Inferência em alguns modelos de processos estocasticamente perturbados

(1)

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s

■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ✲ ❉♦✉t♦r❛❞♦ ❡♠ ❊st❛tíst✐❝❛

❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦

■♥❢❡rê♥❝✐❛ ❡♠ ❆❧❣✉♥s ▼♦❞❡❧♦s ❞❡ Pr♦❝❡ss♦s

❊st♦❝❛st✐❝❛♠❡♥t❡ P❡rt✉r❜❛❞♦s

❆❧✉♥♦✿ ❲❡❝s❧❡② ❖t❡r♦ Pr❛t❡s

✶

❇❡❧♦ ❍♦r✐③♦♥t❡✱ ❏✉♥❤♦ ❞❡ ✷✵✶✻

✶_{❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s}

(2)

■♥❢❡rê♥❝✐❛ ❡♠ ❆❧❣✉♥s ▼♦❞❡❧♦s ❞❡ Pr♦❝❡ss♦s

❊st♦❝❛st✐❝❛♠❡♥t❡ P❡rt✉r❜❛❞♦s

❲❡❝s❧❡② ❖t❡r♦ Pr❛t❡s

❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛ ❞❛

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s

Pr♦❣r❛♠❛✿ Pós✲●r❛❞✉❛çã♦ ❡♠ ❊st❛tíst✐❝❛

❖r✐❡♥t❛❞♦r❛✿ Pr♦❢

❛

_{✳ ❉r}

❛

_{❉❡♥✐s❡ ❉✉❛rt❡}

✷

❈♦✲❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❙♦❦♦❧ ◆❞r❡❝❛

2

❇❡❧♦ ❍♦r✐③♦♥t❡✱ ▼❛✐♦ ❞❡ ✷✵✶✻

✷_{❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s}

(3)

❊st❛ t❡s❡ ❢♦✐ r❡❛❧✐③❛❞❛ ♥♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛ ❞❛

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✱ s♦❜ ❛ ♦r✐❡♥t❛çã♦ ❞❛ Pr♦❢❡ss♦r❛ ❉♦✉t♦r❛ ❉❡♥✐s❡ ❉✉❛rt❡

❡ ♣❡❧❛ ❝♦✲♦r✐❡♥t❛çã♦ ❞♦ Pr♦❢❡ss♦r ❉♦✉t♦r ❙♦❦♦❧ ◆❞r❡❝❛ ❡ ❝♦♥t♦✉ ♥♦s ♣r✐♠❡✐r♦s ✸ ❛♥♦s ❝♦♠ ♦

✜♥❛♥❝✐❛♠❡♥t♦ ❞❛ ❇♦❧s❛ ❘❊❯◆■✳

(4)

❆❣r❛❞❡❝✐♠❡♥t♦s

❆ ♣r❡♦❝✉♣❛çã♦✱ ♦ ♠❡❞♦ ❡ ❛s ✐♥❝❡rt❡③❛s sã♦ ❛ss✉♥t♦s ❝♦♠♣❧❡①♦s ❡ ❡stã♦ ❛ss♦❝✐❛❞♦s ❝♦♠ t❛♥t❛s ❞❡

♥♦ss❛s ❛♥s✐❡❞❛❞❡s ❡ ♣r♦❜❧❡♠❛s q✉❡ é ✐♠♣♦ssí✈❡❧ s❡♣❛rá✲❧♦s ❝♦♠♣❧❡t❛♠❡♥t❡✳

❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s q✉❡ ♠❡ ❞❡✉ ❢♦rç❛✱ ❝♦r❛❣❡♠ ❡ ♣❡rs✐stê♥❝✐❛ ♣❛r❛ ❝❛♥s❛r ❛s

❛❞✈❡rs✐❞❛❞❡s q✉❡ ❡♥❝♦♥tr❡✐ ♣❡❧♦ ❝❛♠✐♥❤♦✳

❆❣r❛❞❡ç♦ ❛ Pr♦❢

❛

_{❡ ❛♠✐❣❛ ❉❡♥✐s❡ ❉✉❛rt❡ ♣❡❧❛ s✉❛ ♦r✐❡♥t❛çã♦✳ ❆ s✉❛ ❣❡♥❡r♦s✐❞❛❞❡ ❡ ❡♥t✉s✐❛s♠♦}

❢♦r❛♠ ❣r❛♥❞❡s ✐♥❝❡♥t✐✈♦s ♥❡st❡s ❛♥♦s✳

❆❣r❛❞❡ç♦ ❛♦ Pr♦❢

♦

_{❡ ❈♦✲♦r✐❡♥t❛❞♦r ❙♦❦♦❧ ◆❞r❡❝❛ ♣❡❧♦ s❡✉ ❛♣♦✐♦ ❡ s✉❛s ✐❞é✐❛s ♣❛r❛ ♦}

❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❙♦✉ ❣r❛t♦ ❛♦s ❝♦❧❡❣❛s P❛✉❧♦ ❈❡rq✉❡✐r❛✱ ❘♦❞r✐❣♦ ❈✐tt♦♥✱ ▲✉✐s

●✉st❛✈♦✱ ❘♦❞♦❧❢♦ ▲♦r❡♥③✉tt✐ ❡ ❙✐❧✈✐♦ ❙♦✉③❛✱ q✉❡ ♣♦ss♦ ❝❤❛♠á✲❧♦s ❞❡ ❛♠✐❣♦s ❡ ❛♣r❡♥❞✐ ❞❡ ❝❛❞❛

✉♠ ❞❡❧❡s ❧✐çõ❡s ❞❡ r❡s♣♦♥s❛❜✐❧✐❞❛❞❡ ❡ ❝♦♠♣❛♥❤❡r✐s♠♦✱ q✉❡ ♠✉✐t❛s ✈❡③❡s ♠❡ ❛❥✉❞❛r❛♠ ❝♦♠ ❡st❛

t❡s❡ ♥❛ ♣❛rt❡ ❝♦♠♣✉t❛❝✐♦♥❛❧✳

❆♦s ♠❡✉s Pr♦❢❡ss♦r❡s q✉❡ ❞♦s ❝♦♥❤❡❝✐♠❡♥t♦s q✉❡ ♠❡ ❢♦r❛♠ ♣❛ss❛❞♦s✱ ♦ ♠❛✐s ✐♠♣♦rt❛♥t❡ é q✉❡

❢❛rí❛♠♦s ♠✉✐t❛s ❝♦✐s❛s s❡ ♥ã♦ ❛s ❥✉❧❣áss❡♠✲♠♦s ♠✉✐t❛s ✈❡③❡s ✐♠♣♦ssí✈❡✐s✳ ❆♦s ❛♠✐❣♦s q✉❡

❝♦♥q✉✐st❡✐ ♥❡ss❛ ❥♦r♥❛❞❛ ❡ q✉❡ ❢♦r❛♠ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♠❛✐s ❡ss❛ ❡t❛♣❛✳

❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛ ♠✐♥❤❛ ❡s♣♦s❛ ❈❛r♦❧✐♥❛ ▼✉❧❡❦ Pr❛t❡s ♣❡❧♦ ❛♣♦✐♦✱ ♣❛❝✐ê♥❝✐❛ ❡ ♣♦r s❡♠♣r❡

❡st❛r ❛♦ ♠❡✉ ❧❛❞♦ ♥♦s ♠♦♠❡♥t♦s ❞❡ ❞✐✜❝✉❧❞❛❞❡s✳

❊ ✉♠ ❣r❛♥❞❡ ❛❣r❛❞❡❝✐♠❡♥t♦ à ♠✐♥❤❛ ❢❛♠í❧✐❛ q✉❡ ♠❡ ❞❡✉ ❛♠♦r✱ ❛♣♦✐♦ ❡ ♠❡ ❡♥❝♦r❛❥❛r❛♠ ❛

r❡❛❧✐③❛r ♠❛✐s ✉♠ s♦♥❤♦✳

(5)

❘❡s✉♠♦

❊♠ ✉♠ ♠♦❞❡❧♦ ❞❡ ♣r♦❝❡ss♦s ❡st♦❝❛st✐❝❛♠❡♥t❡ ♣❡rt✉r❜❛❞♦s ❛s ♦❜s❡r✈❛çõ❡s ❞♦ ♣r♦❝❡ss♦ ♦r✐❣✐♥❛❧ ♣♦❞❡♠ s♦❢r❡r ♣❡r✲ t✉r❜❛çõ❡s✱ ❡♠ ❝❛❞❛ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦✱ ♣♦r ✉♠ r✉í❞♦ ❛❧❡❛t♦r✐♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♦ ♣r♦❝❡ss♦ ♦❜s❡r✈❛❞♦ ♣♦❞❡ ♥ã♦ s❡r ♠❛✐s ✉♠❛ ❛♠♦str❛ ❞♦ ♣r♦❝❡ss♦ ♦r✐❣✐♥❛❧✳

◆❡st❛ t❡s❡ ❛♣r❡s❡♥t❛♠♦s ♠❡t♦❞♦❧♦❣✐❛s ♣❛r❛ ❢❛③❡r ❡st✐♠❛❝ã♦ ❞♦s ♣❛râ♠❡tr♦s ❞❡ ❛❧❣✉♥s ♠♦❞❡❧♦s ❡st♦❝❛st✐❝❛♠❡♥t❡ ♣❡rt✉r❜❛❞♦s t❡♥❞♦ ❝♦♠♦ ❜❛s❡ ♦s ♠♦❞❡❧♦s ♣r♦♣♦st♦s ♣♦r ❬✼❪ ❡ ❬✶✷❪✳ ❆ss✉♠✐♠♦s q✉❡ ♦ ♣r♦❝❡ss♦ ♦r✐❣✐♥❛❧✱ ♦❝✉❧t♦✱ é ✉♠❛ ❝❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ❞❡ ❛❧❝❛♥❝❡ ✈❛r✐á✈❡❧✳ ❊ss❛ ❝❧❛ss❡ ❞❡ ♣r♦❝❡ss♦s ♣❡r♠✐t❡ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s ♣♦r s❡r ♣❛r❝✐♠♦♥✐♦s❛ ❡♠ r❡❧❛❝ã♦ ❛♦ ♥ú♠❡r♦ ❞❡ ♣❛râ♠❡tr♦s ❡ t❛♠❜é♠ ❜❛st❛♥t❡ ♠❛❧❡á✈❡❧✱ ❡♥❣❧♦❜❛♥❞♦ ❛ ❝❧❛ss❡ ❞❛s ❝❛❞❡✐❛s ❞❡ ▼❛r❦♦✈ ❞❡ ♦r❞❡♠ ✜①❛✳

Pr♦♣♦♠♦s ✉♠❛ ❛❞❛♣t❛çã♦ ♥♦ ❛❧❣♦r✐t♠♦ ❞❡ ❇❛✉♠✲❲❡❧❝❤ ❡ ✉♠ ❡st✐♠❛❞♦r ❇■❈ ❜♦♦tstr❛♣ ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞♦s ♠♦❞❡❧♦s ❛♥❛❧✐s❛❞♦s✱ ❝✉❥❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❢♦✐ ❞❡♠♦♥str❛❞❛✱ ❡ ❛tr❛✈és ❞❡ s✐♠✉❧❛çõ❡s✱ ♠♦str❛♠♦s q✉❡ ❛ ♠❡t♦❞♦❧♦❣✐❛ ♣r♦♣♦st❛ é ❝❛♣❛③ ❞❡ r❡❝✉♣❡r❛r ♠✉✐t♦ ❜❡♠ ❛ ✈❡r❞❛❞❡✐r❛ ár✈♦r❡ ❞❡ ❝♦♥t❡①t♦ss ❞❡ ✉♠❛ ❝❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ❝♦♠ ❛❧❝❛♥❞❡ ✈❛r✐á✈❡❧ ❡st♦❝❛st✐❝❛♠❡♥t❡ ♣❡rt✉r❜❛❞❛✱ ❛ss✐♠ ❝♦♠♦ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ❛ss♦❝✐❛❞❛s ❛ ❡ss❛ ár✈♦r❡✱ ❞❡♥tr♦ ❞❡ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ ♥í✈❡✐s ❞❡ ♣❡rt✉r❜❛çã♦✳ ❚❛♠❜é♠ ❝♦♥s❡❣✉✐♠♦s r❡❝✉♣❡r❛r ♦ ❣r❛✉ ❞❡ ♣❡rt✉r❜❛çã♦ q✉❛❧q✉❡r q✉❡ t❡♥❤❛ s✐❞♦✳

Pr♦♣♦♠♦s ✉♠❛ ♠♦❞✐✜❝❛çã♦ ♥♦ ❛❧❣♦r✐t♠♦ ❞❡ ❱✐t❡r❜✐ ♣❛r❛ ❡♥❝♦♥tr❛r ❛ s❡q✉ê♥❝✐❛ ♦❝✉❧t❛ ♠❛✐s ♣r♦✈á✈❡❧ ❞❡ ✉♠❛ ❝❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ❝♦♠ ❛❧❝❛♥❞❡ ✈❛r✐á✈❡❧ ❡st♦❝❛st✐❝❛♠❡♥t❡ ♣❡rt✉r❜❛❞❛✳

❆♣r❡s❡♥t❛♠♦s ✉♠ ❝r✐tér✐♦ ❞❡ s❡❧❡çã♦ ❞❡ ♠♦❞❡❧♦s ♣❛r❛ ✐❞❡♥t✐✜❝❛r ♦ ♠♦❞❡❧♦ ♠❛✐s ❛❞❡q✉❛❞♦✱ ❞❛❞❛ ✉♠❛ ❛♠♦str❛ ♦❜s❡r✈❛❞❛✱ ❞❡♥tr❡ ♦s ❛♥❛❧✐s❛❞♦s ♥❡ss❛ t❡s❡✳

❆♣❧✐❝❛♠♦s ❛ ♠❡t♦❞♦❧♦❣✐❛ ♣r♦♣♦st❛ ❛ ✉♠ ❜❛♥❝♦ ❞❡ ❞❛❞♦s ❞❡ r❡❣✐str♦s ❞❡ ❛t✐✈✐❞❛❞❡ ❞❡ ♥❡✉rô♥✐♦s ❞❡ ✉♠ ❣r✉♣♦ ❞❡ ❝♦r✉❥❛s ❡♠ ✉♠ ❡①♣❡r✐♠❡♥t♦ ❝♦♥tr♦❧❛❞♦ ❡♠ ❧❛❜♦r❛tór✐♦✳ ❖s ❞❛❞♦s ❢♦r❛♠ ❝♦❞✐✜❝❛❞♦s ❡♠ ✷ ❡st❛❞♦s✱ ❞✐s♣❛r♦ ❡ r❡♣♦✉s♦✱ ❡ ♦ ♥♦ss♦ ♦❜❥❡t✐✈♦ é ✐❞❡♥t✐✜❝❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ❞✐❢❡r❡♥t❡s ♣❛❞rõ❡s ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦s ❞❡ss❛ ❛t✐✈✐❞❛❞❡ ♥❡✉r♦♥❛❧✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❧❡✐ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❡st✐♠❛❞❛ ♣❛r❛ ♦ ♣r♦❝❡ss♦✱ ❡♠ r❡❧❛çã♦ ❛♦ t✐♣♦ ❞❡ ❡stí♠✉❧♦ ✈✐s✉❛❧ ❛ q✉❡ ♦ ❣r✉♣♦ ❞❡ ❝♦r✉❥❛s ❢♦✐ s✉❜♠❡t✐❞♦✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ Pr♦❝❡ss♦s ♣❡rt✉r❜❛❞♦s✱ ❈❛❞❡✐❛s ❞❡ ▼❛r❦♦✈ ❞❡ ❆❧❝❛♥❝❡ ❱❛r✐á✈❡❧✱ ár✈♦r❡ ❞❡ ❝♦♥t❡①t♦s✱ ❆❧❣♦r✐t♠♦ ❇■❈✱ ❇♦♦tstr❛♣✱ ❆❧❣♦r✐t♠♦ ❞❡ ❇❛✉♠✲❲❡❧❝❤✳

(6)

❆❜str❛❝t

■♥ ❛ ♠♦❞❡❧ ♦❢ st♦❝❤❛st✐❝❛❧❧② ❞✐st✉r❜❡❞ ♣r♦❝❡ss❡s ❡❛❝❤ ♦❜s❡r✈❛t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣r♦❝❡ss ❝❛♥ ❜❡ ❞✐st✉r❜❡❞ ❛t ❛♥② ♠♦♠❡♥t ♦❢ t✐♠❡ ❜② ❛ r❛♥❞♦♠ ♥♦✐s❡✳ ❚❤✉s t❤❡ ♦❜s❡r✈❡❞ ♣r♦❝❡ss ❝♦✉❧❞ ♥♦t ❜❡ ❛ s❛♠♣❧❡ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣r♦❝❡ss✳ ■♥ t❤✐s t❤❡s✐s ✇❡ ♣r❡s❡♥t ❛ ♠❡t❤♦❞♦❧♦❣② ✐♥ ♦r❞❡r t♦ ❡st✐♠❛t❡ t❤❡ ♣❛r❛♠❡t❡rs ♦❢ s♦♠❡ ❞✐st✉r❜❡❞ st♦❝❤❛st✐❝❛❧❧② ♠♦❞❡❧s ❜❛s❡❞ ♦♥ t❤❡ ♠♦❞❡❧s ♣r♦♣♦s❡❞ ❜② ❬✼❪ ❛♥❞ ❬✶✷❪✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ♦r✐❣✐♥❛❧ ❤✐❞❞❡♥ ♣r♦❝❡ss ✐s ❛ ✈❛r✐❛❜❧❡ ❧❡♥❣t❤ ▼❛r❦♦✈ ❝❤❛✐♥✳ ❚❤✐s ❝❧❛ss ♣r♦❝❡ss❡s ❛❧❧♦✇s ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s s✐♥❝❡ ✐t ✐s ♣❛rs✐♠♦♥✐♦✉s ✐♥ r❡❧❛t✐♦♥ t♦ t❤❡ ♥✉♠❜❡r ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ❛❧s♦ q✉✐t❡ ♠❛❧❧❡❛❜❧❡✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ❝❧❛ss ♦❢ ✜①❡❞✲♦r❞❡r ▼❛r❦♦✈ ❝❤❛✐♥s✳ ❲❡ ♣r♦♣♦s❡ ❛♥ ❛❞❛♣t❛t✐♦♥ ✐♥ t❤❡ ❇❛✉♠✲❲❡❧❝❤ ❛❧❣♦r✐t❤♠ ❛♥❞ ❛ ❜♦♦tstr❛♣ ❇❛②❡s✐❛♥ ■♥❢♦r♠❛t✐♦♥ ❈r✐t❡r✐♦♥ ❛s ❛ ✇❛② t♦ ❡st✐♠❛t❡ t❤❡ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ♠♦❞❡❧s ❛♥❛❧②③❡❞✱ ✇❤♦s❡ ❝♦♥✈❡r❣❡♥❝❡ ✇❛s s❤♦✇♥✱ ❛♥❞ s❤♦✇ t❤r♦✉❣❤ s✐♠✉❧❛t✐♦♥s t❤❛t t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞♦❧♦❣② ✐s ❛❜❧❡ t♦ r❡❝♦✈❡r ✈❡r② ✇❡❧❧ t❤❡ r❡❛❧ ❝♦♥t❡①t tr❡❡ ♦❢ ❛ st♦❝❤❛st✐❝❛❧❧② ❞✐st✉r❜❡❞ ✈❛r✐❛❜❧❡ ❧❡♥❣t❤ ▼❛r❦♦✈ ❝❤❛✐♥ ❛s ✇❡❧❧ ❛s t❤❡ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ tr❡❡✱ ✇✐t❤✐♥ ❛ r❡❛s♦♥❛❜❧❡ r❛♥❣❡ ♦❢ ❞✐st✉r❜❛♥❝❡ ❧❡✈❡❧s✳ ❲❡ ❛❧s♦ ❛❜❧❡ t♦ r❡❝♦✈❡r t❤❡ ❞❡❣r❡❡ ♦❢ ❞✐st✉r❜❛♥❝❡ ✇❤❛t❡✈❡r ✐t ❤❛s ❜❡❡♥✳

❲❡ ♣r♦♣♦s❡ ❛ ♠♦❞✐✜❝❛t✐♦♥ t♦ t❤❡ ❱✐t❡r❜✐ ❛❧❣♦r✐t❤♠ t♦ ✜♥❞ t❤❡ ♠♦st ❛♣♣r♦♣r✐❛t❡ ❤✐❞❞❡♥ s❡q✉❡♥❝❡ ♦❢ ❛ st♦❝❤❛s✲ t✐❝❛❧❧② ❞✐st✉r❜❡❞ ✈❛r✐❛❜❧❡ ❧❡♥❣t❤ ▼❛r❦♦✈ ❝❤❛✐♥✳

❲❡ ♣r❡s❡♥t ❛ ♠♦❞❡❧ s❡❧❡❝t✐♦♥ ❝r✐t❡r✐♦♥ t♦ ✐❞❡♥t✐❢② t❤❡ ♠♦st ❛♣♣r♦♣r✐❛t❡ ♠♦❞❡❧ ❣✐✈❡♥ t❤❡ ♦❜s❡r✈❡❞ s❛♠♣❧❡ ❛♠♦♥❣ t❤♦s❡ ❛♥❛❧②③❡❞ ✐♥ t❤✐s t❤❡s✐s✳

❲❡ ❛♣♣❧② t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞♦❧♦❣② t♦ ❛ ❞❛t❛❜❛s❡ ♦❢ ♥❡✉r♦♥s ❛❝t✐✈✐t② r❡❝♦r❞s ♦❢ ❛ ❣r♦✉♣ ♦❢ ♦✇❧s ✐♥ ❛ ❝♦♥tr♦❧❧❡❞ ❧❛❜♦r❛t♦r② ❡①♣❡r✐♠❡♥t✳ ❉❛t❛ ✇❡r❡ ❝♦❞❡❞ ✐♥ t✇♦ st❛t❡s✱ s♣✐❦❡ ❛♥❞ r❡st✳ ❖✉r ❣♦❛❧ ✐s t♦ ✐❞❡♥t✐❢② t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❞✐✛❡r❡♥t ♣❛tt❡r♥s ♦❢ ❜❡❤❛✈✐♦r t❤❛t ♥❡✉r♦♥❛❧ ❛❝t✐✈✐t② ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❡st✐♠❛t❡❞ ♣r♦❜❛❜✐❧✐t② ❢♦r t❤❡ ♣r♦❝❡ss ✐♥ r❡❧❛t✐♦♥ t♦ t❤❡ t②♣❡ ♦❢ ✈✐s✉❛❧ st✐♠✉❧✉s t❤❛t t❤❡ ❣r♦✉♣ ♦❢ ♦✇❧s ✇❛s s✉❜♠✐tt❡❞✳

❑❡②✇♦r❞s✿ ❉✐st✉r❜❡❞ Pr♦❝❡ss✱ ❱❛r✐❛❜❧❡ ▲❡♥❣t❤ ▼❡♠♦r② ❈❤❛✐♥s✱ ❈♦♥t❡①t tr❡❡✱ ❇♦♦♦tstr❛♣✱ ❇✐❝ ❛❧❣♦r✐t❤♠✱ ❇❛✉♠✲ ❲❡❧❝❤ ❛❧❣♦r✐t❤♠✳

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❮♥❞✐❝❡

✶ ■♥tr♦❞✉çã♦ ✸

✷ ◆♦t❛çõ❡s ❡ ❉❡✜♥✐çõ❡s ✺

✷✳✶ ❈❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ❖❝✉❧t❛ ❝♦♠ ❆❧❝❛♥❝❡ ❱❛r✐á✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✷ ❆❧❣♦r✐t♠♦ ❞❡ ❇❛✉♠✲❲❡❧❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸ ❘❡✈✐sã♦ ❞❡ ❆❧❣✉♥s ▼♦❞❡❧♦s ❞❡ P❡rt✉r❜❛çã♦ ❊st♦❝ást✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✸ ▼♦❞❡❧♦s ❞❡ P❡rt✉r❜❛çã♦ Pr♦♣♦st♦s ✶✵

✸✳✶ ▼♦❞❡❧♦ ❞❡ P❡rt✉r❜❛çã♦ ❚✐♣♦ ❙♦♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✸✳✷ ▼♦❞❡❧♦ ❞❡ P❡rt✉r❜❛çã♦ ❚✐♣♦ Pr♦❞✉t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✹ ▼♦❞❡❧♦ ❞❡ P❡rt✉r❜❛çã♦ ❚✐♣♦ ▼✐st✉r❛ ✶✸

✹✳✶ ❊st✐♠❛çã♦ ✈✐❛ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ P❡r✜❧❛❞❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❚▼❈▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✺ ❆❧❣♦r✐t♠♦s ❡ ❊st✐♠❛❞♦r❡s Pr♦♣♦st♦s ♣❛r❛ ♦s ▼♦❞❡❧♦s ❚❙❈▼ ❡ ❚P❈▼ ✶✻ ✺✳✶ Pr♦❝❡❞✐♠❡♥t♦ ❞❡ ❊st✐♠❛çã♦ ❞❛ ár✈♦r❡ ❞❡ ❝♦♥t❡①t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✺✳✷ ❆❧❣♦r✐t♠♦ ❱✐t❡r❜✐ ▼♦❞✐✜❝❛❞♦ P❛r❛ ♦s ▼♦❞❡❧♦s Pr♦♣♦st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✻ ❙✐♠✉❧❛çã♦ ❡ ❆♥á❧✐s❡ ❞❡ ❙❡♥s✐❜✐❧✐❞❛❞❡ ❞♦ ❘✉í❞♦ ❆❧❡❛tór✐♦ ✷✸ ✻✳✶ Pr✐♠❡✐r♦ ❈❡♥ár✐♦✿ ▼♦❞❡❧♦ ❚❙❈▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✻✳✷ Pr✐♠❡✐r♦ ❈❡♥ár✐♦✿ ▼♦❞❡❧♦ ❚P❈▼✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✻✳✸ ❙❡❣✉♥❞♦ ❈❡♥ár✐♦✿ ▼♦❞❡❧♦ ❚❙❈▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✻✳✹ ❙❡❣✉♥❞♦ ❈❡♥ár✐♦✿ ▼♦❞❡❧♦ ❚P❈▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✼ ❈r✐tér✐♦ ❞❡ ❙❡❧❡çã♦ ❞❡ ▼♦❞❡❧♦s✿ ❚❙❈▼ ♦✉ ❚P❈▼ ✸✹ ✼✳✶ ❙✐♠✉❧❛çã♦ ✶✿ ▼♦❞❡❧♦ ❚❙❈▼ ❝♦♠♦ ✈❡r❞❛❞❡✐r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✼✳✷ ❙✐♠✉❧❛çã♦ ✷✿ ▼♦❞❡❧♦ ❚P❈▼ ❝♦♠♦ ✈❡r❞❛❞❡✐r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✽ ❆♣❧✐❝❛çã♦ ✸✼

✾ ❈♦♥❝❧✉sã♦ ✹✶

✶✵ ▲✐♠✐t❛çõ❡s ❞❛ P❡sq✉✐s❛ ❡ ❙✉❣❡stõ❡s ♣❛r❛ ❚r❛❜❛❧❤♦s ❋✉t✉r♦s ✹✷

✶✶ ❆♣ê♥❞✐❝❡ ✹✸

✶✶✳✶ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ P❡r✜❧❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

(8)

❮◆❉■❈❊

▲✐st❛ ❞❡ ❙í♠❜♦❧♦s

X ✲ ❱▲▼❈ ❖❝✉❧t❛ Y ✲ ❱▲▼❈ ❖❝✉❧t❛ Z ✲ Pr♦❝❡ss♦ P❡rt✉r❜❛❞♦

X∗✲ ❈❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ❞❡ ❖r❞❡♠_k

Z∗ ✲ Pr♦❝❡ss♦ P❡rt✉r❜❛❞♦ ❚r❛♥s❢♦r♠❛❞♦

ξ✲ ❙❡q✉ê♥❝✐❛ ❞❡ ❱❛r✐á✈❡✐s ❆❧❡❛tór✐❛s

ǫ✲ P❛râ♠❡tr♦ ❞❡ P❡rt✉r❜❛çã♦ at ✲ ❊st❛❞♦ ❖❝✉❧t♦ ❞❡X ♥♦ ❚❡♠♣♦t

ct✲ ❊st❛❞♦ ❖❝✉❧t♦ ❞❡Y ♥♦ ❚❡♠♣♦t

bt✲ ❱❛❧♦r ❞❛ ❱❛r✐á✈❡❧ξ♥♦ ❚❡♠♣♦t

zt ✲ ❙í♠❜♦❧♦ ❖❜s❡r✈❛❞♦ ❞❡Z ♥♦ ❚❡♠♣♦t

ω ✲ ❈♦♥t❡①t♦ ν ✲ ❈♦♥t❡①t♦ υ ✲ ❈♦♥t❡①t♦

T ✲ ➪r✈♦r❡ ❞❡ ❝♦♥t❡①t♦s

Tk ✲ ➪r✈♦r❡ ❞❡ ❝♦♥t❡①t♦sk−f ull

Tk ✲ ➪r✈♦r❡ ❞❡ ❝♦♥t❡①t♦s ❚r✉♥❝❛❞❛ ♥❛ ❖r❞❡♠k

ˆ

T ✲ ➪r✈♦r❡ ❞❡ ❝♦♥t❡①t♦s ❊st✐♠❛❞❛

ˆ

Tk ✲ ➪r✈♦r❡ ❞❡ ❝♦♥t❡①t♦sk−f ull❊st✐♠❛❞❛

ˆ

T_k ✲ ➪r✈♦r❡ ❞❡ ❝♦♥t❡①t♦s ❚r✉♥❝❛❞❛ ♥❛ ❖r❞❡♠k❊st✐♠❛❞❛ X ✲ ❆♠♦str❛ ❞❡X

X∗ _{✲ ❆♠♦str❛ ❞❡}_X∗

Z ✲ ❆♠♦str❛ ❞❡Z

Z∗ ✲ ❆♠♦str❛ ❞♦ Pr♦❝❡ss♦ P❡rt✉r❜❛❞♦ ❚r❛♥s❢♦r♠❛❞♦Z∗

ˆ

X ✲ ❆♠♦str❛ ❇♦♦tstr❛♣ ❞❡X∗

A✲ ▼❛tr✐③ ❞❡ ❚r❛♥s✐çã♦ ❞❡ X

A∗ ✲ ▼❛tr✐③ ❞❡ ❚r❛♥s✐çã♦ ❞❡X∗

ˆ

A∗ ✲ ▼❛tr✐③ ❞❡ ❚r❛♥s✐çã♦ ❊st✐♠❛❞❛ ❞❡X∗

B ✲ ❉✐str✐❜✉✐çã♦ ❞❡ ❊♠✐ssã♦ ❡♥tr❡X ❡Z

B∗✲ ❉✐str✐❜✉✐çã♦ ❞❡ ❊♠✐ssã♦ ❡♥tr❡X∗❡_Z∗

ˆ

B∗✲ ❉✐str✐❜✉✐çã♦ ❞❡ ❊♠✐ssã♦ ❊st✐♠❛❞❛ ❡♥tr❡ X∗ ❡_Z∗

p(a|ω)✲ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❚r❛♥s✐çã♦ ❞❡ X

ˆ

p(a|ω)✲ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❚r❛♥s✐çã♦ ❊st✐♠❛❞❛ ❞❡X

p∗₍_ω|ν₎_{✲ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❚r❛♥s✐çã♦ ❞❡}_X∗

ˆ

p∗₍_ω|ν₎_{✲ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❚r❛♥s✐çã♦ ❊st✐♠❛❞❛ ❞❡}_X∗

p(a|ν)✲ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❚r❛♥s✐çã♦ ❞❡Y

bω(zt)✲ ❊❧❡♠❡♥t♦ ❞❡B

bω(zt)∗ ✲ ❊❧❡♠❡♥t♦ ❞❡B∗

ˆ

bω(zt)∗ ✲ ❊❧❡♠❡♥t♦ ❊st✐♠❛❞♦ ❞❡Bˆ∗

π ✲ ❉✐str✐❜✉✐çã♦ ■♥✐❝✐❛❧ ❞❡X π∗ ✲ ❉✐str✐❜✉✐çã♦ ■♥✐❝✐❛❧ ❞❡X∗

✭Z,X✮ ✲ ❈❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ❞❡ ❆❧❝❛♥❝❡ ❱❛r✐á✈❡❧ ❖❝✉❧t❛ λ✲ ❱❡t♦r ❞❡ P❛râ♠❡tr♦s ❞❡ ✭Z,X✮

✭Z∗_,_X∗✮ ✲ ❈❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ❖❝✉❧t❛

λ∗ ✲ ❱❡t♦r ❞❡ P❛râ♠❡tr♦s ❞❡ ✭_Z∗_,_X∗✮

(9)

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❊st❛ t❡s❡ ❛❜♦r❞❛ ❛ q✉❡stã♦ ❞❡ ✐♥❢❡r✐r s❡ ✉♠❛ ❛♠♦str❛ ♦❜s❡r✈❛❞❛ ❢♦✐✱ ❞❡ ❢❛t♦✱ ❣❡r❛❞❛ ♣♦r ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦ ♦✉ s❡ ❡ss❛ ❛♠♦str❛ ❢♦✐ ♣❡rt✉r❜❛❞❛ ♣♦r ✉♠ r✉í❞♦ ❛❧❡❛tór✐♦✳ ◆♦ ❝❛s♦ ♦♥❞❡ ♦ ♣r♦❝❡ss♦ ♦r✐❣✐♥❛❧ q✉❡ ♣♦❞❡ t❡r s♦❢r✐❞♦ ❛ ♣❡rt✉r❜❛çã♦ é ✉♠❛ ❝❛❞❡✐❛ ❞❡ ▼❛r❦♦✈✱ ❡ss❡ ♠♦❞❡❧♦ é ❜❛st❛♥t❡ ❝♦♥❤❡❝✐❞♦ ♥❛ ❧✐t❡r❛t✉r❛ ❝♦♠♦ ▼♦❞❡❧♦ ❞❡ ▼❛r❦♦✈ ❖❝✉❧t♦✱ ❍▼▼✶ _{✐♥tr♦❞✉③✐❞♦ ❡♠ ✶✾✻✻ ♣♦r ❬}_✷_{❪ ❡ t❡♠ ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ tr❛❜❛❧❤♦s ❞❡❞✐❝❛❞♦s ❛ ❡ss❡} t✐♣♦ ❞❡ ♠♦❞❡❧❛❣❡♠ ❞❡✈✐❞♦ ❛ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ❡ ❛♣❧✐❝❛çõ❡s✱ t❛✐s ❝♦♠♦ ❡♠ ♠❛❝❤✐♥❡ ❧❡❛r♥✐♥❣✱ ❣❡♥ét✐❝❛✱ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ✈♦③✱ ❡t❝ ✭❬✷✵❪ ❡ ❬✷✶❪✮✳

❆♥❛❧✐s❛r❡♠♦s ❡ss❡ ♣r♦❜❧❡♠❛ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ♦ ♣r♦❝❡ss♦ ♦❝✉❧t♦ ♦r✐❣✐♥❛❧ ♣❡rt❡♥❝❡ ❛ ✉♠❛ ❣r❛♥❞❡ ❝❧❛ss❡ ❞❡ ♣r♦❝❡ss♦s ♦♥❞❡ ❛ ♦r❞❡♠ ❞❡ ❞❡♣❡♥❞ê♥❝✐❛ ♥♦ ♣❛ss❛❞♦ ♥ã♦ é ✜①❛✱ ♦ q✉❡ ♥ã♦ ❛❝♦♥t❡❝❡ ❡♠ ✉♠❛ ❝❛❞❡✐❛ ❞❡ ▼❛r❦♦✈✳ ❆ q✉❡stã♦ q✉❡ q✉❡r❡♠♦s r❡s♣♦♥❞❡r é✿ ❉❛❞❛ ✉♠❛ ❛♠♦str❛ ❞❡ sí♠❜♦❧♦s ♦❜s❡r✈❛❞♦s ❞❡ ✉♠ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦ é ♣♦ssí✈❡❧ s❛❜❡r s❡ ❛♠♦str❛ ❡stá ♦✉ ♥ã♦ ♣❡rt✉r❜❛❞❛ ♣♦r ❛❧❣✉♠ r✉í❞♦ ❛❧❡❛tór✐♦❄ ❆tr❛✈és ❞❡ss❛ ❛♠♦str❛ ♣❡rt✉r❜❛❞❛✱ é ♣♦ssí✈❡❧ ♠❡♥s✉r❛r ♦ ❣r❛✉ ❞❡ ♣❡rt✉r❜❛çã♦ ❞❡ss❛ ❛♠♦str❛❄ ❊ ❛✐♥❞❛ ❞❡s❝♦❜r✐r ❛ ✈❡r❞❛❞❡✐r❛ ❢♦♥t❡ ❞❛ q✉❛❧ ♦s ❞❛❞♦s ❢♦r❛♠ ❣❡r❛❞♦s✱ ❛♥t❡s ❞❡ t❡r❡♠ s✐❞♦ ♣❡rt✉r❜❛❞♦s❄ ➱ ♣♦ssí✈❡❧ r❡❝✉♣❡r❛r ❛ ❧❡✐ ♦r✐❣✐♥❛❧ ❞♦s ❞❛❞♦s ♣❛r❛ q✉❛❧q✉❡r q✉❡ s❡❥❛ ♦ ❣r❛✉ ❞❡ ♣❡rt✉r❜❛çã♦❄

▼♦❞❡❧♦s ❝♦♠ t❛✐s ❝❛r❛❝t❡ríst✐❝❛s sã♦ ❝❤❛♠❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛ ❞❡ ▼♦❞❡❧♦s ❞❡ ▼❛r❦♦✈ ❖❝✉❧t♦ ❞❡ ❆❧❝❛♥❝❡ ❱❛r✐á✈❡❧ ✭❱▲❍▼▼✮✳ ❖s ❱▲❍▼▼✷ _{❛♣❛r❡❝❡r❛♠ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③✱ s❡❣✉♥❞♦ ❬}_✶✶_{❪✱ ♥❛ ❛♥á❧✐s❡ ❞♦ ♠♦✈✐♠❡♥t♦ ❝♦r♣♦r❛❧ ❤✉♠❛♥♦✱} ❝♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ❡♠ ❬✷✵❪ ❡ ❬✷✶❪✳ ❊♠ ❬✷✶❪✱ ♦ ❛✉t♦r ❛♥❛❧✐s❛ ♦ ♠♦✈✐♠❡♥t♦ ✸❉ ❛tr❛✈és ❞❛ r♦t❛çã♦ ❞❡ ✶✾ ❣r❛♥❞❡s ❛rt✐❝✉❧❛çõ❡s ❞♦ ❝♦r♣♦ ❤✉♠❛♥♦✱ ❡ ❬✷✵❪ ❡♠ s❡❣✉✐❞❛ ✉s❛ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❱▲❍▼▼ ❡♠ q✉❡ ❛ ❝❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ❞❡ ❛❧❝❛♥❝❡ ✈❛r✐á✈❡❧ ✭❱▲▼❈✮✸ _{♦❝✉❧t❛ é ❛ ♣♦s❡ ♥♦ t❡♠♣♦} _n _{❡ ♦s ❞❛❞♦s ♦❜s❡r✈❛❞♦s sã♦ ❛s ♣♦s✐çõ❡s ❞♦ ❝♦r♣♦ ❞❛❞❛s} ♣❡❧❛s r♦t❛çõ❡s ✸❉ ❞♦s ✶✾ ♣♦♥t♦s ♣r✐♥❝✐♣❛✐s✳ ❊❧❡s ❛r❣✉♠❡♥t❛♠ q✉❡ ❱▲❍▼▼ é s✉♣❡r✐♦r ❡♠ ❡✜❝✐ê♥❝✐❛ ❡ ♣r❡❝✐sã♦ ♥❛ ♠♦❞❡❧❛❣❡♠ ♠✉❧t✐✈❛r✐❛❞❛ ❡♠ sér✐❡s t❡♠♣♦r❛✐s ❝♦♠ ❛❧t❛ ✈❛r✐❡❞❛❞❡ ❞✐♥â♠✐❝❛✳

❊①✐st❡♠ ❛❧❣✉♥s tr❛❜❛❧❤♦s ❛♥t❡r✐♦r❡s q✉❡ ❛♥❛❧✐s❛♠ ❡ss❡s ♠♦❞❡❧♦s ❝♦♠ ♣❡rt✉r❜❛çã♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ t❡ór✐❝♦✱ ❡ q✉❡ t♦♠❛♠♦s ❝♦♠♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛✳ ❊♠ ❬✼❪✱ ♦s ❛✉t♦r❡s ❞❡s❝r❡✈❡♠ ✉♠ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦ ♣❡rt✉r❜❛❞♦ ❝♦♠♦ s❡♥❞♦ ✉♠❛ ❢✉♥çã♦ ❞❛ ❢♦♥t❡ ♦r✐❣✐♥❛❧ ❡ ❞❡ ✉♠ r✉í❞♦ ❛❧❡❛tór✐♦✳ ❊❧❡s s✉♣õ❡♠ q✉❡ ❛ ❢♦♥t❡ ♦r✐❣✐♥❛❧ é ✉♠❛ ❝❛❞❡✐❛ ❝♦♠ ♦r❞❡♠ ✐♥✜♥✐t❛✱ ❛ss✉♠✐♥❞♦ ✈❛❧♦r❡s ❡♠ ✉♠ ❛❧❢❛❜❡t♦ ❜✐♥ár✐♦ ❡ q✉❡ ♣♦❞❡ s♦❢r❡r ♣❡rt✉r❜❛çõ❡s ♣♦r ✉♠ r✉í❞♦ ❛❧❡❛tór✐♦ ❇❡r♥♦✉❧❧✐ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❢♦♥t❡ ♦r✐❣✐♥❛❧✳ ❊♠ ❬✶✷❪✱ ♦s ❛✉t♦r❡s ❝♦♥s✐❞❡r❛♠ q✉❡ ❛ ❢♦♥t❡ ♦r✐❣✐♥❛❧ é ✉♠❛ ❱▲▼❈✱ ♦♥❞❡ ❝❛❞❛ sí♠❜♦❧♦ é ♠✉❧t✐♣❧✐❝❛❞♦ ♣♦r ✉♠ r✉í❞♦ ❛❧❡❛tór✐♦ ❇❡r♥♦✉❧❧✐✱ t❛♠❜é♠ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❢♦♥t❡ ♦r✐❣✐♥❛❧✳ ❊❧❡s ❝❤❛♠❛r❛♠ ❡ss❡ ♠♦❞❡❧♦ ❝♦♠ ▼♦❞❡❧♦ ♣❡rt✉r❜❛❞♦ ■♥✢❛❝✐♦♥❛❞♦ ❞❡ ❩❡r♦s✳ ◆❡ss❡ s❡❣✉♥❞♦ tr❛❜❛❧❤♦ t❛♠❜é♠ é ❝♦♥s✐❞❡r❛❞♦ ♦ ❝❛s♦ ❡♠ q✉❡ ❛ ❛♠♦str❛ ♦❜s❡r✈❛❞❛ ♣♦❞❡ t❡r s✐❞♦ ❣❡r❛❞❛ ❞❡ ✉♠❛ ♠✐st✉r❛ ❞❡ ♣r♦❝❡ss♦s ❝♦♠ ♦r❞❡♠ ✈❛r✐á✈❡❧✳

❊♠ ❛♠❜♦s ♦s tr❛❜❛❧❤♦s ♦s ❛✉t♦r❡s ♠♦str❛r❛♠ q✉❡ s❡ ♦ r✉í❞♦ ❛❧❡❛tór✐♦ ❇❡r♥♦✉❧❧✐ ❢♦r ♣❡q✉❡♥♦✱ ❡♥tã♦ ❛ ❛♠♦str❛ ♣❡rt✉r❜❛❞❛ ♣♦❞❡ s❡r ✉s❛❞❛ ♣❛r❛ ❡st✐♠❛r ❛ ♠❛tr✐③ ❞❡ tr❛♥s✐çã♦ ❞♦ ♣r♦❝❡ss♦ ♦r✐❣✐♥❛❧✳ ❊❧❡s ♠♦str❛r❛♠ q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ❞♦ ♣r♦❝❡ss♦ ♣❡rt✉r❜❛❞♦ ❡ ❞♦ ♣r♦❝❡ss♦ ♦r✐❣✐♥❛❧ é ❧✐♠✐t❛❞♦ ♣♦r ✉♠❛ ❝♦♥st❛♥t❡c✱

❡♠ q✉❡cé ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r ❞♦ r✉í❞♦ ❛❧❡❛tór✐♦ ❇❡r♥♦✉❧❧✐ ✭♠❛✐s ❞❡t❛❧❤❡s ❡♠ ❬✼❪ ❡ ❬✶✷❪✮✳ P♦ré♠✱ s❡ ♦ r✉í❞♦ ❛❧❡❛tór✐♦

♥ã♦ ❢♦r ♣❡q✉❡♥♦ s✉✜❝✐❡♥t❡✱ ❡♥tã♦ ❛ ❛♣r♦①✐♠❛çã♦ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ♦❝✉❧t❛ ♣❡❧❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ❡st✐♠❛❞❛s ❞♦ ♣r♦❝❡ss♦ ♣❡rt✉r❜❛❞♦ ♥ã♦ s❡rá s❛t✐s❢❛tór✐❛✱ ✉♠❛ ✈❡③ q✉❡✱ s❡❣✉♥❞♦ ♦s ❛✉t♦r❡s✱ ❛ ♠❡❞✐❞❛ q✉❡

✶_{❍▼▼ é ❛ s✐❣❧❛ ❡♠ ✐♥❣❧ês ♣❛r❛ ❍✐❞❞❡♥ ▼❛r❦♦✈ ▼♦❞❡❧}

✷_{❱▲❍▼▼ é ❛ s✐❣❧❛ ❡♠ ✐♥❣❧ês ♣❛r❛ ❱❛r✐❛❜❧❡ ▲❡♥❣t❤ ❍✐❞❞❡♥ ▼❛r❦♦✈ ▼♦❞❡❧} ✸_{❱▲▼❈ é ❛ s✐❣❧❛ ❡♠ ✐♥❣❧ês ♣❛r❛ ❱❛r✐❛❜❧❡ ▲❡♥❣t❤ ▼❛r❦♦✈ ❈❤❛✐♥}

(10)

❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖

♦ r✉í❞♦ ❛❧❡❛tór✐♦ ❛✉♠❡♥t❛✱ ❛ ❝♦♥st❛♥t❡ c q✉❡ ❧✐♠✐t❛ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ✈❡r❞❛❞❡✐r❛s

❡ ❡st✐♠❛❞❛s ♣❡❧♦ ♣r♦❝❡ss♦ ❝♦♥t❛♠✐♥❞♦ t❛♠❜é♠ ❛✉♠❡♥t❛✳ P♦rt❛♥t♦ é ❝r✉❝✐❛❧ ❡st✐♠❛r ♦ ♣❛râ♠❡tr♦ ❞❡ ♣❡rt✉r❜❛çã♦✱ ❛ ✜♠ ❞❡ s❛❜❡r s❡ t❛❧ ❛♣r♦①✐♠❛çã♦ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛ ♦✉ ♥ã♦✳ ▼❛s✱ ♦s ❛✉t♦r❡s ♥ã♦ ❛❜♦r❞❛r❛♠ ❡ss❡ ♣r♦❜❧❡♠❛ ❞❡ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦✳

❯♠ ✐♠♣♦rt❛♥t❡ r❡s✉❧t❛❞♦ ❡♠ ❡st✐♠❛çã♦ ❞❡ ♣❛râ♠❡tr♦s ♣❛r❛ ✉♠❛ ❝❧❛ss❡ ❞❡ ♠♦❞❡❧♦s ♣❡rt✉r❜❛❞♦s é ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✶✶❪✳ ❆ ❝❧❛ss❡ ❞❡ ♠♦❞❡❧♦s ❞✐s❝✉t✐❞❛ ♥❡ss❡ ❛rt✐❣♦ é ❜❛st❛♥t❡ ❛❜r❛♥❣❡♥t❡✱ ✉♠❛ ✈❡③ q✉❡ ♣❡r♠✐t❡ q✉❡ ♦ r✉í❞♦ ❛❧❡❛tór✐♦ s❡❥❛ ♣r♦✈❡♥✐❡♥t❡ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ♠❛✐♦r ❞❡ ❞✐str✐❜✉✐çõ❡s✱ ♠❛s é r❡str✐t✐✈❛ ❡♠ r❡❧❛çã♦ ❛ ❞✐str✐❜✉✐çã♦ ❝♦♥❞✐❝✐♦♥❛❧ ❡♥tr❡ ♦ ♣r♦❝❡ss♦ ♦❜s❡r✈❛❞♦ ❡ ♦ ♦r✐❣✐♥❛❧✱ ♦✉ s❡❥❛✱ ❛♣❡♥❛s ♦ ú❧t✐♠♦ sí♠❜♦❧♦ ♦❝✉❧t♦ ♥♦ ♣❛ss❛❞♦ ❞♦ ♣r♦❝❡ss♦ ♦❝✉❧t♦ é ❝♦♥s✐❞❡r❛❞♦ ♥❛s ❞✐str✐❜✉✐çõ❡s ❝♦♥❞✐❝✐♦♥❛✐s✱ ❡♥q✉❛♥t♦ q✉❡ ♥❡ss❛ t❡s❡ ❝♦♥s✐❞❡r❛♠♦s q✉❡ ❡ss❛ ❞❡♣❡♥❞ê♥❝✐❛ ♣♦❞❡ s❡r ✉♠ ❝♦♥t❡①t♦✳ ❖ ❛✉t♦r ♣r♦♣õ❡ ✉♠ ❡st✐♠❛❞♦r ❜❛s❡❛❞♦ ❡♠ ✉♠❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❡♥❛❧✐③❛❞❛✱ ❛ss✐♠ ❝♦♠♦ ♥♦ ❈r✐tér✐♦ ❞❡ ■♥❢♦r♠❛çã♦ ❇❛②❡s✐❛♥❛ ✭❇■❈✮✹ _{♣r♦♣♦st♦ ❡♠ ❬}_✾_{❪✱ ♠❛s ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ♣ró♣r✐♦ ❛✉t♦r✱ ♦s r❡s✉❧t❛❞♦s} ❡♠♣ír✐❝♦s ♠♦str❛r❛♠ q✉❡ ♦ ❛❧❣♦r✐t♠♦ ❝♦♠ ❛ ♣❡♥❛❧✐③❛çã♦ ❞♦ ❇■❈ é ♠❛✐s ❡✜❝✐❡♥t❡ ❞♦ q✉❡ ♦ ♣r♦♣♦st♦ ♥♦ ❛rt✐❣♦✳ ❖ ❛✉t♦r ♠♦str❛ q✉❡ ♦ ❡st✐♠❛❞♦r ♣r♦♣♦st♦✱ ❝♦♠ ❡ss❛ ♦✉tr❛ ♣❡♥❛❧✐③❛çã♦✱ é ❢♦rt❡♠❡♥t❡ ❝♦♥s✐st❡♥t❡✳

◆❡ss❛ t❡s❡ ❛♣r❡s❡♥t❛♠♦s ❡st✐♠❛❞♦r❡s ❝♦♥s✐st❡♥t❡s ♣❛r❛ ❛ ár✈♦r❡ ❞❡ ❝♦♥t❡①t♦s✱ ❛ss♦❝✐❛❞❛ ❛ ❱▲▼❈ ♦❝✉❧t❛✱ ❡ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ ❞❡ ♣❡rt✉r❜❛çã♦ ❞♦s ♣r♦❝❡ss♦s ♣❡rt✉r❜❛❞♦s ❝♦♠♦ ❞❡s❝r✐t♦s ❡♠ ❬✼❪ ❡ ❬✶✷❪✳ ❆ s✐♠♣❧✐❝✐❞❛❞❡ ❞❡ss❡s ♠♦❞❡❧♦s ♥♦s ♣❡r♠✐t❡ ❛♣❧✐❝❛r ✉♠ ❛❧❣♦r✐t♠♦ ❊▼ ♣❛r❛ ♦❜t❡r ♦s ❡st✐♠❛❞♦r❡s✳

❆❧é♠ ❞✐ss♦✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❡st✉❞♦ ❞❡ s❡♥s✐❜✐❧✐❞❛❞❡ ❞♦s ❡st✐♠❛❞♦r❡s ♣❛r❛ ✈❡r✐✜❝❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s ❡st✐♠❛❞♦r❡s ♣r♦♣♦st♦s ♥❛ ♠❡❞✐❞❛ ❡♠ q✉❡ ♦ ♥í✈❡❧ ❞❡ ♣❡rt✉r❜❛çã♦ ❛✉♠❡♥t❛✳ ◆♦ss♦ ♦❜❥❡t✐✈♦ ❝♦♠ ❡ss❛ ❛♥á❧✐s❡ ❞❡ s❡♥s✐❜✐❧✐❞❛❞❡ é s❛❜❡r s❡ ❡①✐st❡ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ ♥í✈❡✐s ❞❡ ♣❡rt✉r❜❛çã♦ ❡♠ q✉❡ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡ ❡st✐♠❛çã♦ é ♠❛✐s ❡✜❝✐❡♥t❡✳

❆♣r❡s❡♥t❛♠♦s t❛♠❜é♠ ✉♠ ❝r✐tér✐♦ ❞❡ s❡❧❡çã♦ ❛ ✜♠ ❞❡ ❡s❝♦❧❤❡r✱ ❡♥tr❡ ♠♦❞❡❧♦s ♣❡rt✉r❜❛❞♦s ❞✐s❝✉t✐❞♦s✱ q✉❛❧ é ♦ ♠❛✐s ❛♣r♦♣r✐❛❞♦ ♣❛r❛ ✉♠❛ ❞❛❞❛ ❛♠♦str❛ ♣❡rt✉r❜❛❞❛✳

❈♦♠♦ ❛♣❧✐❝❛çã♦ ❞❛ ♥♦ss❛ ♠❡t♦❞♦❧♦❣✐❛ ❛ ❞❛❞♦s r❡❛✐s r❡❛❧✐③❛♠♦s ✉♠❛ ❛♥á❧✐s❡ ❞❡ ✉♠ ❜❛♥❝♦ ❞❡ ❞❛❞♦s ♠✉✐t♦ ✐♥t❡r❡ss❛♥t❡ q✉❡ ♥♦s ❢♦✐ ❣❡♥t✐❧♠❡♥t❡ ❝❡❞✐❞♦s ♣❡❧♦ ▲❛❜♦r❛tór✐♦ ❞❡ ◆❡✉r♦✜s✐♦❧♦❣✐❛ ❞❛ ❱✐sã♦ ❞❛ ❯❋▼●✱ ❝♦♦r❞❡♥❛❞♦ ♣❡❧♦ ❉r ❏❡r♦♠❡ ❇❛r♦♥✳ ◆❡ss❡ ❜❛♥❝♦ ❞❡ ❞❛❞♦s ❝♦r✉❥❛s sã♦ s✉❜♠❡t✐❞❛s ❛ ❡st✐♠ú❧♦s ✈✐s✉❛✐s ❡♠ ✉♠ ❡①♣❡r✐♠❡♥t♦ ❝♦♥tr♦❧❛❞♦ ❡ ❛s r❡s♣♦st❛s ❞❡ ♥❡✉rô♥✐♦s ❛ ❡ss❡s ❡stí♠✉❧♦s ❢♦r❛♠ ♠❡❞✐❞❛s✳ ❊ss❛s r❡s♣♦st❛s ♥❡✉r♦♥❛✐s sã♦ ❝❤❛♠❛❞❛s ❞❡ ✧s♣✐❦❡s✧q✉❡ ♣♦❞❡♠ s❡r ❝♦♥s✐❞❡r❛❞❛s ❝♦♠♦ ✧❞✐s♣❛r♦s✧❞♦s ♥❡✉rô♥✐♦s✳ ❉❡✈✐❞♦ ❛♦ ❢❛t♦ ❞❡ ❡ss❡s ❞✐s♣❛r♦s ♣♦❞❡r❡♠ s❡r ❡rr♦♥❡❛♠❡♥t❡ ♠❡❞✐❞♦s✱ ♣♦r r❛③õ❡s té❝♥✐❝❛s✱ ❝♦♥s✐❞❡r❛♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❞✐s♣❛r♦s ❞♦s ♥❡✉rô♥✐♦s ♦❜s❡r✈❛❞♦s ♥♦ t❡♠♣♦ ♣♦❞❡ s❡r ♠♦❞❡❧❛❞❛ ❝♦♠♦ ✉♠ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦ q✉❡ ♣♦❞❡ t❡r s♦❢r✐❞♦ ✉♠❛ ♣❡rt✉r❜❛çã♦ ♣♦r ✉♠ r✉í❞♦ ❛❧❡❛tór✐♦✳ ❖s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❝♦♠ ❛ ♠❡t♦❞♦❧♦❣✐❛ ❛q✉✐ ♣r♦♣♦st❛ sã♦ ❜❡♠ ✐♥t❡r❡ss❛♥t❡s ❡ ❝♦❡r❡♥t❡s ❝♦♠ ♦ q✉❡ s❡ ❡s♣❡r❛✈❛ ❡♥❝♦♥tr❛r✳

❊st❛ t❡s❡ ❡stá ♦r❣❛♥✐③❛❞❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ♥♦ ❝❛♣ít✉❧♦ ✷ ❛♣r❡s❡♥t❛♠♦s ❛s ♥♦t❛çõ❡s ❜ás✐❝❛s ❡ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ♣r❡❧✐♠✐♥❛r❡s ❞❡ ♠❡t♦❞♦❧♦❣✐❛s ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❇❛✉♠✲❲❡❧❝❤ ❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❡r✜❧❛❞❛✱ ❢❛r❡♠♦s r❡✈✐sõ❡s ❞♦s ♠♦❞❡❧♦s ❥á ♣r♦♣♦st♦s ♣♦r ❬✼❪ ❡ ❬✶✷❪✳ ◆♦ ❝❛♣ít✉❧♦ ✸ sã♦ ❛♣r❡s❡♥t❛❞♦s ♦s ♠♦❞❡❧♦s ♣r♦♣♦st♦s ♥❡ss❛ t❡s❡ ❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡♥❝♦♥tr❛❞♦s ♣❛r❛ ♦s ♠♦❞❡❧♦s ♣r♦♣♦st♦s✳ ◆♦ ❝❛♣ít✉❧♦ ✹ é ♠♦str❛❞♦ ✉♠ ❞♦s ♠♦❞❡❧♦s ♣r♦♣♦st♦s ♣♦r ❬✶✷❪✱ ♥♦ q✉❛❧ ♠♦str❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ♣r♦♣♦st❛ ❞❡ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ♣❛r❛ ❡ss❡ ♠♦❞❡❧♦✳ ◆♦ ❝❛♣ít✉❧♦ ✺ sã♦ ❛♣r❡s❡♥t❛❞♦s ❛s ♣r♦♣♦st❛s ❞❡ ❛❧❣♦r✐t♠♦s ❡ ❡st✐♠❛❞♦r❡s ♣❛r❛ ♦s ♠♦❞❡❧♦s ❡♠ q✉❡stã♦✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ✉♠❛ ✈❡rsã♦ ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ ❱✐t❡r❜✐ ♣❛r❛ ❱▲▼❈✳ ◆♦ ❝❛♣ít✉❧♦ ✻ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❡st✉❞♦ ❞❡ s✐♠✉❧❛çã♦ ❡ s❡♥s✐❜✐❧✐❞❛❞❡ ♣❛r❛ ♦ r✉í❞♦ ❛❧❡❛tór✐♦ ♣❛r❛ ❛❧❣✉♥s ♠♦❞❡❧♦s ❡♠ q✉❡stã♦✳ ◆♦ ❝❛♣ít✉❧♦ ✼ ❛♣r❡s❡♥t❛♠♦s ❞♦✐s ❝r✐tér✐♦s ❞❡ s❡❧❡çã♦ ❞❡ ♠♦❞❡❧♦s✱ ❛✜♠ ❞❡ ❞❡❝✐❞✐r q✉❛❧ ❞❡ ❞♦✐s ♠♦❞❡❧♦s ❡st✉❞❛❞♦s é ♦ ♠❛✐s ❛❞❡q✉❛❞♦ ♣❛r❛ ♠♦❞❡❧❛r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ sí♠❜♦❧♦s✱ ❞❛❞❛ ✉♠❛ ❛♠♦str❛ ♦❜s❡r✈❛❞❛✳ ◆♦ ❝❛♣ít✉❧♦ ✽ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❛♣❧✐❝❛çã♦✳ ◆♦ ❝❛♣ít✉❧♦ ✾ ❛♣r❡s❡♥t❛♠♦s ❝♦♥❝❧✉sõ❡s ❣❡r❛✐s ❛ r❡s♣❡✐t♦ ❞❛ t❡s❡ r❡❛❧✐③❛❞❛✳ ◆♦ ❝❛♣ít✉❧♦ ✶✵ ❛❜♦r❞❛♠♦s ❧✐♠✐t❛çõ❡s ❞♦s ♠♦❞❡❧♦s ❛❜♦r❞❛❞♦s ❡ tr❛❜❛❧❤♦s ❢✉t✉r♦s✳ ❊ ♥♦ ❆♣ê♥❝✐❝❡ ❛♣r❡s❡♥t❛♠♦s ❛s ♣r♦✈❛s r❡❧❛❝✐♦♥❛❞❛s ❛♦s r❡s✉❧t❛❞♦s ❡♥❝♦♥tr❛❞♦s ♥♦ ❝❛♣ít✉❧♦ ✸ ❡ ✹ ❡ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❜r❡✈❡ ❞❡✜♥✐çã♦ s♦❜r❡ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❞❡ ✉♠ ♠♦❞❡❧♦ ✉s❛♥❞♦ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ P❡r✜❧❛❞❛✱ q✉❡ é ✉♠❛ ❞❛s ♠❡t♦❞♦❧♦❣✐❛s ❞❡ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦ ♣r♦♣♦st♦ ♣♦r ❬✶✷❪ ❛♣r❡s❡♥t❛❞♦ ♥♦ ❝❛♣ít✉❧♦ ✹✳

✹_{❇■❈ é ✉♠❛ s✐❣❧❛ ❡♠ ✐♥❣❧ês ♣❛r❛ ❞❡♥♦t❛r ❇❛②❡s✐❛♥ ■♥❢♦r♠❛t✐♦♥ ❈r✐t❡r✐❛}

(11)

❈❛♣ít✉❧♦ ✷

◆♦t❛çõ❡s ❡ ❉❡✜♥✐çõ❡s

❈♦♥s✐❞❡r❡ ♦ ❛❧❢❛❜❡t♦ ❞✐s❝r❡t♦ ✜♥✐t♦ E ={0,1, ..., N−1} ❝♦♠ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❡ |E| =N✳ ❉❛❞♦s ❞♦✐s ✐♥t❡✐r♦s m, n ∈ Z✱ ❝♦♠ _m _≤_n✱ ✉s❛r❡♠♦s ❛ ♥♦t❛çã♦ _ωn

m ♣❛r❛ ❞❡♥♦t❛r ❛ s❡q✉ê♥❝✐❛ (ωm, ..., ωn) ❞❡ sí♠❜♦❧♦s ❡♠E✱ ❡ s❡❥❛

El(ωn

m)♦ ❝♦♥❥✉♥t♦ q✉❡ ❝♦♥té♠ t❛✐s s❡q✉ê♥❝✐❛s ♦♥❞❡l(ωn

m) =|n−m+ 1| é ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ s❡q✉ê♥❝✐❛ωmn✳ ❯♠❛

s❡q✉ê♥❝✐❛ ✈❛③✐❛ é ❞❡♥♦t❛❞❛ ♣♦r∅❡l(∅) = 0✳

❆ ❝♦♥❝❛♥❡t❛çã♦ ❞❛s s❡q✉ê♥❝✐❛s ω ❡ υ ❝♦♥s✐st❡ ❞♦s sí♠❜♦❧♦s ❞❡ ω s❡❣✉✐❞♦s ♣❡❧♦s sí♠❜♦❧♦s ❞❡ ν✳ ❉❛❞❛s ❞✉❛s

s❡q✉ê♥❝✐❛s ω ❡ υ✱ t❛❧ q✉❡ l(ω) < ∞✱ ❞❡♥♦t❛♠♦s ♣♦r υω ❛ s❡q✉ê♥❝✐❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦ l(υ) +l(ω) ♦❜t✐❞❛ ♣❡❧❛

❝♦♥❝❛t❡♥❛çã♦ ❞❡ss❛s ❞✉❛s s❡q✉ê♥❝✐❛s✳ ❆ ❝♦♥❝❛t❡♥❛çã♦ ♣♦❞❡ s❡r ❡①t❡♥❞✐❞❛ ♣❛r❛ ♦ ❝❛s♦ q✉❛♥❞♦ ❛s s❡q✉ê♥❝✐❛s sã♦ s❡♠✐✲✐♥✜♥✐t❛sυ=...ω−2ω−1✳

❉✐③❡♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ ν é ✉♠ s✉✜①♦ ❞❛ s❡q✉ê♥❝✐❛ ω s❡ ❡①✐st❡ ✉♠❛ s✉❜✲s❡q✉ê♥❝✐❛ η✱ ❝♦♠ l(η) ≥1✱ t❛❧ q✉❡

ω=ην ❡ ❞❡♥♦t❛♠♦sν ω✱ ❡ s❡ν é ✉♠ s✉✜①♦ ♣ró♣r✐♦ ❞❡ ω❡s❝r❡✈❡♠♦sν≺ω✳

✷✳✶

❈❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ❖❝✉❧t❛ ❝♦♠ ❆❧❝❛♥❝❡ ❱❛r✐á✈❡❧

❈♦♥s✐❞❡r❡X ={Xt}t∈Z✉♠ ♣r♦❝❡ss♦ ❡st❛❝✐♦♥ár✐♦ ❡r❣ó❞✐❝♦ ♥♦ ❛❧❢❛❜❡t♦ ❞✐s❝r❡t♦E✳ ❉❛❞❛ ✉♠❛ s❡q✉ê♥❝✐❛ω∈E

∞

❡ ✉♠ sí♠❜♦❧♦a∈E✱ ❞❡♥♦t❛♠♦s

p(a|ω) :=P(X0=a|X−1=ω−1, X−2=ω−2, ...)

❝♦♠♦ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ❞♦ ♣r♦❝❡ss♦X✳ ❊ ♣❛r❛ ✉♠❛ s❡q✉ê♥❝✐❛ ✜♥✐t❛ω∈Ej_{✱ ❞❡♥♦t❛♠♦s}

p(ω) :=P(X₋−_j1=ω).

❉❡✜♥✐çã♦ ✷✳✶✳ ❯♠❛ s❡q✉ê♥❝✐❛ ✜♥✐t❛ω∈ ∪∞

j=1Ej é ✉♠ ❝♦♥t❡①t♦ ❞❡X s❡ s❛t✐s❢❛③✿ ✐✮ P❛r❛ t♦❞❛ s❡q✉ê♥❝✐❛ s❡♠✐✲✐♥✜♥✐t❛ x−−∞1 ❝♦♠ω ❝♦♠♦ s❡♥❞♦ ✉♠ s✉✜①♦✱

P X0=a|X−∞−1 =x−−∞1

=p(a|ω)>0 ✭✷✳✶✮

♣❛r❛ t♦❞♦a∈E✳

✐✐✮ ◆❡♥❤✉♠ s✉✜①♦ ♣ró♣r✐♦ ❞❡ ω s❛t✐s❢❛③ ✭✷✳✶✮✳

❯♠ ❝♦♥t❡①t♦ ✐♥✜♥✐t♦ é ✉♠❛ s❡q✉ê♥❝✐❛ s❡♠✐✲✐♥✜♥✐t❛ω_−∞−1 t❛❧ q✉❡ ♥❡♥❤✉♠ s✉✜①♦ω₋−_j1, j= 1,2, ...é ✉♠ ❝♦♥t❡①t♦✳

❉❡✜♥✐çã♦ ✷✳✷✳ ❖ ❝♦♥❥✉♥t♦T ❞❡ ❝♦♥t❡①t♦s é ❝❤❛♠❛❞♦ ár✈♦r❡ ❞❡ ❝♦♥t❡①t♦s s❡ ♥❡♥❤✉♠ω1∈ T é ✉♠ s✉✜①♦ ♣ró♣r✐♦ ❞❡ ❛❧❣✉♠ ♦✉tr♦ω2∈ T✳ ❉❡✈✐❞♦ á ❝♦♥❞✐çã♦ ✐✐✮ ❛ ár✈♦r❡ ❞❡ ❝♦♥t❡①t♦s é ❝❤❛♠❛❞❛ ❞❡ ✐rr❡❞✉tí✈❡❧✳

❈❛❞❛ ❝♦♥t❡①t♦ ω∈ T ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠ ❝❛♠✐♥❤♦ ❞❡ ✉♠❛ ❢♦❧❤❛ ❛té ❛ r❛✐③ ✭✈❡❥❛ ❋✐❣✉r❛✷✳✶✮✳ ❖s ❣❛❧❤♦s ❞❛

ár✈♦r❡T sã♦ ✐❞❡♥t✐✜❝❛❞♦s ♣❡❧♦s ❝♦♥t❡①t♦s ✭✜♥✐t♦ ♦✉ ✐♥✜♥✐t♦✮ω∈ T✱ ❛ r❛✐③ é ♦ ❝♦♥t❡①t♦ ✈❛③✐♦∅✳

(12)

❈❆P❮❚❯▲❖ ✷✳ ◆❖❚❆➬Õ❊❙ ❊ ❉❊❋■◆■➬Õ❊❙

•

~

•

}

1

•

|

10

000 100

❋✐❣✉r❛ ✷✳✶✳ ár✈♦r❡ ❞❡ ❝♦♥t❡①t♦sT ❝♦♠ k= 3

❆ ❋✐❣✉r❛ ✷✳✶♠♦str❛ ✉♠❛ ár✈♦r❡ ❞❡ ❝♦♥t❡①t♦s ❞❡ ♦r❞❡♠3q✉❡ ❛ss✉♠❡ ✈❛❧♦r❡s ❡♠ ✉♠ ❛❧❢❛❜❡t♦E={0,1}✳

❉❡✜♥✐çã♦ ✷✳✸✳ ❯♠❛ ár✈♦✈r❡ T é ❝♦♠♣❧❡t❛ s❡ ❝❛❞❛ ♥ó ✐♥t❡r♥♦ t❡♠|E|❣❛❧❤♦s ✭♦✉ ✜❧❤♦s✮✳

❉❡✜♥✐çã♦ ✷✳✹✳ ❯♠❛ ár✈♦r❡T é ❝❤❛♠❛❞❛ L✲❢✉❧❧ s❡l(ω) =L,∀ω∈ T✳

❉❡♥♦t❛♠♦s ❛ ♣r♦❢✉♥❞✐❞❛❞❡ ❞❛ ár✈♦r❡T✱d(T) := max{l(ω) :ω∈ T }

❯♠ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦ ❡st❛❝✐♦♥ár✐♦X❡♠Eé ✉♠❛ ❱▲▼❈ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ♦ ♣❛r(T✱p(a|ω))s❡ s❛t✐s❢❛③ ❛ ❉❡✜♥✐çã♦

✷✳✶✳

❉❡✜♥✐çã♦ ✷✳✺✳ ❉❛❞♦ ✉♠ ✐♥t❡✐r♦k✱ ❞❡✜♥✐♠♦s ❛ ár✈♦r❡ tr✉♥❝❛❞❛Tk ❞❡ ♦r❞❡♠k♣♦r

Tk :={ω∈ T ✿ l(ω)≤k}S{ω:l(ω) =k ❡ω≺υ✱ ♣❛r❛ ❛❧❣✉♠ υ∈ T }✳

❉❛❞❛ ✉♠❛ ❛♠♦str❛ ❞❡ ❡st❛❞♦s xT

1 ❞❡ ✉♠❛ ❱▲▼❈ X✱ s❡❥❛ NT(ω, a) ♦ ♥ú♠❡r♦ ❞❡ ♦❝♦rrê♥❝✐❛s ❞❛ s❡q✉ê♥❝✐❛

ω∈ ∪k

j=1Ej s❡❣✉✐❞❛ ♣❡❧♦ sí♠❜♦❧♦a∈ E ♥❛ ❛♠♦str❛x1T ❡ s❡❥❛d(T) =O(logT)✱

NT(ω, a) =

n

i:d(T)< i≤T, xi_i₋−1_l₍_ω₎=ω, xi =a

o

❡ ♦ ♥ú♠❡r♦ ❞❡ ♦❝♦rrê♥❝✐❛s ❞❡ω ❡♠xT

1 é

NT(ω) =

n

i:d(T)< i≤m, xi_i₋−1_l₍_ω₎=ωo

✉♠❛ ár✈♦r❡ ❞❡ ❝♦♥t❡①t♦s ✈✐á✈❡❧ é t❛❧ q✉❡d(T)≤d(T)✱NT(ω)≥1♣❛r❛ t♦❞♦ω∈ T ❡ω′ ❝♦♠NT(ω′)≥1 s✉✜①♦ ❞❡

❛❧❣✉♠ω∈ T✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ ár✈♦r❡s ✈✐á✈❡✐s é ❞❡♥♦t❛❞♦ ♣♦rF(xT

1, d(T))✳

❉❡✜♥✐çã♦ ✷✳✻✳ ❖ ❈r✐tér✐♦ ❞❡ ■♥❢♦r♠❛çã♦ ❇❛②❡s✐❛♥❛ ✭❇■❈✮ ♣❛r❛ ✉♠❛ ár✈♦r❡ ✈✐á✈❡❧ é

BICT =−logM LT(xT1) +

(|E| −1)|T |

2 logT, ✭✷✳✷✮

♦♥❞❡M LT(xT1) =

Y

ω∈τ:NT(aω)≥1

Y

a∈E

NT(ω, a)

NT(ω)

NT(ω,a)

❖ ❚❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ♣r♦✈❛❞♦ ❡♠ ❬✾❪ ✭❚❡♦r❡♠❛ ✷✳✻✮ é ❡♥✉♥❝✐❛❞♦ ❛ s❡❣✉✐r✳ ❚❡♦r❡♠❛ ✷✳✶✳ ❙❡❥❛xT

1 ✉♠❛ ❛♠♦str❛ ❞❡ ✉♠❛ ❱▲▼❈X✳ P❛r❛ d(T)<∞✱ ♦ ❡st✐♠❛❞♦r ❇■❈ ❞❡ T ❞❡✜♥✐❞♦ ♣♦r

ˆ

TBIC xT1

= arg min T ∈F(xT

1, d(T))

BICT(xT1), ✭✷✳✸✮

s❛t✐s❢❛③

ˆ

TBIC xT1

=T

(13)

❈❆P❮❚❯▲❖ ✷✳ ◆❖❚❆➬Õ❊❙ ❊ ❉❊❋■◆■➬Õ❊❙

✱ q✉❛s❡ ❝❡rt❛♠❡♥t❡ q✉❛♥❞♦T → ∞✳

◆♦ ❝❛s♦ ❣❡r❛❧✱ t❡♠✲s❡ q✉❡

ˆ

T_{k BIC} xT

1

=T_k✱

q✉❛s❡ ❝❡rt❛♠❡♥t❡ q✉❛♥❞♦T → ∞✳

❉❡✜♥✐çã♦ ✷✳✼✳ ❯♠❛ ❈❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ❖❝✉❧t❛ ❞❡ ❆❧❝❛♥❝❡ ❱❛r✐á✈❡❧ ✭❱▲❍▼▼✮ é ✉♠ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦ ❜✐✈❛r✐❛❞♦

(X,Z) t❛❧ q✉❡✿

✶✮ N é ♦ ♥ú♠❡r♦ ❞❡ ❡st❛❞♦s ❞❛ ❱▲▼❈ ♦❝✉❧t❛X✱ ❝♦♠ ár✈♦r❡ T❀

✷✮ M é ♦ ♥ú♠❡r♦ ❞❡ ❡st❛❞♦s ❞♦ ♣r♦❝❡ss♦ ♦❜s❡r✈á✈❡❧ Z❝♦♠ ❡s♣❛ç♦ ❞❡ ❡st❛❞♦s O❀

✸✮ A é ❛ ♠❛tr✐③ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ❞♦ ♣r♦❝❡ss♦ ♦❝✉❧t♦ _X ❞❡✜♥✐❞❛ ♣♦r _p₍_a|ω₎✱ _∀ _a _∈_E,_∀ω _{∈ T}✱

♦♥❞❡aé ✉♠ ❡st❛❞♦ ❞❛ ❱▲▼❈ ♦❝✉❧t❛X❀

✹✮ B ✭❉✐str✐❜✉✐çã♦ ❞❡ ❊♠✐ssã♦✮ é ♦ ✈❡t♦r ❞❛s ❞✐str✐❜✉✐çã♦ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❝♦♥❞✐❝✐♦♥❛✐s ♣❛r❛ ❛❧❣✉♠ sí♠❜♦❧♦

❞♦ ♣r♦❝❡ss♦ ♦❜s❡r✈á✈❡❧ ❞❛❞♦ ♦ ❝♦♥t❡①t♦ ω ❞♦ ♣r♦❝❡ss♦ ♦❝✉❧t♦✱ ❞❡✜♥✐❞❛ ♣♦r P(Zt =z|X(tt−l(ω))+1 =ω)✱ ∀ω ∈ T✱

∀z∈O❀

✺✮ πé ♦ ✈❡t♦r ❝♦♠ ❛ ❞✐str✐❜✉✐çã♦ ✐♥✐❝❛❧ ❞♦ ♣r♦❝❡ss♦ ♦❝✉❧t♦✱ ❞❡✜♥✐❞♦ ♣♦r π(ω) =P(X1l(ω)=ω)✱∀ω∈ T✳ ❖❜s❡r✈❛çã♦ ✷✳✶✳ ❙❡ ♦ ♣r♦❝❡ss♦ ♦❝✉❧t♦ X ❢♦r ♠❛r❦♦✈✐❛♥♦ ❡ s❡ t✐✈❡r♠♦s P(Zt = k|X(tt−l(ω))+1 =ω) = P(Zt =

k|Xt=j)✱ ♦✉ s❡❥❛✱ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❡♠✐ssã♦ ♣❡r❞❡ ♠❡♠ór✐❛ ❞❡ t♦❞♦ ♦ ❝♦♥t❡①t♦✱ ❡♥tã♦ ♥❡ss❡ ❝❛s♦✱ t❡♠♦s ✉♠ ❝❛s♦

♣❛rt✐❝✉❧❛r ❞❡ ✉♠ ❱▲❍▼▼ ❜❡♠ ❝♦♥❤❡❝✐❞♦ ♥❛ ❧✐t❡r❛t✉r❛ q✉❡ sã♦ ♦s ♠♦❞❡❧♦s ❞❡ ▼❛r❦♦✈ ♦❝✉❧t♦s ✭❍▼▼✮✳

✷✳✷ ❆❧❣♦r✐t♠♦ ❞❡ ❇❛✉♠✲❲❡❧❝❤

❉❛❞❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♦❜s❡r✈❛çõ❡s ❞❡ t❛♠❛♥❤♦T ∈N✱_Z_{= (}_z₁_{, z}₂_{, ..., z}_T₎✱ ♦ ❛❧❣♦r✐t♠♦ ❊①♣❡❝t❛t✐♦♥✲▼❛①✐♠✐③❛t✐♦♥

❞❡ ❇❛✉♠✲❲❡❧❝❤ ❬✶✼❪ é ✉s❛❞♦ ♣❛r❛ ❡st✐♠❛r ♦ ✈❡t♦r ❞❡ ♣❛râ♠❡tr♦s ❞❡ ✉♠ ❍▼▼✱ ❞❛❞♦ ♣♦r Θ = (A_,B_{, π}₎✱ ♦♥❞❡ A₌_{p_ij_}₌_{P₍_X_t₌_j|X_t₋₁₌_i₎_}✱ ❝♦♠_X s❡♥❞♦ ✉♠❛ ❝❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ❛ss✉♠✐♥❞♦ ✈❛❧♦r❡s ❡♠_E✱B₌_{b_j₍_z_t₎_}₌ {P(Zt=zt|Xt=j)} ❡π={πi}={P(X1=i)}✱ ♣❛r❛ t♦❞♦i, j= 1, ..., N ❡ t♦❞♦t∈Z✳

❈♦♥s✐❞❡r❡ ❛ ✈❛r✐á✈❡❧ ρt(i, j)✱ ❝♦♠♦ s❡♥❞♦

ρt(i, j) = P(Xt=i, Xt+1=j|Z,Θ)

= P(Xt=i, Xt+1=j,Z|Θ)

P(Z|Θ)

= αt(i)aijbj(zt+1)βt+1(j)

N

X

k=1

N

X

l=1

αt(k)aklbl(zt+1)βt+1(l)

♦♥❞❡αt(i)❡βt(i)♣♦❞❡♠ s❡r ❝❛❧❝✉❧❛❞♦s ✉s❛♥❞♦ ♦s ♣r♦❝❡❞✐♠❡♥t♦s ❢♦r✇❛r❞ ❡ ❜❛❝❦✇❛r❞ ❞❡s❝r✐t♦s ❛ s❡❣✉✐r✱ r❡s♣❡❝t✐✲

✈❛♠❡♥t❡✱

αt(i) =P(z1, z2, . . . , zt, Xt=i|Θ), βt(i) =P(zt+1, zt+2, . . . , zT|Xt=i,Θ)

❙❡❥❛γt(i)✱

γt(i) = N

X

j=1

ρt(i, j)

❙♦♠❛♥❞♦γt(i)❡♠t ♦❜t❡♠♦s ♦ ♥ú♠❡r♦ ❡s♣❡r❛❞♦ ❞❡ tr❛♥s✐çõ❡s ❞♦ ❡st❛❞♦i= 1, ..., N✱ T−1

X

t=1

γt(i)✳ ❉♦ ♠❡s♠♦ ♠♦❞♦✱

♦❜t❡♠♦s ♦ ♥ú♠❡r♦ ❡s♣❡r❛❞♦ ❞❡ tr❛♥s✐çã♦ ❞♦ ❡st❛❞♦i♣❛r❛ ♦ ❡st❛❞♦j= 1, ..., N✱

T−1

X

t=1

ρt(i, j)✳ ❖ ✈❡t♦r ❞❡ ♣❛râ♠❡tr♦s

(14)

❈❆P❮❚❯▲❖ ✷✳ ◆❖❚❆➬Õ❊❙ ❊ ❉❊❋■◆■➬Õ❊❙

Θ♣♦❞❡ s❡r ❛t✉❛❧✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛ ˜

πi = γ1(i)

˜

aij =

PT−1

t=1 ρt(i, j)

PT−1

t=1 γt(i)

˜

bi(k) =

PT

t=11{zt=k}γt(i)

PT t=1γt(i)

♦♥❞❡

1{zt=k}= (

1, s❡zt=k

0, ❝❛s♦ ❝♦♥trár✐♦

❆ ♣r♦✈❛ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ss❡ ❛❧❣♦r✐t♠♦ ❊▼ é ❛♣r❡s❡♥t❛❞❛ ❡♠ ❬✷❪✳

✷✳✸ ❘❡✈✐sã♦ ❞❡ ❆❧❣✉♥s ▼♦❞❡❧♦s ❞❡ P❡rt✉r❜❛çã♦ ❊st♦❝ást✐❝❛

◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛r❡♠♦s ♠♦❞❡❧♦s ❛♣r❡s❡♥t❛❞♦s ❡♠ ❬✼❪ ❡ ❬✶✷❪ q✉❡ sã♦ ❛ ❜❛s❡ ❞❡st❛ t❡s❡✳ ❖s ❛✉t♦r❡s ❝♦♥s✐❞❡r❛♠ q✉❡ ❛ ❛♠♦str❛ ♦❜s❡r✈á✈❡❧ ❡stá ♣❡rt✉r❜❛❞❛ ♣♦r ❛❧❣✉♠ t✐♣♦ ❞❡ r✉í❞♦✳ ❊st❡s ♠♦❞❡❧♦s ❞❡ ♣❡rt✉r❜❛çã♦ ❡st♦❝ást✐❝❛ sã♦ ✐♥t❡r❡ss❛♥t❡s ♣♦rq✉❡ ♣♦❞❡♠ s❡r ✉s❛❞♦s ♣❛r❛ ❛♣r♦①✐♠❛r ♠✉✐t♦s ❢❡♥ô♠❡♥♦s ❡♠ q✉❡ ❛ ✈❛r✐á✈❡❧ ❡♠ ❡st✉❞♦ é ❜✐♥ár✐❛ ❡ ♣♦❞❡ s❡r ❧✐❞❛ ❝♦♠ ❡rr♦✳

❊♠ ❬✼❪ ♦s ❛✉t♦r❡s ❛♣r❡s❡♥t❛♠ ✉♠ ♠♦❞❡❧♦ ♦♥❞❡ ❛ ❝❛❞❡✐❛ ❞❡ ♦r❞❡♠ ✐♥✜♥✐t❛ é ❡st♦❝❛st✐❝❛♠❡♥t❡ ♣❡rt✉r❜❛❞❛ ♣♦r ✉♠ r✉í❞♦ ❇❡r♥♦✉❧❧✐✳ ❊❧❡s ❝♦♥s✐❞❡r❛♠X ❝♦♠♦ ✉♠❛ ❝❛❞❡✐❛ ❡st♦❝ást✐❝❛ ❜✐♥ár✐❛ ❞❡ ♦r❞❡♠ ✐♥✜♥✐t❛ ❡ξ❝♦♠♦ ✉♠❛ s❡q✉ê♥❝✐❛

❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❇❡r♥♦✉❧❧✐ t❛❧ q✉❡P(ξt= 0) = 1−ǫ❡ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡X✳ ❈♦♥s✐❞❡r❛♥❞♦

a⊕b=a+b (mod2), a, b∈ {0,1}✳

P❛r❛ ❝❛❞❛ t❡♠♣♦t ♦ ✈❛❧♦r ❞♦ ♣r♦❝❡ss♦ ♣♦❞❡✱ ❛❧❡❛t♦r✐❛♠❡♥t❡ ❡ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✱ ♠✉❞❛r ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ✜①❛✳

❖ ♣r♦❝❡ss♦ ♣❡rt✉r❜❛❞♦Zé ❞❡✜♥✐❞♦ ♣♦r

Zt=Xt⊕ξt t∈Z. ✭✷✳✹✮

❖s ❛✉t♦r❡s ❞❡♠♦♥str❛r❛♠ q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ❞♦ ♣r♦❝❡ss♦ ♦r✐❣✐♥❛❧ ❡ ♦ ♣r♦❝❡ss♦ ♣❡rt✉r❜❛❞♦ é ❧✐♠✐t❛❞♦ ♣♦r ✉♠❛ ❝♦♥st❛♥t❡✱ q✉❡ é ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡ ❞♦ ♣❛râ♠❡tr♦ ❞❡ r✉í❞♦ǫ✳ P♦rt❛♥t♦✱ s❡ ❡st❡

♣❛râ♠❡tr♦ ❞❡ r✉í❞♦ é ♣❡q✉❡♥♦ ♦ s✉✜❝✐❡♥t❡✱ ❡♥tã♦ é ♣♦ssí✈❡❧ ✉t✐❧✐③❛r ❛s ❡st✐♠❛t✐✈❛s ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ❞♦ ♣r♦❝❡ss♦ ♣❡rt✉r❜❛❞♦ ❝♦♠♦ ✉♠❛ ❜♦❛ ❛♣r♦①✐♠❛çã♦ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ❞❡ ♣r♦❝❡ss♦ ♦r✐❣✐♥❛❧✳

❖✉tr♦ r❡s✉❧t❛❞♦ ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✼❪ é q✉❡✱ ♣❛r❛ ✉♠❛ ❛♠♦str❛ ✜♥✐t❛ z1, z2, ..., zn ❞♦ ♣r♦❝❡ss♦ ♣❡rt✉r❜❛❞♦✱ ❛

♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ ár✈♦r❡ ❞❡ ❝♦♥t❡①t♦s ❡st✐♠❛❞❛ tr✉♥❝❛❞❛ ♥❛ ♦r❞❡♠k✱Tˆk✱ s❡r ❞✐❢❡r❡♥t❡ ❞❛ ár✈♦r❡ ❞❡ ❝♦♥t❡①t♦s ✈❡r✲

❞❛❞❡✐r❛ tr✉♥❝❛❞❛ ♥❛ ♦r❞❡♠k✱Tk ❞❡❝r❡s❝❡ ❡①♣♦♥❡♥❝✐❛❧♠❡♥t❡ ❝♦♠♦ ❢✉♥çã♦ ❞♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ❡ ❞♦ ♣❛râ♠❡tr♦

❞❡ r✉í❞♦✳ ❉❡ss❛ ♠❛♥❡✐r❛ ♦s ❛✉t♦r❡s ♦❜tê♠ ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦♥s✐stê♥❝✐❛ ❢♦rt❡✱ ❡♠ q✉❡ ♣❛r❛ ✉♠❛ ❛♠♦str❛ ✐♥✜♥✐t❛

z1, z2, ... ❡①✐st❡ ✉♠n¯ t❛❧ q✉❡ ♣❛r❛ t♦❞♦ n≥n¯ Tˆ

k =T

k✱ q✉❛s❡ ❝❡rt❛♠❡♥t❡✱ ❞❡s❞❡ q✉❡ ❛❧❣✉♠❛s ❝♦♥❞✐çõ❡s s❡❥❛♠

s❛t✐s❢❡✐t❛s ✭♠❛✐s ❞❡t❛❧❤❡s ❡♠ ❬✼❪✮✳

❊♠ ❬✶✷❪ ♦s ❛✉t♦r❡s ❛♣r❡s❡♥t❛♠ ✉♠❛ ♣❡rt✉r❜❛çã♦ ❡st♦❝ást✐❝❛ ❡♠ q✉❡X é ✉♠❛ ❱▲▼❈ ❡ ♦ ♣r♦❝❡ss♦ ♣❡rt✉r❜❛❞♦ Z é ❣❡r❛❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛

Zt=Xt·ξt, ✭✷✳✺✮

♦♥❞❡ξé ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❇❡r♥♦✉❧❧✐ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ X✱ ❝♦♠P(ξt= 1) = 1−ǫ✱ ❡♠ q✉❡ǫé ♦ ♣❛râ♠❡tr♦ ❞❡

r✉í❞♦✳

❆ss✐♠ ❝♦♠♦ ❡♠ ❬✼❪✱ ❡❧❡s t❛♠❜é♠ ♠♦str❛♠ q✉❡ s❡ ♦ ♣❛râ♠❡tr♦ ❞❡ r✉í❞♦ ❢♦r ♣❡q✉❡♥♦ s✉✜❝✐❡♥t❡✱ ❡♥tã♦ ❛s ♣r♦❜❛✲ ❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ❞♦ ♣r♦❝❡ss♦ ♦r✐❣✐♥❛❧ ♣♦❞❡♠ s❡r ❜❡♠ ❛♣r♦①✐♠❛❞❛s ♣♦r ❛q✉❡❧❛s ❞♦ ♣r♦❝❡ss♦ ♣❡rt✉r❜❛❞♦✱ ♣♦✐s é ♣r♦✈❛❞♦ q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❡ss❛s ❞✉❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s sã♦ ❧✐♠✐t❛❞❛s ♣♦r ✉♠❛ ❝♦♥st❛♥t❡ q✉❡ ❝r❡s❝❡ ❝♦♠ ♦ r✉í❞♦ ❛❧❡❛tór✐♦ǫ✳

P❛r❛ ❡ss❡ ♠♦❞❡❧♦✱ ♦s ❛✉t♦r❡s t❛♠❜é♠ ❝♦♥❝❧✉ír❛♠ q✉❡ ♣❛r❛ ✉♠❛ ❛♠♦str❛ ✜♥✐t❛ Zn

1 ❞♦ ♣r♦❝❡ss♦ ♣❡rt✉r❜❛❞♦✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ ár✈♦r❡ ❞❡ ❝♦♥t❡①t♦s ❡st✐♠❛❞❛ tr✉♥❝❛❞❛ ♥❛ ♦r❞❡♠ k✱ Tˆ_k✱ s❡r ❞✐❢❡r❡♥t❡ ❞❛ ár✈♦r❡ ❞❡ ❝♦♥t❡①t♦s

(15)

❈❆P❮❚❯▲❖ ✷✳ ◆❖❚❆➬Õ❊❙ ❊ ❉❊❋■◆■➬Õ❊❙

✈❡r❞❛❞❡✐r❛ tr✉♥❝❛❞❛ ♥❛ ♦r❞❡♠ k✱Tk✱ ❞❡❝r❡s❝❡ ❡①♣♦♥❡♥❝✐❛❧♠❡♥t❡ ❝♦♠ ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ❡ ❝♦♠ ♦ ♣❛râ♠❡tr♦

❞❡ r✉í❞♦✳ ❊ ❛ss✐♠ ❝♦♠♦ ❡♠ ❬✼❪✱ ♦s ❛✉t♦r❡s ♦❜té♠ ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦♥s✐stê♥❝✐❛ ❢♦rt❡✱ ❡♠ q✉❡ ♣❛r❛ ✉♠❛ ❛♠♦str❛ ✐♥✜♥✐t❛ z1, z2, ...❡①✐st❡ ✉♠¯nt❛❧ q✉❡ ♣❛r❛ t♦❞♦n≥n¯ Tˆ

_k =T_k✱ q✉❛s❡ ❝❡rt❛♠❡♥t❡✱ ❞❡s❞❡ q✉❡ ❛❧❣✉♠❛s ❝♦♥❞✐çõ❡s

s❡❥❛♠ s❛t✐s❢❡✐t❛s ✭♠❛✐s ❞❡t❛❧❤❡s ❡♠ ❬✶✷❪✮✳

❖✉tr♦ ♠♦❞❡❧♦ ❝♦♥s✐❞❡r❛❞♦ ❡♠ ❬✶✷❪ é ✉♠❛ ♠✐st✉r❛ ❞❡ ❞✉❛s ❝❛❞❡✐❛s ❞❡ ▼❛r❦♦✈ ❞❡ ❛❧❝❛♥❝❡ ✈❛r✐á✈❡❧ ✭❱▲▼❈✮ ✐♥❞❡♣❡♥❞❡♥t❡s X ❡Y ❛ss✉♠✐♥❞♦ ✈❛❧♦r❡s ❡♠ ✉♠ ❛❧❢❛❜❡t♦ ✜♥✐t♦E ={0,1, ..., N−1}✳ ❖s ❛✉t♦r❡s ❝♦♥s✐❞❡r❛r❛♠ ξ

✉♠❛ ✈✳❛ ❇❡r♥♦✉❧❧✐✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡X ❡Y✱ ❝♦♠P(ξt= 1) = 1−ǫ✱ ♦♥❞❡ǫé ✉♠ ♣❛râ♠❡tr♦ ❞❡ r✉í❞♦ ❝♦♥❤❡❝✐❞♦ ❡

✜①♦ ❡♠(0,1)✳ ❉❡✜♥✐r❛♠ ✉♠ ♠♦❞❡❧♦ ♣❡rt✉r❜❛❞♦ ❞❛❞♦ ♣♦r✿

Zt=

Xt, s❡ ξt= 1

Yt, s❡ ξt= 0.

✭✷✳✻✮

❊ t❛♠❜é♠ ♣❛r❛ ❡ss❡ ♠♦❞❡❧♦ ♦s ❛✉t♦r❡s ❝♦♥❝❧✉ír❛♠ q✉❡ ♣❛r❛ ✉♠❛ ❛♠♦str❛ ✜♥✐t❛ Zn

1 ❞♦ ♣r♦❝❡ss♦ ♣❡rt✉r❜❛❞♦✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ ár✈♦r❡ ❞❡ ❝♦♥t❡①t♦s ❡st✐♠❛❞❛ tr✉♥❝❛❞❛ ♥❛ ♦r❞❡♠ k✱ Tˆk✱ s❡r ❞✐❢❡r❡♥t❡ ❞❛ ár✈♦r❡ ❞❡ ❝♦♥t❡①t♦s

✈❡r❞❛❞❡✐r❛ tr✉♥❝❛❞❛ ♥❛ ♦r❞❡♠k✱T_k✱ ❞❡❝r❡s❝❡ ❡①♣♦♥❡♥❝✐❛❧♠❡♥t❡ ❝♦♠ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ❡ ❝♦♠ ♦ ♣❛râ♠❡tr♦ ❞❡

r✉í❞♦✳ ❊ ♦❜t✐✈❡r❛♠ t❛♠❜é♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦♥s✐stê♥❝✐❛ ❢♦rt❡✱ ❡♠ q✉❡ ♣❛r❛ ✉♠❛ ❛♠♦str❛ ✐♥✜♥✐t❛Z∞

1 ❡①✐st❡ ✉♠n¯ t❛❧ q✉❡ ♣❛r❛ t♦❞♦n≥n¯ Tˆ_k =T_k✱ q✉❛s❡ ❝❡rt❛♠❡♥t❡✱ ❞❡s❞❡ q✉❡ ❛❧❣✉♠❛s ❝♦♥❞✐çõ❡s s❡❥❛♠ s❛t✐s❢❡✐t❛s ✭♠❛✐s ❞❡t❛❧❤❡s

❡♠ ❬✶✷❪✮✳

❆❧❣✉♠❛s ♣❡r❣✉♥t❛s s♦❜r❡ ✐♥❢❡rê♥❝✐❛ ♣❛r❛ ❡st❡s ♠♦❞❡❧♦s ♣❡r♠❛♥❡❝❡♠ s❡♠ r❡s♣♦st❛ ❡ s❡rã♦ ❛❜♦r❞❛❞♦s ♥❡st❛ t❡s❡✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ s❡ ❤♦✉✈❡ ❛❧❣✉♠ t✐♣♦ ❞❡ ♣❡rt✉r❜❛çã♦ ❛♣❧✐❝❛❞❛ ♥♦ ♣r♦❝❡ss♦ ♦❝✉❧t♦ X ❡ s❡ ❡ss❛ ♣❡rt✉r❜❛çã♦ é

♦✉ ♥ã♦ ♣❡q✉❡♥❛✳ ❚❛♠❜é♠ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❡st✐♠❛r ♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦✱ ✐♥❝❧✉s✐✈❡ ♥♦ ❝❛s♦ ❡♠ q✉❡ ♦ ♣❛râ♠❡tr♦ ❞❡ ♣❡rt✉r❜❛çã♦ ♥ã♦ ❢♦r ♣❡q✉❡♥♦ ♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ✉s❛r ❛ ❛♠♦str❛ ♣❛r❛ ❡st✐♠❛r ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ❞♦ ♣r♦❝❡ss♦ ♦❝✉❧t♦X✳

(16)

❈❛♣ít✉❧♦ ✸

▼♦❞❡❧♦s ❞❡ P❡rt✉r❜❛çã♦ Pr♦♣♦st♦s

◆♦ss♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ♥❡st❛ t❡s❡ é ♣r♦♣♦r ♠❡t♦❞♦❧♦❣✐❛s ♣❛r❛ ❡st✐♠❛r ♦s ♣❛râ♠❡tr♦s ❞❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ♠♦❞❡❧♦s ❛ ♣❛rt✐r ❞♦s ♠♦❞❡❧♦s ♣r♦♣♦st♦s ♣♦r ❬✼❪ ❡ ❬✶✷❪✱ ❡ r❡❛❧✐③❛r ✉♠❛ ❛♥á❧✐s❡ ❝✉✐❞❛❞♦s❛ ❞♦s r❡s✉❧t❛❞♦s s❡❣✉✐♥❞♦ ♦ ❡sq✉❡♠❛ ❞❡ ♣❡rt✉r❜❛çã♦ ♣r♦♣♦st❛ ♣♦r ❡ss❡s ❛✉t♦r❡s✳

❆ s❡❣✉✐r ✐r❡♠♦s ♣r♦♣♦r ♦s ♠♦❞❡❧♦s ❞❡ ♣❡rt✉r❜❛çã♦ ❡st♦❝ást✐❝❛ q✉❡ ❛♥❛❧✐③❛r❡♠♦s ♥❡st❛ t❡s❡✳ ❈♦♥s✐❞❡r❛♠♦s X

✉♠❛ ❱▲▼❈✱ ❝♦♠♦ ♥❛ ❉❡✜♥✐çã♦ ✷✳✶✱ ❛ss✉♠✐♥❞♦ ✈❛❧♦r❡s ❡♠ ✉♠ ❛❧❢❛❜❡t♦ ❞✐s❝r❡t♦ E ={0,1, ..., N−1}, N ∈ N❡

ξ={ξt}t∈Z ❝♦♠♦ s❡♥❞♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ❝♦♠P(ξt=i) =ǫi t❛❧ q✉❡ N−1

X

i=0

ǫi= 1✱ ✐♥❞❡♣❡♥❞❡♥t❡

❞❡X✳

❙❡❣✉✐♥❞♦ ❞❡ ♣❡rt♦ ♦s ♠♦❞❡❧♦s ❛♣r❡s❡♥t❛❞♦s ❡♠ ❬✼❪ ❡ ❬✶✷❪ ❝♦♥s✐❞❡r❛♠♦s ♦s ♠♦❞❡❧♦s ❞❡ ♣❡rt✉r❜❛çã♦ ❡st♦❝ást✐❝❛s ❞❡t❛❧❤❛❞♦s ♥❛s ♣ró①✐♠❛s s❡çõ❡s✳

✸✳✶ ▼♦❞❡❧♦ ❞❡ P❡rt✉r❜❛çã♦ ❚✐♣♦ ❙♦♠❛

❊♠ ✉♠ ▼♦❞❡❧♦ ❞❡ P❡rt✉r❜❛çã♦ ❚✐♣♦ ❙♦♠❛✱ q✉❡ ❞❡♥♦t❛r❡♠♦s r❡s✉♠✐❞❛♠❡♥t❡ ♣♦r ❚❙❈▼✶_{✱ ♦ ♣r♦❝❡ss♦ ♣❡rt✉r❜❛❞♦}

Z é ❞❡✜♥✐❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛

Zt=Xt⊕ξt(mod|E|), ✭✸✳✶✮

♦♥❞❡Xé ✉♠❛ ❱▲▼❈✱ ❝♦♠ ár✈♦r❡T ❛ss♦❝✐❛❞❛✱ ❡ ♥ã♦ ✉♠❛ ❝❛❞❡✐❛ ❞❡ ♦r❞❡♠ ✐♥✜♥✐t❛ ❝♦♠♦ ❡♠ ❬✼❪✳ ❖❜s❡r✈❛♠♦s q✉❡ ♦ ❚❙❈▼ ❞❛❞♦ ❡♠ ✭✸✳✶✮ é ✉♠ ♣r♦❝❡ss♦ ❜✐✈❛r✐❛❞♦(Z,X)❝♦♠ ✈❡t♦r ❞❡ ♣❛râ♠❡tr♦sλS = (AS,BS,πS)✱ ♦♥❞❡

AS ={p(a|ω)}=P

X0=a

X−−l1(ω)=ω

,∀a∈ E,∀ω∈ T,

sã♦ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ❞♦ ♣r♦❝❡ss♦ ♦❝✉❧t♦X✱

BS={bω(zt)}=

n

PZt=zt

Xtt−l(ω)+1=ω

o

✱ ∀ω∈El(ω)_✱_∀z

t∈ E✱

é ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞♦ sí♠❜♦❧♦ ♦❜s❡r✈❛❞♦ ❞❛❞❛ ❛ s❡q✉ê♥❝✐❛ ♦❝✉❧t❛ ❞♦ ♣r♦❝❡ss♦ ♦r✐❣✐♥❛❧ ✭❉✐str✐❜✉✐çã♦ ❞❡ ❊♠✐ssã♦✮✳

πS={πω}=

n

PX₋−_l1₍_ω₎=ωo,∀ω∈ T,

é ❛ ❞✐str✐❜✉✐çã♦ ❡st❛❝✐♦♥ár✐❛ ❞♦ ❝♦♥t❡①t♦ω ❞♦ ♣r♦❝❡ss♦ ♦r✐❣✐♥❛❧X✳

❙❡❥❛Z✉♠ ♣r♦❝❡ss♦ ♣❡rt✉r❜❛❞♦ ❛ss✉♠✐♥❞♦ ✈❛❧♦r❡s ❡♠ ✉♠ ❛❧❢❛❜❡t♦ ❞✐s❝r❡t♦E✱ ❡ s❡❥❛λS ♦ ✈❡t♦r ❞❡ ♣❛râ♠❡tr♦s

❞♦ ♣r♦❝❡ss♦ ❜✐✈❛r✐❛❞♦(Z,X)✳ ❆ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛L₍_λS|Z)❞♦ ❝♦♥❥✉♥t♦ ❞♦s ✈❛❧♦r❡s ❞♦ ✈❡t♦r λS✱ ❞❛❞❛

✉♠❛ ❛♠♦str❛ ♣❡rt✉r❜❛❞❛ ❞❡ sí♠❜♦❧♦s ♦❜s❡r✈á✈❡✐sZ ❞❡ t❛♠❛♥❤♦T ∈N✱ é ❞❡✜♥✐❞❛ ♣♦r L₍_λS|Z) =P(Z|λS).

❈♦♥s✐❞❡r❛♥❞♦ ♦ ♠♦❞❡❧♦ ❚❙❈▼✱ t❡♠♦s q✉❡

✶_{❚❙❈▼ é ❛ s✐❣❧❛ ❡♠ ✐♥❣❧ês ♣❛r❛ ❚②♣❡ ❙✉♠ ❈♦♥t❛♠✐♥❛t❡❞ ▼♦❞❡❧}