ❧❛r❣❡ ❧✐t❡r❛t✉r❡ ♦♥ t❤❡ s✉❜❥❡❝t ✫ ♥✉♠❡r♦✉s t②♣❡s ♦❢ P❉▼Ps
• ✉s❡❞ ✐♥ ❛ ✈❛r✐❡t② ♦❢ ✜❡❧❞s ✭♠♦❧❡❝✉❧❛r ❜✐♦❧♦❣②✱ ❝♦♠♠✉♥✐❝❛t✐♦♥

♥❡t✇♦r❦s✱ ✳✳✳✮ ❍❡r❡ ✇❡ r❡str✐❝t ❛tt❡♥t✐♦♥ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ s♣❡❝✐✜❝ ❝❧❛ss✿

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s P❉▼Ps (❩t) ❛♥❞ ( ˜ ❩♥)

■♥tr♦❞✉❝t✐♦♥

• P❉▼Ps ❛r❡ ▼❛r❦♦✈ Pr♦❝❡ss❡s ❣✐✈❡♥ ❜② ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝s

❜❡t✇❡❡♥ r❛♥❞♦♠ s✇✐t❝❤✐♥❣

• ∃ ❧❛r❣❡ ❧✐t❡r❛t✉r❡ ♦♥ t❤❡ s✉❜❥❡❝t ✫ ♥✉♠❡r♦✉s t②♣❡s ♦❢ P❉▼Ps
• ✉s❡❞ ✐♥ ❛ ✈❛r✐❡t② ♦❢ ✜❡❧❞s ✭♠♦❧❡❝✉❧❛r ❜✐♦❧♦❣②✱ ❝♦♠♠✉♥✐❝❛t✐♦♥

♥❡t✇♦r❦s✱ ✳✳✳✮ ❍❡r❡ ✇❡ r❡str✐❝t ❛tt❡♥t✐♦♥ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ s♣❡❝✐✜❝ ❝❧❛ss✿

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s P❉▼Ps (❩t) ❛♥❞ ( ˜ ❩♥)

• ❊ = {✶, . . . , ♠}✱
• ❋ ✶, . . . , ❋ ♠ s♠♦♦t❤ ❜♦✉♥❞❡❞ ✈❡❝t♦r ✜❡❧❞s ♦♥ R❞,
• (Φ✶

t ), . . . , (Φ♠ t ) ✐♥❞✉❝❡❞ ✢♦✇s

✿❞❡t❡r♠✐♥✐st✐❝ ❝♦♠♣♦♥❡♥ts

• ▼ ⊂ R❞ ❂ ❝♦♠♣❛❝t ♣♦s✐t✐✈❡❧② ✐♥✈❛r✐❛♥t s❡t ✉♥❞❡r ❡❛❝❤ Φ✐,
• ❋♦r ① ∈ ▼, (◗✐❥(①)) ❂ ▼❛r❦♦✈ tr❛♥s✐t✐♦♥ ♠❛tr✐① ♦✈❡r ❊

✭✐rr❡❞✉❝✐❜❧❡ ❛♣❡r✐♦❞✐❝ ❛♥❞ ❝♦♥t✐♥✉♦✉s ✐♥ ①✮ ✿s✇✐t❝❤✐♥❣ ♠❡❝❤❛♥✐s♠ ✳

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s P❉▼Ps (❩t) ❛♥❞ ( ˜ ❩♥)

• ❊ = {✶, . . . , ♠}✱
• ❋ ✶, . . . , ❋ ♠ s♠♦♦t❤ ❜♦✉♥❞❡❞ ✈❡❝t♦r ✜❡❧❞s ♦♥ R❞,
• (Φ✶

t ), . . . , (Φ♠ t ) ✐♥❞✉❝❡❞ ✢♦✇s ✿❞❡t❡r♠✐♥✐st✐❝ ❝♦♠♣♦♥❡♥ts

• ▼ ⊂ R❞ ❂ ❝♦♠♣❛❝t ♣♦s✐t✐✈❡❧② ✐♥✈❛r✐❛♥t s❡t ✉♥❞❡r ❡❛❝❤ Φ✐,
• ❋♦r ① ∈ ▼, (◗✐❥(①)) ❂ ▼❛r❦♦✈ tr❛♥s✐t✐♦♥ ♠❛tr✐① ♦✈❡r ❊

✭✐rr❡❞✉❝✐❜❧❡ ❛♣❡r✐♦❞✐❝ ❛♥❞ ❝♦♥t✐♥✉♦✉s ✐♥ ①✮ ✿s✇✐t❝❤✐♥❣ ♠❡❝❤❛♥✐s♠✳

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s P❉▼Ps (❩t) ❛♥❞ ( ˜ ❩♥)

❚❤❡ P❉▼P ❩t = (❳t, ❨t) ∈ ▼ × ❊ ✐s ❝♦♥str✉❝t❡❞ ❛s ❢♦❧❧♦✇s✿ ❙✉♣♣♦s❡ ❩✵ ❳✵ ❨✵ ① ✐ ❇❡❣✐♥ ✿ ❉r❛✇ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❯✶ ✇✐t❤ ❡①♣♦♥❡♥t✐❛❧ ❞✐str✐❜✉t✐♦♥ P ❯✶ t ❡

t

❋♦❧❧♦✇

✐ ❞✉r✐♥❣ t✐♠❡ ❯✶

❳t

✐ t ①

❢♦r t ❯✶ ❨t ❨✵ ✐ ❢♦r t ❯✶ P✐❝❦ ❥ ❊ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ◗✐❥ ❳❯✶ ❛♥❞ s❡t ❨❯✶ ❥ ❘❡♣❡❛t ❊♥❞

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s P❉▼Ps (❩t) ❛♥❞ ( ˜ ❩♥)

❚❤❡ P❉▼P ❩t = (❳t, ❨t) ∈ ▼ × ❊ ✐s ❝♦♥str✉❝t❡❞ ❛s ❢♦❧❧♦✇s✿

• ❙✉♣♣♦s❡ ❩✵ = (❳✵, ❨✵) = (①, ✐).

❇❡❣✐♥ ✿ ❉r❛✇ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❯✶ ✇✐t❤ ❡①♣♦♥❡♥t✐❛❧ ❞✐str✐❜✉t✐♦♥ P ❯✶ t ❡

t

❋♦❧❧♦✇

✐ ❞✉r✐♥❣ t✐♠❡ ❯✶

❳t

✐ t ①

❢♦r t ❯✶ ❨t ❨✵ ✐ ❢♦r t ❯✶ P✐❝❦ ❥ ❊ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ◗✐❥ ❳❯✶ ❛♥❞ s❡t ❨❯✶ ❥ ❘❡♣❡❛t ❊♥❞

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s P❉▼Ps (❩t) ❛♥❞ ( ˜ ❩♥)

❚❤❡ P❉▼P ❩t = (❳t, ❨t) ∈ ▼ × ❊ ✐s ❝♦♥str✉❝t❡❞ ❛s ❢♦❧❧♦✇s✿

• ❙✉♣♣♦s❡ ❩✵ = (❳✵, ❨✵) = (①, ✐).

❇❡❣✐♥ ✿ − ❉r❛✇ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❯✶ ✇✐t❤ ❡①♣♦♥❡♥t✐❛❧ ❞✐str✐❜✉t✐♦♥ P(❯✶ > t) = ❡−λt ❋♦❧❧♦✇

✐ ❞✉r✐♥❣ t✐♠❡ ❯✶

❳t

✐ t ①

❢♦r t ❯✶ ❨t ❨✵ ✐ ❢♦r t ❯✶ P✐❝❦ ❥ ❊ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ◗✐❥ ❳❯✶ ❛♥❞ s❡t ❨❯✶ ❥ ❘❡♣❡❛t ❊♥❞

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s P❉▼Ps (❩t) ❛♥❞ ( ˜ ❩♥)

❚❤❡ P❉▼P ❩t = (❳t, ❨t) ∈ ▼ × ❊ ✐s ❝♦♥str✉❝t❡❞ ❛s ❢♦❧❧♦✇s✿

• ❙✉♣♣♦s❡ ❩✵ = (❳✵, ❨✵) = (①, ✐).

❇❡❣✐♥ ✿ − ❉r❛✇ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❯✶ ✇✐t❤ ❡①♣♦♥❡♥t✐❛❧ ❞✐str✐❜✉t✐♦♥ P(❯✶ > t) = ❡−λt − ❋♦❧❧♦✇ Φ✐ ❞✉r✐♥❣ t✐♠❡ ❯✶ : ❳t = Φ✐

t(①) ❢♦r t ≤ ❯✶; ❨t = ❨✵ = ✐, ❢♦r t<❯✶.

P✐❝❦ ❥ ❊ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ◗✐❥ ❳❯✶ ❛♥❞ s❡t ❨❯✶ ❥ ❘❡♣❡❛t ❊♥❞

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s P❉▼Ps (❩t) ❛♥❞ ( ˜ ❩♥)

❚❤❡ P❉▼P ❩t = (❳t, ❨t) ∈ ▼ × ❊ ✐s ❝♦♥str✉❝t❡❞ ❛s ❢♦❧❧♦✇s✿

• ❙✉♣♣♦s❡ ❩✵ = (❳✵, ❨✵) = (①, ✐).

❇❡❣✐♥ ✿ − ❉r❛✇ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❯✶ ✇✐t❤ ❡①♣♦♥❡♥t✐❛❧ ❞✐str✐❜✉t✐♦♥ P(❯✶ > t) = ❡−λt − ❋♦❧❧♦✇ Φ✐ ❞✉r✐♥❣ t✐♠❡ ❯✶ : ❳t = Φ✐

t(①) ❢♦r t ≤ ❯✶; ❨t = ❨✵ = ✐, ❢♦r t<❯✶.

− P✐❝❦ ❥ ∈ ❊ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ◗✐❥(❳❯✶) ❛♥❞ s❡t ❨❯✶ = ❥. ❘❡♣❡❛t ❊♥❞

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s P❉▼Ps (❩t) ❛♥❞ ( ˜ ❩♥)

❚❤❡ P❉▼P ❩t = (❳t, ❨t) ∈ ▼ × ❊ ✐s ❝♦♥str✉❝t❡❞ ❛s ❢♦❧❧♦✇s✿

• ❙✉♣♣♦s❡ ❩✵ = (❳✵, ❨✵) = (①, ✐).

❇❡❣✐♥ ✿ − ❉r❛✇ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❯✶ ✇✐t❤ ❡①♣♦♥❡♥t✐❛❧ ❞✐str✐❜✉t✐♦♥ P(❯✶ > t) = ❡−λt − ❋♦❧❧♦✇ Φ✐ ❞✉r✐♥❣ t✐♠❡ ❯✶ : ❳t = Φ✐

t(①) ❢♦r t ≤ ❯✶; ❨t = ❨✵ = ✐, ❢♦r t<❯✶.

− P✐❝❦ ❥ ∈ ❊ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ◗✐❥(❳❯✶) ❛♥❞ s❡t ❨❯✶ = ❥. − ❘❡♣❡❛t ❊♥❞

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s P❉▼Ps (❩t) ❛♥❞ ( ˜ ❩♥)

❚❤✐s ♠❛❦❡s (❩t) ❛ ❝♦♥t✐♥✉♦✉s t✐♠❡ ❋❡❧❧❡r ▼❛r❦♦✈ ♣r♦❝❡ss✱ ❛♥❞ (˜ ❩♥) = (❩❚♥) ❛ ❞✐s❝r❡t❡ t✐♠❡ ❋❡❧❧❡r ▼❛r❦♦✈ ❝❤❛✐♥✱ ❚♥ = ❯✶ + . . . + ❯♥ ❛r❡ t❤❡ ❥✉♠♣ t✐♠❡s✳ ❋❡❧❧❡r ✿ Pt❢ ③

③ ❢ ❩t

P❢ ③

③ ❢ ❩✶

♠❛♣ ❈❜ ▼ ✐♥t♦ ✐ts❡❧❢✳

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s P❉▼Ps (❩t) ❛♥❞ ( ˜ ❩♥)

❚❤✐s ♠❛❦❡s (❩t) ❛ ❝♦♥t✐♥✉♦✉s t✐♠❡ ❋❡❧❧❡r ▼❛r❦♦✈ ♣r♦❝❡ss✱ ❛♥❞ (˜ ❩♥) = (❩❚♥) ❛ ❞✐s❝r❡t❡ t✐♠❡ ❋❡❧❧❡r ▼❛r❦♦✈ ❝❤❛✐♥✱ ❚♥ = ❯✶ + . . . + ❯♥ ❛r❡ t❤❡ ❥✉♠♣ t✐♠❡s✳ ❋❡❧❧❡r ✿ Pt❢ (③) = E③(❢ (❩t)), ˜ P❢ (③) = E③(❢ (˜ ❩✶)) ♠❛♣ ❈❜(▼) ✐♥t♦ ✐ts❡❧❢✳

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s

▼❛✐♥ ●♦❛❧s

❖✉r ♠❛✐♥ ❣♦❛❧ ✐s t♦ ■♥✈❡st✐❣❛t❡ t❤❡ ❧♦♥❣ t❡r♠ ❜❡❤❛✈✐♦r ♦❢ (❩t), (❩♥) ❉❡s❝r✐❜❡ t❤❡ s✉♣♣♦rt ♦❢ t❤❡✐r ❧❛✇s✱ t❤❡✐r ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s ❛♥❞✱ t❤❡✐r s✉♣♣♦rts

• ✐✈❡ ❝♦♥❞✐t✐♦♥s ❡♥s✉r✐♥❣

❯♥✐q✉❡♥❡ss ✭♦❢ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡✮✱ ❊r❣♦❞✐❝✐t②✱ ❊①♣♦♥❡♥t✐❛❧ ❝♦♥✈❡r❣❡♥❝❡✱ ✳✳✳

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s

▼❛✐♥ ●♦❛❧s

❖✉r ♠❛✐♥ ❣♦❛❧ ✐s t♦ ■♥✈❡st✐❣❛t❡ t❤❡ ❧♦♥❣ t❡r♠ ❜❡❤❛✈✐♦r ♦❢ (❩t), (❩♥) ❉❡s❝r✐❜❡ t❤❡

• s✉♣♣♦rt ♦❢ t❤❡✐r ❧❛✇s✱

t❤❡✐r ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s ❛♥❞✱ t❤❡✐r s✉♣♣♦rts

• ✐✈❡ ❝♦♥❞✐t✐♦♥s ❡♥s✉r✐♥❣

❯♥✐q✉❡♥❡ss ✭♦❢ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡✮✱ ❊r❣♦❞✐❝✐t②✱ ❊①♣♦♥❡♥t✐❛❧ ❝♦♥✈❡r❣❡♥❝❡✱ ✳✳✳

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s

▼❛✐♥ ●♦❛❧s

❖✉r ♠❛✐♥ ❣♦❛❧ ✐s t♦ ■♥✈❡st✐❣❛t❡ t❤❡ ❧♦♥❣ t❡r♠ ❜❡❤❛✈✐♦r ♦❢ (❩t), (❩♥) ❉❡s❝r✐❜❡ t❤❡

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

t❤❡✐r s✉♣♣♦rts

• ✐✈❡ ❝♦♥❞✐t✐♦♥s ❡♥s✉r✐♥❣

❯♥✐q✉❡♥❡ss ✭♦❢ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡✮✱ ❊r❣♦❞✐❝✐t②✱ ❊①♣♦♥❡♥t✐❛❧ ❝♦♥✈❡r❣❡♥❝❡✱ ✳✳✳

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s

▼❛✐♥ ●♦❛❧s

❖✉r ♠❛✐♥ ❣♦❛❧ ✐s t♦ ■♥✈❡st✐❣❛t❡ t❤❡ ❧♦♥❣ t❡r♠ ❜❡❤❛✈✐♦r ♦❢ (❩t), (❩♥) ❉❡s❝r✐❜❡ t❤❡

• s✉♣♣♦rt ♦❢ t❤❡✐r ❧❛✇s✱
• t❤❡✐r ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s ❛♥❞✱
• t❤❡✐r s✉♣♣♦rts
• ✐✈❡ ❝♦♥❞✐t✐♦♥s ❡♥s✉r✐♥❣

❯♥✐q✉❡♥❡ss ✭♦❢ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡✮✱ ❊r❣♦❞✐❝✐t②✱ ❊①♣♦♥❡♥t✐❛❧ ❝♦♥✈❡r❣❡♥❝❡✱ ✳✳✳

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s

▼❛✐♥ ●♦❛❧s

❖✉r ♠❛✐♥ ❣♦❛❧ ✐s t♦ ■♥✈❡st✐❣❛t❡ t❤❡ ❧♦♥❣ t❡r♠ ❜❡❤❛✈✐♦r ♦❢ (❩t), (❩♥) ❉❡s❝r✐❜❡ t❤❡

• s✉♣♣♦rt ♦❢ t❤❡✐r ❧❛✇s✱
• t❤❡✐r ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s ❛♥❞✱
• t❤❡✐r s✉♣♣♦rts
• ✐✈❡ ❝♦♥❞✐t✐♦♥s ❡♥s✉r✐♥❣
• ❯♥✐q✉❡♥❡ss ✭♦❢ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡✮✱

❊r❣♦❞✐❝✐t②✱ ❊①♣♦♥❡♥t✐❛❧ ❝♦♥✈❡r❣❡♥❝❡✱ ✳✳✳

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s

▼❛✐♥ ●♦❛❧s

❖✉r ♠❛✐♥ ❣♦❛❧ ✐s t♦ ■♥✈❡st✐❣❛t❡ t❤❡ ❧♦♥❣ t❡r♠ ❜❡❤❛✈✐♦r ♦❢ (❩t), (❩♥) ❉❡s❝r✐❜❡ t❤❡

• s✉♣♣♦rt ♦❢ t❤❡✐r ❧❛✇s✱
• t❤❡✐r ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s ❛♥❞✱
• t❤❡✐r s✉♣♣♦rts
• ✐✈❡ ❝♦♥❞✐t✐♦♥s ❡♥s✉r✐♥❣
• ❯♥✐q✉❡♥❡ss ✭♦❢ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡✮✱
• ❊r❣♦❞✐❝✐t②✱

❊①♣♦♥❡♥t✐❛❧ ❝♦♥✈❡r❣❡♥❝❡✱ ✳✳✳

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s

▼❛✐♥ ●♦❛❧s

❖✉r ♠❛✐♥ ❣♦❛❧ ✐s t♦ ■♥✈❡st✐❣❛t❡ t❤❡ ❧♦♥❣ t❡r♠ ❜❡❤❛✈✐♦r ♦❢ (❩t), (❩♥) ❉❡s❝r✐❜❡ t❤❡

• s✉♣♣♦rt ♦❢ t❤❡✐r ❧❛✇s✱
• t❤❡✐r ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s ❛♥❞✱
• t❤❡✐r s✉♣♣♦rts
• ✐✈❡ ❝♦♥❞✐t✐♦♥s ❡♥s✉r✐♥❣
• ❯♥✐q✉❡♥❡ss ✭♦❢ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡✮✱
• ❊r❣♦❞✐❝✐t②✱
• ❊①♣♦♥❡♥t✐❛❧ ❝♦♥✈❡r❣❡♥❝❡✱ ✳✳✳

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆♥♦t❤❡r ✭❧❡ss tr✐✈✐❛❧✮ ❡①❛♠♣❧❡ ❆ ♥♦♥ tr✐✈✐❛❧ ❡①❛♠♣❧❡

❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡

❊ = {✶}, ▼ = ❙✶ = R/Z, Φt(①) = (① + t) ♠♦❞ ✶. ✱

❋✐❣✉r❡✿ ❞❡s✐❣♥❡❞ ❜② ❢r❡❡♣✐❝❦

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆♥♦t❤❡r ✭❧❡ss tr✐✈✐❛❧✮ ❡①❛♠♣❧❡ ❆ ♥♦♥ tr✐✈✐❛❧ ❡①❛♠♣❧❡

❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡

❊ = {✶}, ▼ = ❙✶ = R/Z, Φt(①) = (① + t) ♠♦❞ ✶. ✱

❋✐❣✉r❡✿ ❞❡s✐❣♥❡❞ ❜② ❢r❡❡♣✐❝❦

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆♥♦t❤❡r ✭❧❡ss tr✐✈✐❛❧✮ ❡①❛♠♣❧❡ ❆ ♥♦♥ tr✐✈✐❛❧ ❡①❛♠♣❧❡

❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡

❊ = {✶}, ▼ = ❙✶ = R/Z, Φt(①) = (① + t) ♠♦❞ ✶. ❳✵ = ① ⇒ ❳t = Φt(①), ˜ ❳♥ = (① + ❚♥) ♠♦❞ ✶ ✇✐t❤ ❚♥ = ❯✶ + . . . + ❯♥ ∼ Γ(♥). ❇♦t❤ ❩ ❛♥❞ ❩ ❛r❡ ✉♥✐q✉❡❧② ❡r❣♦❞✐❝ ✭✉♥✐q✉❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ✐s t❤❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ♦♥ ❙✶✮ ▲❛✇ ❩t

t ①

▲❛✇ ❩♥ ① ♥ ♠♦❞ ✶

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆♥♦t❤❡r ✭❧❡ss tr✐✈✐❛❧✮ ❡①❛♠♣❧❡ ❆ ♥♦♥ tr✐✈✐❛❧ ❡①❛♠♣❧❡

❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡

❊ = {✶}, ▼ = ❙✶ = R/Z, Φt(①) = (① + t) ♠♦❞ ✶. ❳✵ = ① ⇒ ❳t = Φt(①), ˜ ❳♥ = (① + ❚♥) ♠♦❞ ✶ ✇✐t❤ ❚♥ = ❯✶ + . . . + ❯♥ ∼ Γ(♥). ❇♦t❤ ❩ ❛♥❞ (˜ ❩) ❛r❡ ✉♥✐q✉❡❧② ❡r❣♦❞✐❝ ✭✉♥✐q✉❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ✐s t❤❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ µ ♦♥ ❙✶✮ ▲❛✇(❩t) = δΦt(①), ▲❛✇(˜ ❩♥) = (① + Γ(♥)) ♠♦❞ ✶ → µ.

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆♥♦t❤❡r ✭❧❡ss tr✐✈✐❛❧✮ ❡①❛♠♣❧❡ ❆ ♥♦♥ tr✐✈✐❛❧ ❡①❛♠♣❧❡

❆♥♦t❤❡r ✭❧❡ss tr✐✈✐❛❧✮ ❡①❛♠♣❧❡

❊ = {✶, ✷}, ▼ ⊂ R✷ ❋ ✶(①) = ❆①, ❋ ✷(①) = ❆(① − ❡) ❆ = −✶ ✶ ✶ −✶

• , ❡ =

✶ ✵

• .

0.1 0.2 0.3 0.4

• 0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

• 0.4
• 0.3
• 0.2
• 0.1

0.0

✱

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆♥♦t❤❡r ✭❧❡ss tr✐✈✐❛❧✮ ❡①❛♠♣❧❡ ❆ ♥♦♥ tr✐✈✐❛❧ ❡①❛♠♣❧❡

❆ ♥♦♥ tr✐✈✐❛❧ ❡①❛♠♣❧❡ ✭❢r♦♠ ❇❡♥❛✐♠ ❛♥❞ ▲♦❜r②✱ ✶✺✮

❊ = {✶, ✷}, ▼ ⊂ R+ × R+. ❋ ✶, ❋ ✷ t✇♦ ▲♦t❦❛✲❱♦❧t❡rr❛ ✈❡❝t♦r ✜❡❧❞s ❋ ✐(①, ②) = α✐①(✶ − ❛✐① − ❜✐②) β✐②(✶ − ❝✐① − ❞✐②) ❢❛✈♦r❛❜❧❡ t♦ t❤❡ s❛♠❡ s♣❡❝✐❡s ①

✐ ✐ ❛✐ ❜✐

✵ ❛✐ ❝✐ ❛♥❞ ❜✐ ❞✐

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆♥♦t❤❡r ✭❧❡ss tr✐✈✐❛❧✮ ❡①❛♠♣❧❡ ❆ ♥♦♥ tr✐✈✐❛❧ ❡①❛♠♣❧❡

❆ ♥♦♥ tr✐✈✐❛❧ ❡①❛♠♣❧❡ ✭❢r♦♠ ❇❡♥❛✐♠ ❛♥❞ ▲♦❜r②✱ ✶✺✮

❊ = {✶, ✷}, ▼ ⊂ R+ × R+. ❋ ✶, ❋ ✷ t✇♦ ▲♦t❦❛✲❱♦❧t❡rr❛ ✈❡❝t♦r ✜❡❧❞s ❋ ✐(①, ②) = α✐①(✶ − ❛✐① − ❜✐②) β✐②(✶ − ❝✐① − ❞✐②) ❢❛✈♦r❛❜❧❡ t♦ t❤❡ s❛♠❡ s♣❡❝✐❡s ① α✐, β✐, ❛✐, ❜✐ > ✵, ❛✐ < ❝✐ ❛♥❞ ❜✐ < ❞✐.

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆♥♦t❤❡r ✭❧❡ss tr✐✈✐❛❧✮ ❡①❛♠♣❧❡ ❆ ♥♦♥ tr✐✈✐❛❧ ❡①❛♠♣❧❡

✱

❋✐❣✉r❡✿ P❤❛s❡ ♣♦rtr❛✐ts ♦❢ ❋ ✶ ❛♥❞ ❋ ✷

◗ ① ✶ ♣ ♣ ✶ ♣ ♣ ❉✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ ♣ ❧❡❛❞ t♦ ❞✐✛❡r❡♥t ❜❡❤❛✈✐♦rs

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆♥♦t❤❡r ✭❧❡ss tr✐✈✐❛❧✮ ❡①❛♠♣❧❡ ❆ ♥♦♥ tr✐✈✐❛❧ ❡①❛♠♣❧❡

✱

❋✐❣✉r❡✿ P❤❛s❡ ♣♦rtr❛✐ts ♦❢ ❋ ✶ ❛♥❞ ❋ ✷

◗(①) = ✶ − ♣ ♣ ✶ − ♣ ♣

• .

❉✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ ♣, λ ❧❡❛❞ t♦ ❞✐✛❡r❡♥t ❜❡❤❛✈✐♦rs ֒ →

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆♥♦t❤❡r ✭❧❡ss tr✐✈✐❛❧✮ ❡①❛♠♣❧❡ ❆ ♥♦♥ tr✐✈✐❛❧ ❡①❛♠♣❧❡

❋✐❣✉r❡✿ ❡①t✐♥❝t✐♦♥ ♦❢ ✷

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆♥♦t❤❡r ✭❧❡ss tr✐✈✐❛❧✮ ❡①❛♠♣❧❡ ❆ ♥♦♥ tr✐✈✐❛❧ ❡①❛♠♣❧❡

❋✐❣✉r❡✿ P❡rs✐st❡♥❝❡

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆♥♦t❤❡r ✭❧❡ss tr✐✈✐❛❧✮ ❡①❛♠♣❧❡ ❆ ♥♦♥ tr✐✈✐❛❧ ❡①❛♠♣❧❡

❋✐❣✉r❡✿ P❡rs✐st❡♥❝❡

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆♥♦t❤❡r ✭❧❡ss tr✐✈✐❛❧✮ ❡①❛♠♣❧❡ ❆ ♥♦♥ tr✐✈✐❛❧ ❡①❛♠♣❧❡

❋✐❣✉r❡✿ ❊①t✐♥❝t✐♦♥ ♦❢ ✶

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

■♥tr♦❞✉❝t✐♦♥

• ♦❛❧s

❙♦♠❡ ❊①❛♠♣❧❡s ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆ tr✐✈✐❛❧ ✭❜✉t ✐♥str✉❝t✐✈❡✮ ❡①❛♠♣❧❡ ❆♥♦t❤❡r ✭❧❡ss tr✐✈✐❛❧✮ ❡①❛♠♣❧❡ ❆ ♥♦♥ tr✐✈✐❛❧ ❡①❛♠♣❧❡

❋✐❣✉r❡✿ ❊①t✐♥❝t✐♦♥ ♦❢ ✶ ♦r ✷

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ❙❖▼❊ ▼❆❚❍❙

❙♦♠❡ ▼❛t❤s

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡

❆ s✉♣♣♦rt ❚❤❡♦r❡♠

❝♦(❋)(①) := ❝♦♥✈(❋ ✶(①), . . . , ❋ ♠(①)),

• ❙① t❤❡ s❡t ♦❢ ✭❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s✮ ♠❛♣s η : R+ → R❞

s♦❧✉t✐♦♥s t♦ ˙ η ∈ ❝♦(❋)(η), η(✵) = ① ❈❧❡❛r❧② ❳✵ ① ❳t ❙① ❜✉t ♠♦r❡ ❝❛♥ ❜❡ s❛✐❞ ❚❤❡♦r❡♠ ✭❇❡♥❛✐♠✱ ▲❡❜♦r❣♥❡✱ ▼❛❧r✐❡✉✱ ❩✐tt✱ ✶✺✮ ■❢ ❳✵ ① t❤❡♥✱ t❤❡ s✉♣♣♦rt ♦❢ t❤❡ ❧❛✇ ♦❢ ❳t ❡q✉❛❧s ❙①

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡

❆ s✉♣♣♦rt ❚❤❡♦r❡♠

❝♦(❋)(①) := ❝♦♥✈(❋ ✶(①), . . . , ❋ ♠(①)),

• ❙① t❤❡ s❡t ♦❢ ✭❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s✮ ♠❛♣s η : R+ → R❞

s♦❧✉t✐♦♥s t♦ ˙ η ∈ ❝♦(❋)(η), η(✵) = ① ❈❧❡❛r❧② ❳✵ = ① ⇒ (❳t) ∈ ❙① ❜✉t ♠♦r❡ ❝❛♥ ❜❡ s❛✐❞ ❚❤❡♦r❡♠ ✭❇❡♥❛✐♠✱ ▲❡❜♦r❣♥❡✱ ▼❛❧r✐❡✉✱ ❩✐tt✱ ✶✺✮ ■❢ ❳✵ ① t❤❡♥✱ t❤❡ s✉♣♣♦rt ♦❢ t❤❡ ❧❛✇ ♦❢ ❳t ❡q✉❛❧s ❙①

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡

❆ s✉♣♣♦rt ❚❤❡♦r❡♠

❝♦(❋)(①) := ❝♦♥✈(❋ ✶(①), . . . , ❋ ♠(①)),

• ❙① t❤❡ s❡t ♦❢ ✭❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s✮ ♠❛♣s η : R+ → R❞

s♦❧✉t✐♦♥s t♦ ˙ η ∈ ❝♦(❋)(η), η(✵) = ① ❈❧❡❛r❧② ❳✵ = ① ⇒ (❳t) ∈ ❙① ❜✉t ♠♦r❡ ❝❛♥ ❜❡ s❛✐❞ ❚❤❡♦r❡♠ ✭❇❡♥❛✐♠✱ ▲❡❜♦r❣♥❡✱ ▼❛❧r✐❡✉✱ ❩✐tt✱ ✶✺✮ ■❢ ❳✵ = ① t❤❡♥✱ t❤❡ s✉♣♣♦rt ♦❢ t❤❡ ❧❛✇ ♦❢ (❳t) ❡q✉❛❧s ❙①.

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

■♥✈❛r✐❛♥t ▼❡❛s✉r❡s

❆ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ ▼ × ❊ ✐s ❝❛❧❧❡❞ ✐♥✈❛r✐❛♥t ❢♦r (❩t) ✇❤❡♥❡✈❡r ▲❛✇(❩✵) = µ ⇒ ▲❛✇(❩t) = µ ⇔

• P③(❩t ∈ ·)µ(❞③) = µ(·)

⇔

• Pt❢ (③)µ(❞③) =
• ❢ µ(❞③).

❙✐♠✐❧❛r ❞❡✜♥✐t✐♦♥ ❢♦r ❩♥ ❊r❣♦❞✐❝ ♠❡❛s✉r❡ ❂ ❡①tr❡♠❛❧ ✐♥✈❛r✐❛♥t

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

■♥✈❛r✐❛♥t ▼❡❛s✉r❡s

❆ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ ▼ × ❊ ✐s ❝❛❧❧❡❞ ✐♥✈❛r✐❛♥t ❢♦r (❩t) ✇❤❡♥❡✈❡r ▲❛✇(❩✵) = µ ⇒ ▲❛✇(❩t) = µ ⇔

• P③(❩t ∈ ·)µ(❞③) = µ(·)

⇔

• Pt❢ (③)µ(❞③) =
• ❢ µ(❞③).

❙✐♠✐❧❛r ❞❡✜♥✐t✐♦♥ ❢♦r (˜ ❩♥) ❊r❣♦❞✐❝ ♠❡❛s✉r❡ ❂ ❡①tr❡♠❛❧ ✐♥✈❛r✐❛♥t

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

■♥✈❛r✐❛♥t ▼❡❛s✉r❡s

❆ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ ▼ × ❊ ✐s ❝❛❧❧❡❞ ✐♥✈❛r✐❛♥t ❢♦r (❩t) ✇❤❡♥❡✈❡r ▲❛✇(❩✵) = µ ⇒ ▲❛✇(❩t) = µ ⇔

• P③(❩t ∈ ·)µ(❞③) = µ(·)

⇔

• Pt❢ (③)µ(❞③) =
• ❢ µ(❞③).

❙✐♠✐❧❛r ❞❡✜♥✐t✐♦♥ ❢♦r (˜ ❩♥) ❊r❣♦❞✐❝ ♠❡❛s✉r❡ ❂ ❡①tr❡♠❛❧ ✐♥✈❛r✐❛♥t

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

• P✐♥✈, (P❡r❣) := ✐♥✈❛r✐❛♥t ✭❡r❣♦❞✐❝✮ ♠❡❛s✉r❡s ❢♦r (❩t)
• ˜

P✐♥✈, (˜ P❡r❣) := ✐♥✈❛r✐❛♥t ✭❡r❣♦❞✐❝✮ ♠❡❛s✉r❡s ❢♦r (❩t) ❈♦♠♣❛❝t♥❡ss ❛♥❞ ❋❡❧❧❡r ❈♦♥t✐♥✉✐t② ⇒ P✐♥✈, ˜ P✐♥✈ ❛r❡ ♥♦♥ ❡♠♣t②✳ Pr♦♣♦s✐t✐♦♥ ✭❈♦rr❡s♣♦♥❞❡♥❝❡ ❢♦r ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s✱ ❇▲▼❩✱ ✶✺✮ ✭✐✮

✐♥✈ ❡r❣

❛♥❞

✐♥✈ ❡r❣

❛r❡ ❤♦♠❡♦♠♦r♣❤✐❝ ✭✐✐✮ ❚❤✐s ❤♦♠❡♦♠♦r♣❤✐s♠ ♣r❡s❡r✈❡s t❤❡ s✉♣♣♦rt ✭✐✐✐✮ ■❢ ✐s ❡r❣♦❞✐❝ ✐t ✐s ❡✐t❤❡r ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ♦r s✐♥❣✉❧❛r ❖♣❡♥ ♣r♦❜❧❡♠ ◆♦t❤✐♥❣ ✐s ❦♥♦✇♥ ✐♥ ❣❡♥❡r❛❧ ✭✐♥ t❤❡ ❛✲❝ ❝❛s❡✮ ♦♥ t❤❡ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ❞❡♥s✐t② ❡①❝❡♣t ✐♥ ❞✐♠❡♥s✐♦♥ ✶ ✿ ✭❨✳ ❇❛❦❤t✐♥✱ ❚✳ ❍✉rt ❛♥❞ ❏✲ ▼❛tt✐♥❣❧②✱ ◆♦♥❧✐♥❡❛r✐t②✱ ✷✵✶✺✮

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

• P✐♥✈, (P❡r❣) := ✐♥✈❛r✐❛♥t ✭❡r❣♦❞✐❝✮ ♠❡❛s✉r❡s ❢♦r (❩t)
• ˜

P✐♥✈, (˜ P❡r❣) := ✐♥✈❛r✐❛♥t ✭❡r❣♦❞✐❝✮ ♠❡❛s✉r❡s ❢♦r (❩t) ❈♦♠♣❛❝t♥❡ss ❛♥❞ ❋❡❧❧❡r ❈♦♥t✐♥✉✐t② ⇒ P✐♥✈, ˜ P✐♥✈ ❛r❡ ♥♦♥ ❡♠♣t②✳ Pr♦♣♦s✐t✐♦♥ ✭❈♦rr❡s♣♦♥❞❡♥❝❡ ❢♦r ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s✱ ❇▲▼❩✱ ✶✺✮ ✭✐✮ ˜ P✐♥✈, (˜ P❡r❣) ❛♥❞ ˜ P✐♥✈, (˜ P❡r❣) ❛r❡ ❤♦♠❡♦♠♦r♣❤✐❝ ✭✐✐✮ ❚❤✐s ❤♦♠❡♦♠♦r♣❤✐s♠ ♣r❡s❡r✈❡s t❤❡ s✉♣♣♦rt ✭✐✐✐✮ ■❢ µ ✐s ❡r❣♦❞✐❝ ✐t ✐s ❡✐t❤❡r ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ♦r s✐♥❣✉❧❛r ❖♣❡♥ ♣r♦❜❧❡♠ ◆♦t❤✐♥❣ ✐s ❦♥♦✇♥ ✐♥ ❣❡♥❡r❛❧ ✭✐♥ t❤❡ ❛✲❝ ❝❛s❡✮ ♦♥ t❤❡ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ❞❡♥s✐t② ❡①❝❡♣t ✐♥ ❞✐♠❡♥s✐♦♥ ✶ ✿ ✭❨✳ ❇❛❦❤t✐♥✱ ❚✳ ❍✉rt ❛♥❞ ❏✲ ▼❛tt✐♥❣❧②✱ ◆♦♥❧✐♥❡❛r✐t②✱ ✷✵✶✺✮

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

• P✐♥✈, (P❡r❣) := ✐♥✈❛r✐❛♥t ✭❡r❣♦❞✐❝✮ ♠❡❛s✉r❡s ❢♦r (❩t)
• ˜

P✐♥✈, (˜ P❡r❣) := ✐♥✈❛r✐❛♥t ✭❡r❣♦❞✐❝✮ ♠❡❛s✉r❡s ❢♦r (❩t) ❈♦♠♣❛❝t♥❡ss ❛♥❞ ❋❡❧❧❡r ❈♦♥t✐♥✉✐t② ⇒ P✐♥✈, ˜ P✐♥✈ ❛r❡ ♥♦♥ ❡♠♣t②✳ Pr♦♣♦s✐t✐♦♥ ✭❈♦rr❡s♣♦♥❞❡♥❝❡ ❢♦r ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s✱ ❇▲▼❩✱ ✶✺✮ ✭✐✮ ˜ P✐♥✈, (˜ P❡r❣) ❛♥❞ ˜ P✐♥✈, (˜ P❡r❣) ❛r❡ ❤♦♠❡♦♠♦r♣❤✐❝ ✭✐✐✮ ❚❤✐s ❤♦♠❡♦♠♦r♣❤✐s♠ ♣r❡s❡r✈❡s t❤❡ s✉♣♣♦rt ✭✐✐✐✮ ■❢ µ ✐s ❡r❣♦❞✐❝ ✐t ✐s ❡✐t❤❡r ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ♦r s✐♥❣✉❧❛r ❖♣❡♥ ♣r♦❜❧❡♠ ◆♦t❤✐♥❣ ✐s ❦♥♦✇♥ ✐♥ ❣❡♥❡r❛❧ ✭✐♥ t❤❡ ❛✲❝ ❝❛s❡✮ ♦♥ t❤❡ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ❞❡♥s✐t② ❡①❝❡♣t ✐♥ ❞✐♠❡♥s✐♦♥ ✶ ✿ ✭❨✳ ❇❛❦❤t✐♥✱ ❚✳ ❍✉rt ❛♥❞ ❏✲ ▼❛tt✐♥❣❧②✱ ◆♦♥❧✐♥❡❛r✐t②✱ ✷✵✶✺✮

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

❆❝❝❡ss✐❜❧❡ ❙❡t

• (Ψt) t❤❡ s❡t ✈❛❧✉❡❞ s❡♠✐ ✢♦✇ ✐♥❞✉❝❡❞ ❜② ˙

η ∈ ❝♦(❋)(η) : Ψt(①) = {η(t) : η ∈ ❙①}. ❖♠❡❣❛ ❧✐♠✐t s❡t✿ ωΨ(①) =

• t≥✵

{Ψs(①), s ≥ t} ❆❝❝❡ss✐❜❧❡ ❙❡t ✿ Γ =

• ①∈▼

ωΨ(①) ✐s ❛ ❝♦♠♣❛❝t ♣♦ss✐❜❧② ❡♠♣t② s❡t

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

❆❝❝❡ss✐❜❧❡ ❙❡t

• (Ψt) t❤❡ s❡t ✈❛❧✉❡❞ s❡♠✐ ✢♦✇ ✐♥❞✉❝❡❞ ❜② ˙

η ∈ ❝♦(❋)(η) : Ψt(①) = {η(t) : η ∈ ❙①}. ❖♠❡❣❛ ❧✐♠✐t s❡t✿ ωΨ(①) =

• t≥✵

{Ψs(①), s ≥ t} ❆❝❝❡ss✐❜❧❡ ❙❡t ✿ Γ =

• ①∈▼

ωΨ(①) Γ ✐s ❛ ❝♦♠♣❛❝t ♣♦ss✐❜❧② ❡♠♣t② s❡t

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

Pr♦♣♦s✐t✐♦♥ ✭Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❛❝❝❡ss✐❜❧❡ s❡t✱ ❇▲▼❩✱ ✶✺✮ ✭✐✮ Γ = ωΨ(♣) ❢♦r ❛❧❧ ♣ ∈ Γ ✭✐✐✮ Γ ✐s ❝♦♠♣❛❝t ❝♦♥♥❡❝t❡❞ ✐♥✈❛r✐❛♥t ✭∀t ≥ ✵Ψt(Γ) = Γ). ✭✐✐✐✮ ❊✐t❤❡r Γ ❤❛s ❡♠♣t② ✐♥t❡r✐♦r ♦r ✐ts ✐♥t❡r✐♦r ✐s ❞❡♥s❡ ✐♥ Γ.

0.1 0.2 0.3 0.4

• 0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

• 0.4
• 0.3
• 0.2
• 0.1

0.0

✱

❋✐❣✉r❡✿ ❊①❛♠♣❧❡ ♦❢ ❛❝❝❡ss✐❜❧❡ s❡t

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

Pr♦♣♦s✐t✐♦♥ ✭❆❝❝❡ss✐❜❧❡ s❡t ❛♥❞ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s✱ ❇▲▼❩✱ ✶✺✮ ✭✐✮ ■❢ Γ = ∅ t❤❡♥ Γ × ❊ ⊂ s✉♣♣(µ) ❢♦r ❛❧❧ µ ∈ P✐♥✈ ❛♥❞ t❤❡r❡ ❡①✐sts µ ∈ P✐♥✈ s✉❝❤ t❤❛t s✉♣♣(µ) = Γ × ❊. ✭✐✐✮ ■❢ Γ ❤❛s ♥♦♥ ❡♠♣t② ✐♥t❡r✐♦r✱ t❤❡♥ Γ × ❊ = s✉♣♣(µ) ❢♦r ❛❧❧ µ ∈ P✐♥✈✳ ✭✐✐✐✮ ❙✉♣♣♦s❡ t❤❛t t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ❢♦r ❩ ✭♦r ˜ ❩✮ π✱ t❤❡♥ s✉♣♣(π) = Γ × ❊✳ ❲❤❡♥ ❤❛s ❡♠♣t② ✐♥t❡r✐♦r✱ ✐♥❝❧✉s✐♦♥ ❊ s✉♣♣ ❝❛♥ ❜❡ str✐❝t ✦

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

Pr♦♣♦s✐t✐♦♥ ✭❆❝❝❡ss✐❜❧❡ s❡t ❛♥❞ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s✱ ❇▲▼❩✱ ✶✺✮ ✭✐✮ ■❢ Γ = ∅ t❤❡♥ Γ × ❊ ⊂ s✉♣♣(µ) ❢♦r ❛❧❧ µ ∈ P✐♥✈ ❛♥❞ t❤❡r❡ ❡①✐sts µ ∈ P✐♥✈ s✉❝❤ t❤❛t s✉♣♣(µ) = Γ × ❊. ✭✐✐✮ ■❢ Γ ❤❛s ♥♦♥ ❡♠♣t② ✐♥t❡r✐♦r✱ t❤❡♥ Γ × ❊ = s✉♣♣(µ) ❢♦r ❛❧❧ µ ∈ P✐♥✈✳ ✭✐✐✐✮ ❙✉♣♣♦s❡ t❤❛t t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ❢♦r ❩ ✭♦r ˜ ❩✮ π✱ t❤❡♥ s✉♣♣(π) = Γ × ❊✳ ❲❤❡♥ Γ ❤❛s ❡♠♣t② ✐♥t❡r✐♦r✱ ✐♥❝❧✉s✐♦♥ Γ × ❊ ⊂ s✉♣♣(µ) ❝❛♥ ❜❡ str✐❝t ✦

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

❋✐❣✉r❡✿ Γ ⊂ R+ × {✵} Γ × ❊ = s✉♣♣(µ)

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

Pr♦♣♦s✐t✐♦♥ ✭❆❝❝❡ss✐❜❧❡ s❡t ❛♥❞ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s✱ ❇▲▼❩✱ ✶✺✮ ✭✐✮ ■❢ Γ = ∅ t❤❡♥ Γ × ❊ ⊂ s✉♣♣(µ) ❢♦r ❛❧❧ µ ∈ P✐♥✈ ❛♥❞ t❤❡r❡ ❡①✐sts µ ∈ P✐♥✈ s✉❝❤ t❤❛t s✉♣♣(µ) = Γ × ❊. ✭✐✐✮ ■❢ Γ ❤❛s ♥♦♥ ❡♠♣t② ✐♥t❡r✐♦r✱ t❤❡♥ Γ × ❊ = s✉♣♣(µ) ❢♦r ❛❧❧ µ ∈ P✐♥✈✳ ✭✐✐✐✮ ❙✉♣♣♦s❡ t❤❛t t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ❢♦r ❩ ✭♦r ˜ ❩✮ π✱ t❤❡♥ s✉♣♣(π) = Γ × ❊✳ ❲❤❡♥ ❤❛s ♥♦♥ ❡♠♣t② ✐♥t❡r✐♦r✱ t❤❡r❡ ♠❛② ❜❡ s❡✈❡r❛❧ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s ✦ ✭❋✉rst❡♥❜❡r❣✬s t②♣❡ ❡①❛♠♣❧❡✮

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

Pr♦♣♦s✐t✐♦♥ ✭❆❝❝❡ss✐❜❧❡ s❡t ❛♥❞ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s✱ ❇▲▼❩✱ ✶✺✮ ✭✐✮ ■❢ Γ = ∅ t❤❡♥ Γ × ❊ ⊂ s✉♣♣(µ) ❢♦r ❛❧❧ µ ∈ P✐♥✈ ❛♥❞ t❤❡r❡ ❡①✐sts µ ∈ P✐♥✈ s✉❝❤ t❤❛t s✉♣♣(µ) = Γ × ❊. ✭✐✐✮ ■❢ Γ ❤❛s ♥♦♥ ❡♠♣t② ✐♥t❡r✐♦r✱ t❤❡♥ Γ × ❊ = s✉♣♣(µ) ❢♦r ❛❧❧ µ ∈ P✐♥✈✳ ✭✐✐✐✮ ❙✉♣♣♦s❡ t❤❛t t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ❢♦r ❩ ✭♦r ˜ ❩✮ π✱ t❤❡♥ s✉♣♣(π) = Γ × ❊✳ ❲❤❡♥ Γ ❤❛s ♥♦♥ ❡♠♣t② ✐♥t❡r✐♦r✱ t❤❡r❡ ♠❛② ❜❡ s❡✈❡r❛❧ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s ✦ ✭❋✉rst❡♥❜❡r❣✬s t②♣❡ ❡①❛♠♣❧❡✮

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

❲❤❛t ✐❢ Γ = ∅ ❄

■❢ Γ = ∅ s♦♠❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s r❡s✉❧t st✐❧❧ ❤♦❧❞ ❜② ❝♦♥s✐❞❡r✐♥❣ ■♥✈❛r✐❛♥t ❈♦♥tr♦❧ ❙❡ts ❋ ✶ ❋ ♠ ✐s s❛✐❞ ❧♦❝❛❧❧② ❛❝❝❡ss✐❜❧❡ ✐♥ ▼ ✐❢ ❢♦r ❛❧❧ ① ▼ ❛♥❞ ❚ ✵ ❖❚ ①

t ①

✵ t ❚ ❛♥❞ ❖❚ ① ② ▼ ①

t ②

❢♦r s♦♠❡ ✵ t ❚ ❤❛✈❡ ♥♦♥ ❡♠♣t② ✐♥t❡r✐♦r

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

❲❤❛t ✐❢ Γ = ∅ ❄

■❢ Γ = ∅ s♦♠❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s r❡s✉❧t st✐❧❧ ❤♦❧❞ ❜② ❝♦♥s✐❞❡r✐♥❣ ■♥✈❛r✐❛♥t ❈♦♥tr♦❧ ❙❡ts

• {❋ ✶, . . . , ❋ ♠} ✐s s❛✐❞ ❧♦❝❛❧❧② ❛❝❝❡ss✐❜❧❡ ✐♥ ▼ ✐❢ ❢♦r ❛❧❧ ① ∈ ▼ ❛♥❞

❚ > ✵ ❖+

❚ (①) = {Ψt(①) : ✵ ≤ t ≤ ❚}

❛♥❞ ❖−

❚ (①) = {② ∈ ▼ : ① ∈ Ψt(②) ❢♦r s♦♠❡ ✵ ≤ t ≤ ❚}

❤❛✈❡ ♥♦♥ ❡♠♣t② ✐♥t❡r✐♦r

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❆❝❝❡ss✐❜❧❡ ❙❡t

Pr♦♣♦s✐t✐♦♥ ✭❇❡♥❛✐♠✱ ❈♦❧♦♥✐✉s ✫ ▲❡tt❛✉✱ ✶✻✮ ❙✉♣♣♦s❡ {❋ ✶, . . . , ❋ ♠} ❧♦❝❛❧❧② ❛❝❝❡ss✐❜❧❡ ✐♥ ▼. ✭✐✮ ❚❤❡r❡ ❛r❡ ✜♥✐t❡❧② ♠❛♥② ✐♥✈❛r✐❛♥t ❝♦♥tr♦❧ s❡ts Γ✶, . . . , Γ❧ ✭✐✐✮ ❚❤❡② ❝♦✐♥❝✐❞❡ ✇✐t❤ ♠✐♥✐♠❛❧ ❝♦♠♣❛❝t ✐♥✈❛r✐❛♥t s❡ts ❢♦r Ψ ❛♥❞ ❤❛✈❡ ♥♦♥ ❡♠♣t② ✐♥t❡r✐♦rs ✭✐✐✐✮ ❋♦r ❡✈❡r② µ ∈ P✐♥✈ s✉♣♣(µ) ⊂ ❧

✐=✶ Γ✐ × ❊.

✭✐✈✮ ❋♦r ❡✈❡r② µ ∈ P❡r❣ t❤❡r❡ ✐s s♦♠❡ ✐ ❢♦r ✇❤✐❝❤ s✉♣♣(µ) = Γ✐ × ❊.

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ❇r❛❝❦❡t ❝♦♥❞✐t✐♦♥s ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐s❝r❡t❡ t✐♠❡ ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡

❯♥✐q✉❡♥❡ss ❛♥❞ ❈♦♥✈❡r❣❡♥❝❡

❚❤❡ ❇r❛❝❦❡t ❝♦♥❞✐t✐♦♥s ❲❡❛❦ ❇r❛❝❦❡t ❛t ① ∈ ▼ : ❚❤❡ ▲✐❡ ❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② {❋ ✐, ✐ = ✶, . . . , ♠} ❤❛s ❢✉❧❧ r❛♥❦ ❛t ① ❙tr♦♥❣ ❇r❛❝❦❡t ❛t ① ∈ ▼ :

• ✵ = {❋ ✐ − ❋ ❥ : ✐, ❥ = ✶, . . . ♠} ●❦+✶ = ●❦ ∪ {[❋ ✐, ❱ ] : ❱ ∈ ●❦}

❤❛s ❢✉❧❧ r❛♥❦ ❛t ① ❢♦r s♦♠❡ ❦.

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ❇r❛❝❦❡t ❝♦♥❞✐t✐♦♥s ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐s❝r❡t❡ t✐♠❡ ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡

❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐s❝r❡t❡ t✐♠❡

❚❤❡♦r❡♠ ✭❇▲▼❩✱ ✶✺✮ ❙✉♣♣♦s❡ ∃① ∈ Γ ❛t ✇❤✐❝❤ t❤❡ ✇❡❛❦ ❜r❛❝❦❡t ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✳ ❚❤❡♥ ˜ ❩ ❛❞♠✐ts ❛ ✉♥✐q✉❡ ✐♥✈❛r✐❛♥t ♣r♦❜❛❜✐❧✐t② ˜ π✱ ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ λ▼×❊ ♦♥ ▼ × ❊❀ ❛♥❞ P(˜ ❩♥ ∈ ·) − ˜ π ≤ ❝ρ♥ ❢♦r s♦♠❡ ❝ > ✶ ❛♥❞ ρ ∈ (✵, ✶). ❍❡r❡ · = t♦t❛❧ ✈❛r✐❛t✐♦♥ ♥♦r♠✳

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ❇r❛❝❦❡t ❝♦♥❞✐t✐♦♥s ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐s❝r❡t❡ t✐♠❡ ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡

❈♦r♦❧❧❛r② ❙✉♣♣♦s❡ ∃① ∈ Γ ❛t ✇❤✐❝❤ t❤❡ ✇❡❛❦ ❜r❛❝❦❡t ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✳ ❚❤❡♥ ❩ ❛❞♠✐ts ❛ ✉♥✐q✉❡ ✐♥✈❛r✐❛♥t ♣r♦❜❛❜✐❧✐t② π✱ ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ λ▼×❊ ♦♥ ▼ × ❊❀ ❋♦❧❧♦✇s ❛❧s♦ ❢r♦♠ r❡s✉❧ts ❜② ✭❇❛❦t❤✐♥ ✫ ❍✉rt❤✱ ✷✵✶✷✮ ✇❤❡♥ ◗ ① ◗ ❲❡❛❦ ❇r❛❝❦❡t ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❧❛✇ ♦❢ ❩t ✦

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ❇r❛❝❦❡t ❝♦♥❞✐t✐♦♥s ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐s❝r❡t❡ t✐♠❡ ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡

❈♦r♦❧❧❛r② ❙✉♣♣♦s❡ ∃① ∈ Γ ❛t ✇❤✐❝❤ t❤❡ ✇❡❛❦ ❜r❛❝❦❡t ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✳ ❚❤❡♥ ❩ ❛❞♠✐ts ❛ ✉♥✐q✉❡ ✐♥✈❛r✐❛♥t ♣r♦❜❛❜✐❧✐t② π✱ ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ λ▼×❊ ♦♥ ▼ × ❊❀

• ❋♦❧❧♦✇s ❛❧s♦ ❢r♦♠ r❡s✉❧ts ❜② ✭❇❛❦t❤✐♥ ✫ ❍✉rt❤✱ ✷✵✶✷✮ ✇❤❡♥

◗(①) = ◗ ❲❡❛❦ ❇r❛❝❦❡t ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❧❛✇ ♦❢ ❩t ✦

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ❇r❛❝❦❡t ❝♦♥❞✐t✐♦♥s ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐s❝r❡t❡ t✐♠❡ ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡

❈♦r♦❧❧❛r② ❙✉♣♣♦s❡ ∃① ∈ Γ ❛t ✇❤✐❝❤ t❤❡ ✇❡❛❦ ❜r❛❝❦❡t ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✳ ❚❤❡♥ ❩ ❛❞♠✐ts ❛ ✉♥✐q✉❡ ✐♥✈❛r✐❛♥t ♣r♦❜❛❜✐❧✐t② π✱ ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ λ▼×❊ ♦♥ ▼ × ❊❀

• ❋♦❧❧♦✇s ❛❧s♦ ❢r♦♠ r❡s✉❧ts ❜② ✭❇❛❦t❤✐♥ ✫ ❍✉rt❤✱ ✷✵✶✷✮ ✇❤❡♥

◗(①) = ◗

• ❲❡❛❦ ❇r❛❝❦❡t ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❧❛✇ ♦❢ (❩t) ✦

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ❇r❛❝❦❡t ❝♦♥❞✐t✐♦♥s ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐s❝r❡t❡ t✐♠❡ ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡

▼ = ❙✶ = R/Z, Φt(①) = (① + t) ♠♦❞ ✶. ✱

❋✐❣✉r❡✿ ❞❡s✐❣♥❡❞ ❜② ❢r❡❡♣✐❝❦

❩t = Φt(①), ˜ ❩♥ ∼ ① + Γ(♥)) ♠♦❞ ✶ ❝♦♥✈❡r❣❡s ✐♥ ❞✐str✐❜✉t✐♦♥

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ❇r❛❝❦❡t ❝♦♥❞✐t✐♦♥s ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐s❝r❡t❡ t✐♠❡ ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡

❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡

❚❤❡♦r❡♠ ✭❇▲▼❩✱ ✶✺✮ ❙✉♣♣♦s❡ ∃♣ ∈ Γ ❛t ✇❤✐❝❤ t❤❡ str♦♥❣ ❜r❛❝❦❡t ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✳ ▲❡t π ❜❡ t❤❡ ✐♥✈❛r✐❛♥t ♣r♦❜❛❜✐❧✐t② ♦❢ ❩. ❚❤❡♥ P(❩t ∈ ·) − π ≤ ❈❡−κt ❢♦r s♦♠❡ ❈ > ✶ ❛♥❞ κ > ✵

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ❇r❛❝❦❡t ❝♦♥❞✐t✐♦♥s ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐s❝r❡t❡ t✐♠❡ ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡

❲❤❛t ✐❢ Γ = ∅ ❄

Pr♦♣♦s✐t✐♦♥ ✭❇❡♥❛✐♠✱ ❈♦❧♦♥✐✉s ✫ ▲❡tt❛✉✱ ✶✻✮ ❙✉♣♣♦s❡ t❤❡ ✇❡❛❦ ❜r❛❝❦❡t ❝♦♥❞✐t✐♦♥ ❤♦❧❞s ❢♦r ❛❧❧ ① ∈ ▼. ✭✐✮ ❚❤❡r❡ ❛r❡ ✜♥✐t❡❧② ♠❛♥② ❝♦♠♣❛❝t ✐♥✈❛r✐❛♥t ❝♦♥tr♦❧ s❡ts Γ✶, . . . , Γ❧ ✭✐✐✮ ˜ P❡r❣ = {˜ π✶, . . . , ˜ π❧} ❛♥❞ s✉♣♣(˜ π✐) = Γ✐ × ❊. ✭✐✐✐✮ ❖♥ Γ✐ t❤❡ ❧❛✇ ♦❢ ˜ ❩ ❝♦♥✈❡r❣❡s ❡①♣♦♥❡♥t✐❛❧❧② ❢❛st t♦✇❛r❞ ˜ π✐.

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ❇r❛❝❦❡t ❝♦♥❞✐t✐♦♥s ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐s❝r❡t❡ t✐♠❡ ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡

❙✐♠✐❧❛r❧② Pr♦♣♦s✐t✐♦♥ ✭❇❡♥❛✐♠✱ ❈♦❧♦♥✐✉s ✫ ▲❡tt❛✉✱ ✶✻✮ ❙✉♣♣♦s❡ t❤❡ str♦♥❣ ❜r❛❝❦❡t ❝♦♥❞✐t✐♦♥ ❤♦❧❞s ❢♦r ❛❧❧ ① ▼ ✭✐✮ ❚❤❡r❡ ❛r❡ ✜♥✐t❡❧② ♠❛♥② ❝♦♠♣❛❝t ✐♥✈❛r✐❛♥t ❝♦♥tr♦❧ s❡ts

✶ ❧

✭✐✐✮

❡r❣ ✶ ❧

❛♥❞ s✉♣♣

✐ ✐

❊ ✭✐✐✐✮ ❖♥

✐ t❤❡ ❧❛✇ ♦❢ ❩ ❝♦♥✈❡r❣❡s ❡①♣♦♥❡♥t✐❛❧❧② ❢❛st

t♦✇❛r❞

✐

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ❇r❛❝❦❡t ❝♦♥❞✐t✐♦♥s ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐s❝r❡t❡ t✐♠❡ ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡

❙✐♠✐❧❛r❧② Pr♦♣♦s✐t✐♦♥ ✭❇❡♥❛✐♠✱ ❈♦❧♦♥✐✉s ✫ ▲❡tt❛✉✱ ✶✻✮ ❙✉♣♣♦s❡ t❤❡ str♦♥❣ ❜r❛❝❦❡t ❝♦♥❞✐t✐♦♥ ❤♦❧❞s ❢♦r ❛❧❧ ① ∈ ▼. ✭✐✮ ❚❤❡r❡ ❛r❡ ✜♥✐t❡❧② ♠❛♥② ❝♦♠♣❛❝t ✐♥✈❛r✐❛♥t ❝♦♥tr♦❧ s❡ts Γ✶, . . . , Γ❧ ✭✐✐✮ P❡r❣ = {π✶, . . . , π❧} ❛♥❞ s✉♣♣(π✐) = Γ✐ × ❊. ✭✐✐✐✮ ❖♥ Γ✐ t❤❡ ❧❛✇ ♦❢ ❩ ❝♦♥✈❡r❣❡s ❡①♣♦♥❡♥t✐❛❧❧② ❢❛st t♦✇❛r❞ π✐.

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s

❆ s✉♣♣♦rt ❚❤❡♦r❡♠ ■♥✈❛r✐❛♥t ▼❡❛s✉r❡s ❛♥❞ ❆❝❝❡ss✐❜❧❡ ❙❡t ❯♥✐q✉❡♥❡ss ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ❇r❛❝❦❡t ❝♦♥❞✐t✐♦♥s ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐s❝r❡t❡ t✐♠❡ ❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡

❚✐♠❡ t♦ ❙✇✐t❝❤ t♦ ▲✉♥❝❤✳✳✳

▼✐❝❤❡❧ ❇❡♥❛ï♠ ✭◆❡✉❝❤ât❡❧✮ ❖♥ P✐❡❝❡✇✐s❡ ❉❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ Pr♦❝❡ss❡s