■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛
❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦
■♥❢❡rê♥❝✐❛ ❡st❛tíst✐❝❛ ♥♦ ❞♦♠í♥✐♦ ❞❡ ❋♦✉r✐❡r
♣❛r❛ ♦ ❡st✉❞♦ ❞❛ ❞✐♥â♠✐❝❛ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡
♣r♦❝❡ss♦s ❞✐❢✉s✐✈♦s ❛♥ô♠❛❧♦s
♣♦r
❘❛✉❧ ❨✉❦✐❤✐r♦ ▼❛ts✉s❤✐t❛
♣❛r❛ ♦ ❡st✉❞♦ ❞❛ ❞✐♥â♠✐❝❛ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡
♣r♦❝❡ss♦s ❞✐❢✉s✐✈♦s ❛♥ô♠❛❧♦s
♣♦r
❘❛✉❧ ❨✉❦✐❤✐r♦ ▼❛ts✉s❤✐t❛
❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ❋ís✐❝❛✳ ➪r❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ❋ís✐❝❛ ❊st❛tíst✐❝❛
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❆♥♥✐❜❛❧ ❉✐❛s ❞❡ ❋✐❣✉❡✐r❡❞♦ ◆❡t♦
s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ❉♦✉t♦r ❡♠ ❋ís✐❝❛✳
❆♣r♦✈❛❞❛ ♣♦r✿
Pr♦❢✳ ❆♥♥✐❜❛❧ ❉✐❛s ❞❡ ❋✐❣✉❡✐r❡❞♦ ◆❡t♦ ❖r✐❡♥t❛❞♦r✱ ■❋✴❯♥❇
Pr♦❢✳ ❚❛r❝ís✐♦ ▼❛r❝✐❛♥♦ ❞❛ ❘♦❝❤❛ ❋✐❧❤♦ ■❋✴❯♥❇
Pr♦❢✳ ❋á❜✐♦ ▼❛❝ê❞♦ ▼❡♥❞❡s ●❛♠❛✴❯♥❇
Pr♦❢✳ P✉s❤♣❛ ◆❛r❛②❛♥ ❘❛t❤✐❡ ❊❙❚✴❯♥❇
Pr♦❢✳ ■r❛♠ ▼❛r❝❡❧♦ ●❧ér✐❛ ■❋✴❯❋❆▲
P(Xt= 0,85Xt−1+ 0,04Yt−1, Yt=−0,04Xt−1+ 0,85Yt−1+ 1) = 0,85✱
P(Xt= 0,20Xt−1−0,26Yt−1, Yt= 0,26Xt−1+ 0,22Yt−1+ 1.8) = 0,07❡
P(Xt=−0,15Xt−1+ 0,28Yt−1, Yt= 0,26Xt−1+ 0,24Yt−1+ 1) = 0,07✱
❡♠ q✉❡X0=Y0= 1❡t≥1✳✑
✭▼✐❝❤❛❡❧ ❇❛r♥s❧❡②✮
❨❛s✉❦♦ ❡ ▼❛s❛r✉
❛ ❆♥♥✐❜❛❧ ❉✐❛s ❞❡ ❋✐❣✉❡✐r❡❞♦ ◆❡t♦✱ ▼❛r❝♦ ❆♥tô♥✐♦ ❆♠❛t♦✱ ❚❛r❝ís✐♦ ▼❛r❝✐❛♥♦ ❞❛ ❘♦❝❤❛ ❋✐❧❤♦✱ ❆♥t♦♥② ▼❛r❝♦ ▼♦t❛ P♦❧✐t♦✱ ❆♠✐❧❝❛r ❘❛❜❡❧♦ ❞❡ ◗✉❡✐r♦③✱ ❋á❜✐♦ ▼❛❝ê❞♦ ▼❡♥❞❡s ❡ ❆❞❡♠✐r ❊✉❣ê♥✐♦ ❞❡ ❙❛♥t❛♥❛❀ ❛♦s ♣r❡③❛❞♦s ♣r♦❢❡ss♦r❡s ■r❛♠ ▼❛r❝❡❧♦ ●❧ér✐❛ ✭■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛✱ ❯❋❆▲✮✱ ❊r❛❧❞♦ ❙ér❣✐♦ ❇❛r❜♦s❛ ❉❛ ❙✐❧✈❛ ✭❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊❝♦♥♦♠✐❛✱ ❯❋❙❈✮ ❡ P✉s❤♣❛ ◆❛r❛②❛♥ ❘❛t❤✐❡ ✭❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛✱ ❯♥❇✮❀ ❛♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛r✲ t❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥❇❀ ❡ ❛♦s ❝♦❧❡❣❛s ❆♥❞ré ❚❡❧❧❡s✱ ❘❡❣✐♥❛ ❋♦♥s❡❝❛ ❡ ▼ár❝✐♦ ❞❡ ❈❛str♦✳
❘❡s✉♠♦
❙✐st❡♠❛s ❝♦♠♣❧❡①♦s s♦❜ r❡❣✐♠❡ ❞✐❢✉s✐✈♦ ❛♥ô♠❛❧♦ ♣♦❞❡♠ s❡r ❞❡s❝r✐t♦s ♣♦r ❞✐str✐❜✉✐çõ❡s tr✉♥❝❛❞❛s ❞❡ ▲é✈②✳ Pr♦❜❧❡♠❛s ❞❡ ✐♥❢❡rê♥❝✐❛ ❡st❛tíst✐❝❛ ♥❡ss❡ ❛♠❜✐❡♥t❡ ♥ã♦ ❣❛✉ss✐❛♥♦ ♣♦✲ ❞❡♠ s❡r ❛❜♦r❞❛❞♦s ✈✐❛ tr❛♥s❢♦r♠❛❞❛s ❞❡ ❋♦✉r✐❡r✱ ❝♦♠♦ ❛s ❢✉♥çõ❡s ❝❛r❛❝t❡ríst✐❝❛s✳ ❊st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ ✉♠❛ ❡①♣❛♥sã♦ ❛❧t❡r♥❛t✐✈❛ ❞❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ q✉❡ s❡ ♠♦str♦✉ út✐❧ ♣❛r❛ ❛ ❡st✐♠❛çã♦ ♣♦r ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞♦s ♣❛râ♠❡tr♦s ❞❛s ❞✐str✐❜✉✐çõ❡s s♦❜ ❛ ❤✐✲ ♣ót❡s❡ ❞❡ ❡st❛❜✐❧✐❞❛❞❡✳ P❛r❛ ✐❧✉str❛r✱ ❝♦♥s✐❞❡r❛♠♦s ❛s sér✐❡s t❡♠♣♦r❛✐s ❞♦ í♥❞✐❝❡ ❞❛ ❇♦❧s❛ ❞❡ ❱❛❧♦r❡s ❞❡ ❙ã♦ P❛✉❧♦✱ ❞♦ í♥❞✐❝❡ ❉♦✇ ❏♦♥❡s ■♥❞✉str✐❛❧ ❆✈❡r❛❣❡ ❞❛ ❇♦❧s❛ ❞❡ ❱❛❧♦r❡s ❞❡ ◆♦✈❛ ■♦rq✉❡ ✭◆❨❙❊✮ ✖ ❝♦♥t❡♠♣❧❛♥❞♦ ♦ ❡✈❡♥t♦ ❞❡♥♦♠✐♥❛❞♦ ✢❛s❤ ❝r❛s❤ ♦❝♦rr✐❞♦ ❡♠ ✻ ❞❡ ♠❛✐♦ ❞❡ ✷✵✶✵ ✖✱ ❞❛s t❛①❛s ❞❡ ❝â♠❜✐♦ ❞❛s ♣r✐♥❝✐♣❛✐s ♠♦❡❞❛s ❢r❡♥t❡ ❛♦ ❞ó❧❛r ♥♦rt❡ ❛♠❡r✐✲ ❝❛♥♦✱ ❡ ❞♦s ♣r❡ç♦s ❞❡ ❛❧❣✉♠❛s ❛çõ❡s ♥❡❣♦❝✐❛❞❛s ♥❛ ◆❨❙❊ q✉❡ s♦❢r❡r❛♠ ♠✐♥✐✲✢❛s❤ ❝r❛s❤❡s ❡♠ ✷✵✶✶✳ ❊♠ ❣❡r❛❧✱ ❡ss❡s ❞❛❞♦s ♣♦❞❡♠ s❡r ♠♦❞❡❧❛❞♦s ♣♦r ❞✐str✐❜✉✐çõ❡s tr✉♥❝❛❞❛s✱ ❡ ❛ ❧❡♥t✐❞ã♦ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ss❡s ♣r♦❝❡ss♦s ♣❛r❛ ❛ ❣❛✉ss✐❛♥❛ s❡ ❡①♣❧✐❝❛ ♣❡❧❛ ❞❡♣❡♥❞ê♥❝✐❛ s❡r✐❛❧ ❞❡ ❝✉rt♦ ❡ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡✳ ❖❜s❡r✈❛♠♦s t❛♠❜é♠ q✉❡ ❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❡♠✲ ♣ír✐❝❛ s♦❢r❡ tr✉♥❝❛♠❡♥t♦ ❞❡✈✐❞♦ à ✜♥✐t✉❞❡ ❞❛ ❛♠♦str❛✱ ❤❛✈❡♥❞♦ q✉❡❜r❛ ❞❡ s❝❛❧✐♥❣ s❡♠♣r❡ ♥♦ ♠❡s♠♦ ♣❛t❛♠❛r✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞❛ ❢♦r♠❛ ❞❛ ❞✐str✐❜✉✐çã♦ ❞♦s ❞❛❞♦s✳ ❋✐♥❛❧♠❡♥t❡✱ ✐♥tr♦❞✉③✐♠♦s ✉♠ ♥♦✈♦ ♠ét♦❞♦ ❛ss✐♥tót✐❝♦ q✉❡ ♣❡r♠✐t❡ t❡st❛r ❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s ❞❡ ❞❛❞♦s✳ ◆♦ss♦ t❡st❡ é ❞♦ t✐♣♦ ❈r❛♠ér✲✈♦♥ ▼✐s❡s✱ ❡♠ q✉❡ ♦ ♣r♦❝❡ss♦ ❡♠♣ír✐❝♦ é ♦❜t✐❞♦ ❝♦♠ ❜❛s❡ ♥❛ ❞✐✈❡r❣ê♥❝✐❛ ❞❡ ❑✉❧❧❜❛❝❦✲▲❡✐❜❧❡r✱ ❡ s❡ ♠♦str♦✉ ❡st❛t✐st✐❝❛✲ ♠❡♥t❡ ♣♦❞❡r♦s♦ ♣❛r❛ ❞❡t❡❝t❛r ❞❡♣❡♥❞ê♥❝✐❛ ♥ã♦ ❧✐♥❡❛r ❢♦r❛ ❞♦ ❛♠❜✐❡♥t❡ ❣❛✉ss✐❛♥♦✳
❆❜str❛❝t
❈♦♠♣❧❡① s②st❡♠s ✉♥❞❡r ❛♥♦♠❛❧♦✉s ❞✐✛✉s✐✈❡ r❡❣✐♠❡ ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❧② ❞❡s❝r✐❜❡❞ ❜② tr✉♥❝❛t❡❞ ▲é✈② ✢✐❣❤ts✳ ▼❛♥② ❞✐✣❝✉❧t st❛t✐st✐❝❛❧ ✐ss✉❡s ✐♥ t❤✐s ♥♦♥✲●❛✉ss✐❛♥ ❡♥✈✐r♦♥♠❡♥t ❝❛♥ ❜❡ ❛♠❡♥❛❜❧❡ t♦ s♦❧✉t✐♦♥ ❜② t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♠❡t❤♦❞s✱ ❛s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝✲ t✐♦♥s✳ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ♣✉t ❢♦r✇❛r❞ ❛♥ ❛❧t❡r♥❛t✐✈❡ ❡①♣❛♥s✐♦♥ ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ♣r♦✈❡❞ ✉s❡❢✉❧ ❢♦r t❤❡ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs ✉♥❞❡r t❤❡ st❛❜✐❧✐t② ❤②♣♦t❤❡s✐s✳ ❖✉r ❛♣♣r♦❛❝❤ ✐s ❡①❡♠♣❧✐✜❡❞ ✇✐t❤ t❤❡ ❙❛♦ P❛✉❧♦ ❙t♦❝❦ ❊①❝❤❛♥❣❡ ✐♥❞❡① t✐♠❡ s❡r✐❡s✱ t❤❡ ❤✐❣❤✲❢r❡q✉❡♥❝② ❞❛t❛ ❢r♦♠ t❤❡ ❉♦✇ ❏♦♥❡s ■♥❞✉str✐❛❧ ❆✈❡r❛❣❡ ✐♥❞❡① ✖ ✇❤✐❝❤ ❡♥❝♦♠♣❛ss t❤❡ r❡❝❡♥t ❡♣✐s♦❞❡ ❦♥♦✇♥ ❛s t❤❡ ✢❛s❤ ❝r❛s❤ ♦❢ ▼❛② ✻✱ ✷✵✶✵ ✖✱ t❤❡ ❢♦r❡✐❣♥ ❡①❝❤❛♥❣❡ r❛t❡ ❞❛t❛✱ ❛♥❞ t❤❡ ❤✐❣❤✲❢r❡q✉❡♥❝② ❞❛t❛ ❢r♦♠ st♦❝❦s ❧✐st❡❞ ♦♥ t❤❡ ◆❨❙❊ t❤❛t r❡❝❡♥t❧② ❡①♣❡r✐❡♥❝❡❞ s♦✲❝❛❧❧❡❞ ♠✐♥✐✲✢❛s❤ ❝r❛s❤❡s✳ ❲❡ ❝♦♥✜r♠ t❤❛t t❤❡ s❧✉❣❣✐s❤ ❝♦♥✲ ✈❡r❣❡♥❝❡ ♦❢ t❤❡ tr✉♥❝❛t❡❞ ▲é✈② ✢✐❣❤ts t♦ ❛ ●❛✉ss✐❛♥ ❝❛♥ ❜❡ ❡①♣❧❛✐♥❡❞ ❜② t❤❡ ♣r❡s❡♥❝❡ ♦❢ s❤♦rt r❛♥❣❡ ❛♥❞ ❧♦♥❣ r❛♥❣❡ s❡r✐❛❧ ❞❡♣❡♥❞❡♥❝❡ ✐♥ t❤❡s❡ ❞❛t❛✳ ❲❡ ❛❧s♦ ✐♥✈❡st✐❣❛t❡❞ t❤❡ tr✉♥❝❛t✐♦♥ ♣❤❡♥♦♠❡♥♦♥ ♦❢ t❤❡ ❡♠♣✐r✐❝❛❧ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ✭❊❈❋✮ ❞✉❡ t♦ t❤❡ s❛♠♣❧❡ ✜♥✐t✉❞❡✳ ❘❡❣❛r❞❧❡ss ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ s❤❛♣❡✱ t❤❡ ❊❈❋ s❝❛❧✐♥❣ ❜r❡❛❦s ❞♦✇♥ ❛❧✇❛②s ❛t t❤❡ s❛♠❡ ❧❡✈❡❧✱ ❞❡♣❡♥❞✐♥❣ ♦♥❧② ♦♥ t❤❡ s❛♠♣❧❡ s✐③❡✳ ❋✐♥❛❧❧②✱ ✇❡ ❞❡✈✐s❡ ❛ ♥♦✈❡❧ ❛s②♠♣t♦t✐❝ st❛t✐st✐❝❛❧ t❡st t♦ ❛ss❡ss ✐♥❞❡♣❡♥❞❡♥❝❡ ✐♥ ❜✐✈❛r✐❛t❡ ❞❛t❛ s❡t✳ ❖✉r ❛♣♣r♦❛❝❤ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s t❡st✱ ❛♥❞ ♣r♦✈❡❞ ❛❜❧❡ t♦ ❞❡t❡❝t ♥♦♥❧✐♥❡❛r ❞❡♣❡♥❞❡♥❝❡ ❡✈❡♥ ✐❢ t❤❡ ❡♥✈✐r♦♥♠❡♥t ✐s ♥♦♥✲●❛✉ss✐❛♥✳
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶
✶✳✶ ❈♦♥s✐❞❡r❛çõ❡s ✐♥✐❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❯♠❛ ❜r❡✈❡ r❡tr♦s♣❡❝t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✸ ❖❜❥❡t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✹ ❉❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✹✳✶ ❖ ■❇♦✈❡s♣❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✹✳✷ ❖ ❮♥❞✐❝❡ ❉❏■❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✹✳✸ Pr❡ç♦s ❞❡ ❛❧❣✉♠❛s ❛çõ❡s ♥❡❣♦❝✐❛❞❛s ♥❛ ◆❨❙❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✹✳✹ ❚❛①❛s ❞❡ ❝â♠❜✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✺ ❊s❜♦ç♦ ❞♦ tr❛❜❛❧❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✷ ❆ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❡ ❛s ❞✐stâ♥❝✐❛s ❡♥tr❡ ❞✐str✐❜✉✐çõ❡s ✷✸ ✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷ ❆ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷✳✶ Pr♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷✳✷ ❯♠❛ ❡①♣❛♥sã♦ ❛❧t❡r♥❛t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷✳✸ ❉✐str✐❜✉✐çõ❡s s✐♠étr✐❝❛s ❡♠ t♦r♥♦ ❞❡ ③❡r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷✳✹ ❘❡❧❛çõ❡s ❝♦♠ r❡s♣❡✐t♦ ❛♦ ✈❡t♦r ❞❡ ♣❛râ♠❡tr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✸ ❉✐stâ♥❝✐❛s ❡♥tr❡ ❞✉❛s ❞✐str✐❜✉✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸✳✶ ❆ ❞✐stâ♥❝✐❛L2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸✳✷ ❆ ❞✐✈❡r❣ê♥❝✐❛ ❞❡ ❑✉❧❧❜❛❝❦✲▲❡✐❜❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✷✳✸✳✹ ❘❡❧❛çã♦ ❝♦♠ ❛ ♠❡❞✐❞❛ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❋✐s❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✹ ❈♦♥s✐❞❡r❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✸ ❆s ❞✐str✐❜✉✐çõ❡s ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡✐s ❡ ❛s ❡stá✈❡✐s ✸✼ ✸✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✷ ❉✐str✐❜✉✐çõ❡s ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✸ ❖ ♣r♦❝❡ss♦ ❞❡ ▲é✈② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✹ ❆ ❞✐str✐❜✉✐çã♦ ❡stá✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✹✳✶ ❖ ♣r♦❝❡ss♦ ❞❡ ▲é✈② ❡stá✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✺ ❖ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✻ ❈♦♥s✐❞❡r❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✹ ❆ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❡♠♣ír✐❝❛ ✺✾
✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✹✳✷ ❉❡✜♥✐çã♦ ❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❋❈❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✹✳✸ ❖ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ❡♠♣ír✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✹✳✹ ❆ ❋❈❊ tr✉♥❝❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✹✳✺ ❆ ❋❈❊ ❞❡ ✉♠❛ s♦♠❛ ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✹✳✻ ❊st✉❞♦ ♣♦r s✐♠✉❧❛çõ❡s ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✹✳✼ ■❧✉str❛çã♦✿ ❞❛❞♦s ❞♦ ■❇♦✈❡s♣❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✹✳✽ ❈♦♥s✐❞❡r❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼
✺ ❊st✐♠❛çã♦ ♣♦r ❢✉♥çõ❡s ❝❛r❛❝t❡ríst✐❝❛s ✽✺
✺✳✺ ❈♦♥s✐❞❡r❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶
✻ ❚❡st❡ ❞❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ✶✶✶
✻✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✶ ✻✳✷ ❆ ❋❈ ♠✉❧t✐✈❛r✐❛❞❛ ❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✸ ✻✳✸ ❖ t❡st❡ ❞❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✹ ✻✳✹ ❱❛❧♦r❡s ❝rít✐❝♦s ❛ss✐♥tót✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾ ✻✳✺ ❱❛❧✐❞❛çã♦ ❡ ♦ ♣♦❞❡r ❞♦ t❡st❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✵ ✻✳✻ ■❧✉str❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✸ ✻✳✻✳✶ ❆çõ❡s ❞❛ ❜♦❧s❛ ❞❡ ◆♦✈❛ ■♦rq✉❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✸ ✻✳✻✳✷ ❚❛①❛s ❞❡ ❝â♠❜✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✹ ✻✳✼ ❉✐s❝✉ssã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✻
✼ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✶✷✾
✼✳✶ P❡rs♣❡❝t✐✈❛s ♣❛r❛ tr❛❜❛❧❤♦s ❢✉t✉r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✸ ✼✳✶✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❡♠ sér✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✸ ✼✳✶✳✷ ❊st✉❞♦ ❞❛ ♦r✐❣❡♠ ❞♦ ❛❣r✉♣❛♠❡♥t♦ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡s ❡ ❞❛s ❝♦rr❡❧❛çõ❡s
❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✺ ✼✳✶✳✸ ❆ ❋❈❊ ♠✉❧t✐✈❛r✐❛❞❛ ❡ ♦✉tr❛s ♠❡❞✐❞❛s ❞❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✼ ✼✳✶✳✹ ❊①t❡♥sõ❡s ❞♦ t❡st❡ ❞❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✽
❆ ❆❞❞❡♥❞✉♠ ♠❛t❡♠át✐❝♦ ✶✹✶
❆✳✶ ■♥t❡❣r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✶ ❆✳✷ ❋✉♥çã♦ ❣❛♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✷ ❆✳✸ ❉❡r✐✈❛❞❛s ❞❛ ❢✉♥çã♦ ❞❡❧t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✸ ❆✳✹ ❈♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✸
❇ ❯♠❛ r❡❧❛çã♦ ❜✐♥♦♠✐❛❧ ❞❛ ❢✉♥çã♦ ❡s❝♦r❡ ✶✹✺
❊ ❙✐♠✉❧❛çã♦ ❞❡ ✉♠❛ ❱❆ ❡stá✈❡❧ ✶✺✺
❋ ❉❡t❛❧❤❛♠❡♥t♦ ❞♦ ❈❛♣✳ ✻ ✶✺✼
❋✳✶ ❋♦r♠❛ ❣❡r❛❧ ❞❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❞❡B ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✼
❆❜r❡✈✐❛çõ❡s ❡ ❙✐❣❧❛s
❆❆❙ ❛♠♦str❛ ❛❧❡❛tór✐❛ s✐♠♣❧❡s ❈♦✈ ❝♦✈❛r✐â♥❝✐❛
❈❱ ✈❛❧✐❞❛çã♦ ❝r✉③❛❞❛
❉❏■❆ ✭í♥❞✐❝❡✮ ❉♦✇ ❏♦♥❡s ■♥❞✉str✐❛❧ ❆✈❡r❛❣❡ ❋❈ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛
❋❈❊ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❡♠♣ír✐❝❛ ❋❉ ❢✉♥çã♦ ❞❡ ❞❡♥s✐❞❛❞❡
❋❉❆ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❛❝✉♠✉❧❛❞❛
H0 ❤✐♣ót❡s❡ ♥✉❧❛
H1 ❤✐♣ót❡s❡ ❛❧t❡r♥❛t✐✈❛
❍❇❑❘ ✭t❡st❡ ❞❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❞❡✮ ❍♦❡✛❞✐♥❣✱ ❇❧✉♠✱ ❑✐❡❢❡r ❡ ❘♦s❡♥❜❧❛tt ■❇♦✈❡s♣❛ í♥❞✐❝❡ ❞❛ ❇♦❧s❛ ❞❡ ❱❛❧♦r❡s ❞❡ ❙ã♦ P❛✉❧♦
■■❉ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐str✐❜✉í❞❛s ▼▼❱ ♠ét♦❞♦ ❞❛ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛
▼❙❊ ♠❡❛♥ sq✉❛r❡❞ ❡rr♦rs ▼❱ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ◆❨❙❊ ◆❡✇ ❨♦r❦ ❙t♦❝❦ ❊①❝❤❛♥❣❡
P✲✈❛❧✉❡ ♥í✈❡❧ ❞❡s❝r✐t✐✈♦ ❞❡ ✉♠ t❡st❡ ❞❡ ❤✐♣ót❡s❡s ❚▲❋ tr✉♥❝❛t❡❞ ▲é✈② ✢✐❣❤ts
❱❆ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❱❛r ✈❛r✐â♥❝✐❛
▲✐st❛ ❞❡ ❙í♠❜♦❧♦s ❡ ◆♦t❛çõ❡s
bk ❝♦❡✜❝✐❡♥t❡✱ ❊q✳ ✭✷✳✸✵✮ ck ❝♦❡✜❝✐❡♥t❡✱ ❊q✳ ✭✷✳✶✷✮
dk ❝♦❡✜❝✐❡♥t❡✱ ❊q✳ ✭✺✳✶✼✮
ˆ
ck ❡st✐♠❛t✐✈❛ ❞❡ ck
f(x;θ) ❢✉♥çã♦ ❞❡ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♥♦ ♣♦♥t♦x ♣❛r❛♠❡tr✐③❛❞❛ ♣♦r θ
f =f(0;θ)
f(k)(x;θ) = dk
dxkf(x;θ)
f(0)(x;θ) =f(x;θ)
f(k) =f(k)(0;θ) = dk
dxkf(x;θ)
x=0
ˆ
f(k) ❡st✐♠❛t✐✈❛ ❞❡ f(k)
gj(x;θ) = dθdjf(x;θ)✱ ❡♠ q✉❡ θj ∈θ
g(jk) =g(jk)(0;θ)
hj(q;θ) = dθdjφX(q;θ)
hj,2(q;θ) ♣❛rt❡ r❡❛❧ ❞❡hj(q;θ)
hj,1(q;θ) ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❞❡ hj(q;θ)
i ✉♥✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛✱ i2 =−1
i, j, k, l, s, t í♥❞✐❝❡s
n t❛♠❛♥❤♦ ❞❡ ✉♠❛ ❛♠♦str❛
p ❞✐♠❡♥sã♦ ❞❡θ
q ♣♦♥t♦ ❞♦ s✉♣♦rt❡ ❞❡ ✉♠❛ ❋❈ sj(x;θ) ❢✉♥çã♦ ❡s❝♦r❡✱ ✭✷✳✹✶✮
u, v, x, y, z ♣♦ssí✈❡✐s r❡❛❧✐③❛çõ❡s ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s w∆t(q) ✈❡r ❊q✳ ✭✶✳✼✮
z ❝♦♠♣❧❡①♦ ❝♦♥❥✉❣❛❞♦ ❞❡ ✉♠ ♥ú♠❡r♦z ∈C
zns q✉❛♥t✐❧ r❡❧❛t✐✈♦ ❛♦ ♥í✈❡❧ ❞❡ s✐❣♥✐✜❝â♥❝✐❛ ♥s✱ ✐✳❡✳✱ns = P(|Z|> zns)
A1,k ✈❡r ❊q✳ ✭✸✳✹✾✮
A2,k ✈❡r ❊q✳ ✭✸✳✺✵✮
B2,0 ❢❛t♦r ❞❡ ✐♥✢❛çã♦✱ ❊q✳ ✭✸✳✻✵✮
B ❡st❛tíst✐❝❛ ❞♦ t❡st❡ ❞❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ C+ ❡ C− ❝♦❡✜❝✐❡♥t❡s ❞❡ ❛ss✐♠❡tr✐❛
Dk
q ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ❝♦♠ r❡s♣❡✐t♦ ❛q✱Dkq = d
k
dqk
DKL ❞✐stâ♥❝✐❛ ❞❡ ❑✉❧❧❜❛❝❦✲▲❡✐❜❧❡r✱ ❊q✳ ✭✷✳✸✼✮
F(x;θ) ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❛❝✉♠✉❧❛❞❛
ˆ
F(x) ❡st✐♠❛t✐✈❛ ❞❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❛❝✉♠✉❧❛❞❛
Fn(x) ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❛❝✉♠✉❧❛❞❛ ❡♠♣ír✐❝❛
H ❡①♣♦❡♥t❡ ❞❡ ❍✉rst H(·) ❡♥tr♦♣✐❛
I(·) ❢✉♥çã♦ ✐♥❞✐❝❛❞♦r❛
IF(θ) ♠❡❞✐❞❛ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❋✐s❤❡r
K(u) ❢✉♥çã♦ ❑❡r♥❡❧
L2 ❞✐stâ♥❝✐❛ L2✱ ❊q✳ ✭✷✳✸✺✮
M(u) ❊q✳ ✭✸✳✸✮ P(·) ♣r♦❜❛❜✐❧✐❞❛❞❡
Q ✐♥t❡r✈❛❧♦ ♦✉ ❢❛✐①❛ ❞❡ ♦♣❡r❛çã♦ ❞❛ ❋❈❊ Rj(q;θ) ✈❡r ❊q✳✭✷✳✺✹✮
S∆t =X1+· · ·+X∆t✱ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦
SR
∆t ♣❛ss❡✐♦ ❛❧❡❛t♦r✐③❛❞♦ ❊q✳ ✭✹✳✹✽✮
S0
∆t ♣❛ss❡✐♦ ♥ã♦ ❛❧❡❛t♦r✐③❛❞♦ ❊q✳ ✭✹✳✹✾✮
Wt ✈❛❧♦r ❞❡ ✉♠ í♥❞✐❝❡ ✭♦✉ ✐♥❞✐❝❛❞♦r ♦✉ ♣r❡ç♦ ❞❡ ✉♠ ❛t✐✈♦✮ ♥♦ ✐♥st❛♥t❡t
Xt =Rt−µ✱ r❡t♦r♥♦ ❝❡♥tr❛❞♦
X, Y, Z ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✭❧❡tr❛s ♠❛✐ús❝✉❧❛s✮ α í♥❞✐❝❡ ❞❡ ❡st❛❜✐❧✐❞❛❞❡✱0< α≤2
β ♣❛râ♠❡tr♦ ❞❡ ❛ss✐♠❡tr✐❛✱ |β| ≤1
βα =βtanπα2 ✭❛ss✐♠❡tr✐❛ ❡❢❡t✐✈❛✮
δ(·) ❢✉♥çã♦ ❞❡❧t❛ ❞❡ ❉✐r❛❝
φ(q;θ) ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ✭❋❈✮✱ ❊q✳ ✭✷✳✶✮
ˆ
φ(q) ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❡♠♣ír✐❝❛ ✭❋❈❊✮
φ2(q;θ) ♣❛rt❡ r❡❛❧ ❞❛ ❋❈
φ1(q;θ) ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❞❛ ❋❈
φ0,n ♣♦♥t♦ ❝rít✐❝♦ ❞❛ ❋❈❊
ˆ
φ2(q) ♣❛rt❡ r❡❛❧ ❞❛ ❋❈❊
ˆ
φ1(q) ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❞❛ ❋❈❊
ˆ
φ∗(q) ❋❈❊ tr✉♥❝❛❞❛
φ(k)(q;θ) k✲és✐♠❛ ❞❡r✐✈❛❞❛ ❞❛ ❋❈ ❝♦♠ r❡s♣❡✐t♦ ❛ q
ϕ(q;θ) = lnφ(q;θ)
γ ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛✱γ >0
γ∗ =γ·p1 +β2
α
γ∗,0 ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛ ✐♥✢❛❝✐♦♥❛❞♦✱ ❊q✳ ✭✸✳✻✵✮
η ❝♦♥st❛♥t❡ ❞❡ ♥♦r♠❛❧✐③❛çã♦ λj j✲és✐♠♦ ❛✉t♦✈❛❧♦r
µ ♣❛râ♠❡tr♦ ❞❡ ❧♦❝❛çã♦✱ µ∈R
θ ✈❡t♦r ❞❡ ♣❛râ♠❡tr♦s✱ θ∈Rp
ˆ
θ ❡st✐♠❛t✐✈❛ ❞❡ θ
θj ❡❧❡♠❡♥t♦ ❞♦ ✈❡t♦r θ
ϑ(u) ♠❡❞✐❞❛ ❞❡ ▲é✈②✱ ❊q✳ ✭✸✳✸✮
ρ ❝♦rr❡❧❛çã♦ ❧✐♥❡❛r ❞❡ P❡❛rs♦♥
σ =γα1
ωj(q;θ) tr❛♥s❢♦r♠❛❞❛ ✐♥✈❡rs❛ ❞♦ ❡s❝♦r❡✱ ✭✷✳✹✹✮
ζ ♣♦♥t♦ ❞❡ tr✉♥❝❛♠❡♥t♦ ❞❛ ❚▲❋
∆t t❛♠❛♥❤♦ ❞♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦
∆(q;θ) ❊qs✳ ✭✹✳✶✼✮ ❡ ✭✹✳✶✽✮
Φ(Dq;θ) ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦✱ ❊q✳ ✭✷✳✶✻✮
Φ2(Dq;θ) ♣❛rt❡ r❡❛❧ ❞♦ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦
Φ1(Dq;θ) ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❞♦ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦
ˆ
Φ(Dq) ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ❡♠♣ír✐❝♦
Γ(·) ❢✉♥çã♦ ❣❛♠♠❛
Σ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛s Ψ(Dq;θ) ❊q✳ ✭✺✳✶✻✮
hXi ✈❛❧♦r ❡s♣❡r❛❞♦ ❞❡X
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
✶✳✶ ❈♦♥s✐❞❡r❛çõ❡s ✐♥✐❝✐❛✐s
❯♠ s✐st❡♠❛ ❡❝♦♥ô♠✐❝♦ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡①♦ ❛❜❡rt♦✱ ❡♠ q✉❡ ❤á ✐♥ú♠❡r❛s ❢♦r♠❛s ❞❡ ✐♥t❡r❛çã♦ ❡♥tr❡ s❡✉s ❝♦♠♣♦♥❡♥t❡s ❬✶✼✱ ✶✶✹❪✳ ❆ ❞✐♥â♠✐❝❛ q✉❡ r❡❣❡ ❡ss❡ s✐st❡♠❛ ❛✐♥❞❛ ♥ã♦ é ❝♦♠♣❧❡t❛♠❡♥t❡ ❝♦♥❤❡❝✐❞❛✱ ♦ q✉❡ ❛tr❛✐ ♠✉✐t♦s ♣❡sq✉✐s❛❞♦r❡s ♣❛r❛ ♦ ❞❡s❛✜♦ ❞❡ ❞❡s✈❡♥❞á✲❧❛ ❛♦s ♣♦✉❝♦s ♠❡❞✐❛♥t❡ ❡st✉❞♦s ❡♠♣ír✐❝♦s✳ ❊♠ ✜♥❛♥ç❛s✱ ❛s r❡❣✉❧❛r✐❞❛❞❡s ❡st❛tíst✐❝❛s ♦❜s❡r✈❛❞❛s ❡♠♣✐r✐❝❛♠❡♥t❡ ❡♠ sér✐❡s t❡♠♣♦r❛✐s ❞❡ r❡t♦r♥♦s ✜✲ ♥❛♥❝❡✐r♦s ❞❡♥♦♠✐♥❛♠✲s❡ ❢❛t♦s ❡st✐❧✐③❛❞♦s ❬✸✽✱ ✶✵✺✱ ✶✶✾❪✳ ❆ ♥ã♦ ❣❛✉ss✐❛♥✐❞❛❞❡ ❡ ❛ ♣r❡s❡♥ç❛ ❞❡ ❛❣r✉♣❛♠❡♥t♦s ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡s✱ ♣♦r ❡①❡♠♣❧♦✱ s❡ ❡♥❝♦♥tr❛♠ ❡♥tr❡ ♦s ❢❛t♦s ♠❛✐s ❝♦♥❤❡✲ ❝✐❞♦s✳ ❆ ♣❛rt✐r ❞❡ss❛s ♦❜s❡r✈❛çõ❡s ❡♠♣ír✐❝❛s✱ ♠♦❞❡❧♦s t❡ór✐❝♦s ♣♦❞❡♠ s❡r s✉❣❡r✐❞♦s ♣❛r❛ s❡ ❞❡s❝r❡✈❡r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ss❡ s✐st❡♠❛ ❬✶✼✱ ✼✺✱ ✶✶✹✱ ✶✶✽❪ ✳
◆❛ ❞é❝❛❞❛ ❞❡ ✶✾✻✵✱ ❇✳ ▼❛♥❞❡❧❜r♦t ♦❜s❡r✈♦✉ q✉❡ ❛s ❞✐str✐❜✉✐çõ❡s ❞❛s ✈❛r✐❛çõ❡s ❞❡ ♣r❡ç♦s ✭❝♦♠♦ ❛ ❞♦ ❛❧❣♦❞ã♦✮ ♥ã♦ s❡ ❛❥✉st❛✈❛♠ ❛ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❣❛✉ss✐❛♥❛✱ ♣♦✐s ❡❧❛s ❛♣r❡s❡♥t❛✈❛♠ ❡①❝❡ss♦ ❞❡ ❝✉rt♦s❡ ❡ ❝❛✉❞❛s ♠❛✐s ♣❡s❛❞❛s ❬✻✾✱ ✼✵✱ ✼✶✱ ✼✷❪✳ ❊❧❡ t❛♠❜é♠ ♥♦t♦✉ q✉❡ ❛ ❞✐str✐❜✉✐çã♦ X ❞❛s ✈❛r✐❛çõ❡s ❞✐ár✐❛s s❡ r❡❧❛❝✐♦♥❛✈❛ ❝♦♠ ❛ ❞❛s ✈❛r✐❛çõ❡s ♠❡♥s❛✐s
♠❡❞✐❛♥t❡ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❡s❝❛❧❛✳ ❖✉ s❡❥❛✱ X s❡❣✉✐❛ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✉♠❛ ❧❡✐ ❞❡
♣♦tê♥❝✐❛ ✭♣♦✇❡r ❧❛✇✮ ♥❛ ❢♦r♠❛ f(γ−1/αx) = γ1/αf(x)✱ ❡♠ q✉❡ f(x) r❡♣r❡s❡♥t❛ ❛ ❢✉♥çã♦
é ♦ í♥❞✐❝❡ ❞❡ ❡st❛❜✐❧✐❞❛❞❡✳ ❆ss✐♠✱ ▼❛♥❞❡❧❜r♦t ♦❜s❡r✈♦✉ q✉❡ ❛s ❞✐str✐❜✉✐çõ❡s ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡✐s ❡ ❡stá✈❡✐s ❡r❛♠ ♣♦ssí✈❡✐s ♠♦❞❡❧♦s ❝❛♥❞✐❞❛t♦s ♣❛r❛ ❞❡s❝r❡✈❡r s✉❛s ❞❡s❝♦❜❡rt❛s✳
❯♠❛ ❞✐str✐❜✉✐çã♦ X é ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡❧ s❡✱ ♣❛r❛ q✉❛❧q✉❡r n ≥ 1✱ ❡①✐st✐r ✉♠❛
❞✐str✐❜✉✐çã♦Xnt❛❧ q✉❡Xé ❛ ❝♦♥✈♦❧✉çã♦ ❞❡n❝ó♣✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡Xn❬✹✼✱ ✹✽✱ ✺✼✱ ✶✵✸✱
✶✵✹❪✳ ❊✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ✉♠❛ ❞✐str✐❜✉✐çã♦ ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡❧ X é ❡stá✈❡❧ s❡ ✖ ❛ ♠❡♥♦s
❞❡ ✉♠ ♣❛râ♠❡tr♦ ❞❡ ❧♦❝❛çã♦ µ ∈ R ❡ ❞❡ ❡s❝❛❧❛ γ > 0 ✖ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞✐str✐❜✉❝✐♦♥❛✐s
sã♦ ♣r❡s❡r✈❛❞❛s ❛♣ós ❝♦♥✈♦❧✉çõ❡s ❞❡ ❝ó♣✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡ X ❬✸✱ ✻✹✱ ✶✵✸✱ ✶✵✹❪✳ P♦r
❡①❡♠♣❧♦✱ s❡ X1 ❡ X2 sã♦ ❝ó♣✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❡stá✈❡❧X✱ ❡♥tã♦
X s❡ r❡❧❛❝✐♦♥❛ ❝♦♠ s✉❛s ❝ó♣✐❛s X1 ❡ X2 ♠❡❞✐❛♥t❡ ✉♠❛ ❝♦♥✈♦❧✉çã♦ ♥❛ ❢♦r♠❛ γX+µ =
γ1X1+γ2X2✱ ❡♠ q✉❡γ1, γ2 >0 t❛♠❜é♠ sã♦ ♣❛râ♠❡tr♦s ❞❡ ❡s❝❛❧❛✳
❆s ❝❛✉❞❛s ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡stá✈❡❧ s❡❣✉❡♠ ✉♠❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ♥❛ ❢♦r♠❛ f(|x|)∝
|x|−(α+1) ✭0 < α ≤ 2✮✱ ❡✱ ❛❧é♠ ❞✐ss♦✱ h|X|qi = ∞✱ s❡ q ≥ α✱ ❡♥q✉❛♥t♦ h|X|qi < ∞✱ s❡
q < α ❬✷✱ ✹✽✱ ✻✶✱ ✶✵✸❪✳ ❆ss✐♠✱ ✉♠ ❢❡♥ô♠❡♥♦ ❞❡s❝r✐t♦ ♣♦r ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡stá✈❡❧ ❝♦♠ α < 2 ♥ã♦ ♣♦ss✉✐ ❡s❝❛❧❛ ❝❛r❛❝t❡ríst✐❝❛ ♥❡♠ s❡❣✉♥❞♦ ♠♦♠❡♥t♦❀ ❡✱ s❡ α < 1✱ t❛♠♣♦✉❝♦
❛ ♠é❞✐❛ ❡①✐st❡✳ ❉❡ss❡ ♠♦❞♦✱ ❛ t❡♦r✐❛ ❞❛s ❞✐str✐❜✉✐çõ❡s ❡stá✈❡✐s✱ ✐♥tr♦❞✉③✐❞❛ ❡♥tr❡ ✶✾✷✹ ❡ ✶✾✸✻ ♣♦r P✳ ▲é✈② ❡ ❆✳ ❑❤✐♥❝❤✐♥❡ ❬✻✹✱ ✶✵✸✱ ✶✵✹❪✱ r❡♠❡t❡ ♥❛t✉r❛❧♠❡♥t❡ ❛ ✉♠ t❡♦r❡♠❛ ❧✐♠✐t❡ ❝❡♥tr❛❧ ❣❡♥❡r❛❧✐③❛❞♦✱ ❥á q✉❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡stá✈❡❧ s❡ r❡❧❛❝✐♦♥❛ ❝♦♠ ✉♠❛ s♦♠❛ ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♠ ✈❛r✐â♥❝✐❛s ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✜♥✐t❛s✳
❆♣❡s❛r ❞❛s ❞❡s❝♦❜❡rt❛s ❞❡ ▼❛♥❞❡❧❜r♦t✱ ❛s ❞✐str✐❜✉✐çõ❡s ❡stá✈❡✐s ❞❡ ▲é✈② ❢♦r❛♠ ♠❛♥✲ t✐❞❛s à ♠❛r❣❡♠ ❞❛ ár❡❛ ♣r✐♥❝✐♣❛❧ ❡♠ ✜♥❛♥ç❛s ❬✶✼✱ ✶✵✺❪✳ ❊♥tr❡ ❛s ♣♦ssí✈❡✐s r❛③õ❡s✱ ❛ ✐♥❡✲ ①✐stê♥❝✐❛ ❞♦ ❞❡s✈✐♦ ♣❛❞rã♦ ❝♦♠♦ ♠❡❞✐❞❛ ❞❡ ✈♦❧❛t✐❞❛❞❡ ❞❛ ❞✐str✐❜✉✐çã♦ é ✉♠ ✐♥❝♦✈❡♥✐❡♥t❡✱ ♣♦✐s ❡❧❡ r❡♣r❡s❡♥t❛ ✉♠❛ ♠❡❞✐❞❛ ❞❡ r✐s❝♦ ✜♥❛♥❝❡✐r♦✳ P♦r ❡①❡♠♣❧♦✱ ✉♠❛ ❣r❛♥❞❡ ✈❛r✐❛çã♦ ♠é❞✐❛ ❞❡ ✉♠❛ sér✐❡ ❞❡ r❡t♦r♥♦s ❡♠ ❝❡rt♦ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ ✐♥❞✐❝❛ ♠❛✐♦r ❡①♣♦s✐çã♦ ❞♦ ✐♥✈❡st✐❞♦r ❛ ♣❡r❞❛s ♦✉ ❣❛♥❤♦s ❝♦♥s✐❞❡rá✈❡✐s✳
❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ▲é✈② s✐♠étr✐❝❛✳ ❆ ♥♦✈✐❞❛❞❡✱ ♥♦ ❡♥t❛♥t♦✱ ❢♦✐ ❛ ♦❜s❡r✈❛çã♦ ❞❡ q✉❡✲ ❜r❛s ♥❛s ❧❡✐s ❞❡ ❡s❝❛❧❛ s✉❣❡r✐❞❛s ♣♦r ▼❛♥❞❡❧❜r♦t✱ ❞❡ ♠♦❞♦ q✉❡ ❡ss❡s ❞❛❞♦s ♥ã♦ ♣♦❞❡r✐❛♠ s❡r ❞❡ ❢❛t♦ ❡stá✈❡✐s ♥❡♠ ♣♦ss✉✐r ♠♦♠❡♥t♦s ✐♥✜♥✐t♦s✳ ❆ss✐♠✱ ❡ss❡s ❛✉t♦r❡s ♣r♦♣✉s❡r❛♠ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ▲é✈② tr✉♥❝❛❞❛ ✭tr✉♥❝❛t❡❞ ▲❡✈② ✢✐❣❤ts✱ ❚▲❋✮ ♣❛r❛ ❝♦♥t❡♠♣❧❛r ❡ss❡ ♥♦✈♦ ❢❛t♦ ❡st✐❧✐③❛❞♦✳ ❚r❛❜❛❧❤♦s s✉❜s❡q✉❡♥t❡s ♠♦str❛r❛♠ r❡s✉❧t❛❞♦s s✐♠✐❧❛r❡s ❡♠ ❞✐✈❡rs❛s ♦✉✲ tr❛s sér✐❡s ✜♥❛♥❝❡✐r❛s✱ ❝♦♠♦ ❛s ❞♦ í♥❞✐❝❡ ❞❛ ❇♦❧s❛ ❞❡ ❱❛❧♦r❡s ❞❡ ❙ã♦ P❛✉❧♦ ❬✹✻❪✱ ❛s ❞♦s í♥❞✐❝❡s ❞❡ ♦✉tr❛s ❜♦❧s❛ ❞❡ ✈❛❧♦r❡s ❬✹✺✱ ✽✽✱ ✶✵✾❪ ❡ ❛s ❞❛s t❛①❛s ❞❡ ❝â♠❜✐♦ ❬✸✵✱ ✾✵❪✳
✶✳✷ ❯♠❛ ❜r❡✈❡ r❡tr♦s♣❡❝t✐✈❛
❈♦♥s✐❞❡r❡ ♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦
S∆t=X1+· · ·+X∆t, ✭✶✳✶✮
❡♠ q✉❡ {Xk}k=1,···,∆t ❝♦♥st✐t✉✐ ✉♠❛ ❛♠♦str❛ ❛❧❡❛tór✐❛ r❡t✐r❛❞❛ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ X✱
❡stá✈❡❧ ❡ s✐♠étr✐❝❛ ❡♠ t♦r♥♦ ❞❡ ③❡r♦✱ ❝✉❥♦s ♣❛râ♠❡tr♦s sã♦ r❡♣r❡s❡♥t❛❞♦s ♣❡❧♦ ✈❡t♦r θs =
(α, γ)′✳ ◆❡ss❛ s✐t✉❛çã♦✱ ❛ ❢✉♥çã♦ ❞❡ ❞❡♥s✐❞❛❞❡ ❞❡ S
∆t ♥♦ ♣♦♥t♦ u∈R é ❬✷✱ ✸✵✱ ✼✺✱ ✶✵✸❪
fS∆t(u;θs) =
1
π
Z +∞
0
e−γ∆tqαcos(qu)dq, ✭✶✳✷✮
❡ s✉❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ♥♦ ♣♦♥t♦ q∈R é
φS∆t(q;θs) =
eiqS∆t = e−γ∆tqα. ✭✶✳✸✮ P❡❧❛ ❡st❛❜✐❧✐❞❛❞❡✱ s❡∆t= 1✱ ❛s ❡①♣r❡ssõ❡s ❛❝✐♠❛ r❡♣r❡s❡♥t❛♠ ❛ ❞✐str✐❜✉✐çã♦ ❞❡X✳ ❆❣♦r❛✱
❝♦♥s✐❞❡r❡ ✉♠❛ s♦♠❛ ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s
S∆′ t=X1′ +· · ·+X∆′ t, ✭✶✳✹✮
❡♠ q✉❡ ❝❛❞❛ Xk′ s❡❣✉❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❛❜r✉♣t❛♠❡♥t❡ tr✉♥❝❛❞❛ ✭❚▲❋✮ ♥❛ ❢♦r♠❛ ❬✼✸✱ ✼✺❪ fX′
k(u;
θT LF) = ηI(|u| ≤ζ)fX(u;θs), ✭✶✳✺✮
❝♦♠ θT LF = (α, γ, η, ζ)′✱ η > 0 é ♦ ♣❛râ♠❡tr♦ ❞❡ ♥♦r♠❛❧✐③❛çã♦✱ ζ > 0 é ♦ ♣♦♥t♦ ❞❡
tr✉♥❝❛♠❡♥t♦✱ ❡ I(|u| ≤ ζ) = 1✱ s❡ |u| ≤ ζ✱ ❡ I(|u| ≤ ζ) = 0✱ s❡ |u| > ζ✳ P❛r❛ ∆t → 1✱
❡♠❜♦r❛ s❡❥❛ tr✉♥❝❛❞♦✱ ♦ ♣r♦❝❡ss♦ S∆′ t ♣♦❞❡ s❡r ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❞❡s❝r✐t♦ ♣❡❧❛s ❢♦r♠❛s
❡stá✈❡✐s ❝♦rr❡s♣♦♥❞❡♥t❡s ✭✶✳✷✮ ♦✉ ✭✶✳✸✮✳ P♦ré♠✱ ❡s♣❡r❛✲s❡ q✉❡ S∆′ t s❡ ❛♣r♦①✐♠❡ ❞❡ ✉♠❛
❞✐str✐❜✉✐çã♦ ❣❛✉ss✐❛♥❛ à ♠❡❞✐❞❛ q✉❡ ∆t ❛✉♠❡♥t❛✱ ❥á q✉❡ ❛ ❚▲❋ ♥ã♦ é ❡stá✈❡❧ ❡ ♣♦ss✉✐
♠♦♠❡♥t♦s ✜♥✐t♦s ❬✸✵❪✳ ❈♦♥s✐❞❡r❡ ❡♥tã♦ ❛ ✈❛r✐á✈❡❧ r❡❞✉③✐❞❛ ✭♦✉ ♣❛❞r♦♥✐③❛❞❛✮
¯
S∆′ t = S
′
∆t−
S∆′ t σ∆t
, ✭✶✳✻✮
❡♠ q✉❡ σ2 ∆t =
(S∆′ t)2−S′
∆t
2
r❡♣r❡s❡♥t❛ ❛ ✈❛r✐â♥❝✐❛ ❞♦ ♣r♦❝❡ss♦ tr✉♥❝❛❞♦ S∆′ t✳ ◆❡ss❡
❝❛s♦✱ ❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❞❛ ✈❛r✐á✈❡❧ r❡❞✉③✐❞❛ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♥❛ ❢♦r♠❛ ❬✸✵✱ ✸✶✱ ✻✹❪
φS¯′
∆t(q) = e
−q2(1+w
∆t(q))/2, ✭✶✳✼✮
❡♠ q✉❡ w∆t(q) é ✉♠❛ ❢✉♥çã♦ t❛❧ q✉❡w(0) = 0✳
❙❡ {Xk′} ❢♦r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❝ó♣✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❝♦♠ ♠é❞✐❛ µ ❡ ✈❛r✐â♥❝✐❛ σ2✱ ❡♥tã♦ S′
∆t
= ∆tµ❡ σ2
∆t= ∆tσ2✳ ◆❡ss❡ ❝❛s♦✱
¯
S∆′ t = S
′
∆t−∆tµ
√
∆tσ
= √1 ∆t
∆t
X
k=1
¯
Xk′.
❆ss✐♠✱
φS¯′
∆t(q) =
D
eiqS¯′∆t
E
=Dei√q∆t P∆t
k=1X¯
′
k
E
=Dei√q∆tX¯
′E∆t
= e−q2(1+w1(q/∆t))/2 ,
❞❡ ♠♦❞♦ q✉❡ φS¯′
∆t(q)→e
−q2/2
à ♠❡❞✐❞❛ q✉❡∆t→ ∞✱ ❡♠ q✉❡e−q2/2
r❡♣r❡s❡♥t❛ ❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❞❛ ❣❛✉ss✐❛♥❛ ♣❛❞r♦♥✐③❛❞❛✳
▼❛s✱ s❡{Xk′}♥ã♦ ❢♦r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❤á r❡❞✉çã♦
♥❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡w∆t(q) ♣❛r❛ ③❡r♦ à ♠❡❞✐❞❛ q✉❡∆t ❛✉♠❡♥t❛ ❬✸✵✱ ✸✶✱ ✸✷✱
✸✸✱ ✸✹✱ ✹✺❪✳ ❆ss✐♠✱ ❡♥q✉❛♥t♦ ❤♦✉✈❡r ♠❡♠ór✐❛ s❡r✐❛❧ s✐❣♥✐✜❝❛t✐✈❛✱ ♦ t❡r♠♦ w∆t(q) ♣♦❞❡
✈❛r✐❛r ❧❡♥t❛♠❡♥t❡✱ ❞❡ ♠♦❞♦ q✉❡
w∆t(q)≈w(q) ✭✶✳✽✮
♣❛r❛ ❛❧❣✉♠ ✐♥t❡r✈❛❧♦ ∆t0 ≤ ∆t ≤ ∆t1✳ ❊ss❛ ❡st❛❜✐❧✐❞❛❞❡ ♠♦♠❡♥tâ♥❡❛ ❢♦✐ ❞❡♥♦♠✐♥❛❞❛ q✉❛s❡✲❡st❛❜✐❧✐❞❛❞❡ ♣♦r ❆✳ ❋✐❣✉❡✐r❡❞♦ ❡ s❡✉s ❝♦❧❛❜♦r❛❞♦r❡s ❬✸✵✱ ✸✶✱ ✹✺❪✱ t❡♥❞♦ s✐❞♦ ♦❜s❡r✲ ✈❛❞❛ ❡♠♣✐r✐❝❛♠❡♥t❡ ❡♠ sér✐❡s ❞♦s r❡t♦r♥♦s ❞❡ t❛①❛s ❞❡ ❝â♠❜✐♦ ❡ ❞❡ ❜♦❧s❛s ❞❡ ✈❛❧♦r❡s✳ ❙❡ ♦s r❡t♦r♥♦s Xt ❞❡ ❞❡t❡r♠✐♥❛❞♦ ❛t✐✈♦ ✜♥❛♥❝❡✐r♦ ❢♦ss❡♠ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ✐❞❡♥t✐❝❛♠❡♥t❡
❞✐str✐❜✉í❞♦s ✭■■❉✮ s❡❣✉♥❞♦ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ▲é✈② s✐♠étr✐❝❛ tr✉♥❝❛❞❛✱ ♣❡❧♦ t❡♦r❡♠❛ ❧✐✲ ♠✐t❡ ❝❡♥tr❛❧✱ ❛s s♦♠❛s ♣❛r❝✐❛✐s ❞❡ss❡s r❡t♦r♥♦s✱S∆t=X1+· · ·+X∆t✱ ❞❡✈❡r✐❛♠ ❝♦♥✈❡r❣✐r
r❛♣✐❞❛♠❡♥t❡ ♣❛r❛ ❛ ❣❛✉ss✐❛♥❛✳ P♦ré♠✱ ❤❛✈❡♥❞♦ ❝♦rr❡❧❛çõ❡s✱ ♦❜s❡r✈♦✉✲s❡ q✉❡ ❤á ✉♠ ✐♥t❡r✲ ✈❛❧♦ ∆t0 ≤∆t ≤∆t1 ❡♠ q✉❡ ♦ ♣r♦❝❡ss♦ {S∆t} é ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❡stá✈❡❧ ♣❡❧❛ ❧❡♥t✐❞ã♦
❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ❛ ❣❛✉ss✐❛♥❛✳ ▼❡s♠♦ q✉❡ ❛s ❛✉t♦❝♦rr❡❧❛çõ❡s ❧✐♥❡❛r❡s ❡♠ ✉♠❛ sér✐❡ t❡♠♣♦r❛❧ ✜♥❛♥❝❡✐r❛ s❡ ❡♥❝♦♥tr❡♠ ♥♦ ♥í✈❡❧ ❞❡ r✉í❞♦✱ ❢♦r♠❛s ♥ã♦ ❧✐♥❡❛r❡s ❞❡ ❛✉t♦❝♦rr❡❧❛çã♦ ❜❡♠ ❝♦♠♦ t✐♣♦s ♣❛rt✐❝✉❧❛r❡s ❞❡ ♥ã♦ ❡st❛❝✐♦♥❛r✐❡❞❛❞❡ t❛♠❜é♠ ♣♦❞❡♠ ❝♦♥tr✐❜✉✐r ♣❛r❛ ❛ ♣❡r♠❛♥ê♥❝✐❛ ❞❡ S∆t ♥♦ r❡❣✐♠❡ ❞❡ ▲é✈② ❬✸✷✱ ✸✺❪✳
❙❡ ❤♦✉✈❡r q✉❛s❡✲❡st❛❜✐❧✐❞❛❞❡✱ ❛ r❡❣✐ã♦ ♠♦❞❛❧ ❞❛ ❞❡♥s✐❞❛❞❡ ❡♠♣ír✐❝❛ ♣♦❞❡ s❡r ❛♣r♦✲ ①✐♠❛❞❛♠❡♥t❡ ❞❡s❝r✐t❛ ♣♦r ✉♠❛ ❞✐str✐❜✉✐çà♦ ❡stá✈❡❧✳ ❋♦r❛ ❞❛ r❡❣✐ã♦ ♠♦❞❛❧✱ ♣♦ré♠✱ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❡♠♣ír✐❝♦ ❞❛s ❝❛✉❞❛s ♣♦❞❡ s❡ ❞❡s✈✐❛r ❞♦ q✉❡ s❡ ❡s♣❡r❛ ❞❡ ✉♠❛ ❞✐str✐❜✉✐✲ çã♦ ❡stá✈❡❧✳ ❆ss✐♠✱ s♦❜ ❛ ❤✐♣ót❡s❡ ❞❡ q✉❡ ♦s ♣r♦❝❡ss♦s r❡❛✐s sã♦ ❧✐♠✐t❛❞♦s ♣❡❧❛ ✜♥✐t✉❞❡ ❞♦s r❡❝✉rs♦s ❬✼✸✱ ✼✺❪✱ ♦✉tr❛s ❢♦r♠❛s ❞❡ tr✉♥❝❛♠❡♥t♦ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ▲é✈② ♣♦❞❡♠ s❡r s✉❣❡✲ r✐❞❛s✱ ❝♦♠♦ ♦ tr✉♥❝❛♠❡♥t♦ s✉❛✈❡ ❬✾✷❪✱ ♦ ❣r❛❞✉❛❧ ❬✺✵✱ ✺✶❪ ❡ ♦ ❡①♣♦♥❡♥❝✐❛❧♠❡♥t❡ ❛♠♦rt❡❝✐❞♦ ❬✽✹✱ ✽✺✱ ✹✺❪✳ ❊ss❛s ♠♦❞✐✜❝❛çõ❡s r❡s✉❧t❛♠ ❡♠ ❞✐str✐❜✉✐çõ❡s ♥ã♦ ❡stá✈❡✐s ❝♦♠ ♠♦♠❡♥t♦s ✜♥✐✲ t♦s✱ ❡ ♣❡r♠✐t❡♠ ❡①♣❧✐❝❛r✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ♣r❡s❡♥ç❛ ❞❡ ♠✉❧t✐s❝❛❧✐♥❣ ♥♦s ♠♦♠❡♥t♦s ❛❜s♦❧✉t♦s ❞❛s s♦♠❛s ♣❛r❝✐❛✐s S∆t✳
à ♣❛rt❡✳ ❖s r❡t♦r♥♦s ❞❡ss❛ t❛①❛ ❞❡ ❝â♠❜✐♦ ❛♣r❡s❡♥t❛♠ ✉♠❛ ❡str✉t✉r❛ ❢r❛❝t❛❧ tí♣✐❝❛ ❞❡ ✉♠ ❥♦❣♦ ❝❛ót✐❝♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ tr✐â♥❣✉❧♦ ❞❡ ❙✐❡r♣✐♥s❦✐✱ ❡♠ q✉❡ ❛s r❡❣r❛s ❞❡t❡r♠✐♥íst✐❝❛s ❝♦❡①✐st❡♠ ❝♦♠ ❛s ❡st♦❝ást✐❝❛s ❬✽✶✱ ✽✸✱ ✶✶✸✱ ✶✶✽❪✳ ❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❡st♦❝ást✐❝♦✱ ❛ ❞✐♠❡♥sã♦ ❢r❛❝t❛❧ D ❞❡ ✉♠ ♣r♦❝❡ss♦ s❡ r❡❧❛❝✐♦♥❛ ❝♦♠ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ ♠❡❞✐❞❛ ❝♦♠
❜❛s❡ ♥♦ ❡①♣♦❡♥t❡ H ❞❡ ❍✉rst ❬✺✱ ✶✻✱ ✹✽✱ ✼✷✱ ✽✷❪✱ ❞❡✜♥✐❞♦ ❝♦♠♦
σ∆t∼∆tH = ∆t2−D.
P❛r❛ ✉♠ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ ❝♦♠ ✐♥❝r❡♠❡♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s✱ t❡♠✲s❡ H = 0,5✳ ❖ ❡①♣♦✲
❡♥t❡ ❞❡ ❍✉rst ♣♦❞❡ s❡r ❡st✐♠❛❞♦ ❝♦♠ ❜❛s❡ ♥❛ ❡st❛tíst✐❝❛ R/S ✭r❡s❝❛❧❡❞ r❛♥❣❡ ❛♥❛❧②s✐s✱
❬✺✱ ✶✹✱ ✼✷❪✮✱ ♥♦ ♠ét♦❞♦ ❉❋❆ ✭❞❡tr❡♥❞❡❞ ✢✉❝t✉❛t✐♦♥ ❛♥❛❧②s✐s✱ ❬✶✷✹❪✮ ♦✉ ❉▼❆ ✭❞❡tr❡♥❞❡❞ ♠♦✈✐♥❣ ❛✈❡r❛❣❡✱ ❬✶✷✱ ✽✷✱ ✶✷✹❪✮✳ ❊♠ ✜♥❛♥ç❛s✱ ♦ ❡①♣♦❡♥t❡ ❞❡ ❍✉rst ✖ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❛ ❞✐♠❡♥sã♦ ❢r❛❝t❛❧ ✖ ♣❡r♠✐t❡ ❛✈❛❧✐❛r ❛ ❤✐♣ót❡s❡ ❞♦ ♠❡r❝❛❞♦ ❡✜❝✐❡♥t❡✳ ❙❡❣✉♥❞♦ ❡ss❛ ❤✐✲ ♣ót❡s❡✱ ❝♦♠ ❜❛s❡ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✐♥❢♦r♠❛çõ❡s ♣✉❜❧✐❝❛♠❡♥t❡ ❞✐s♣♦♥í✈❡✐s à ❝♦♠✉♥✐❞❛❞❡ ✜♥❛♥❝❡✐r❛✱ ✉♠ ✐♥✈❡st✐❞♦r ♥ã♦ é ❝❛♣❛③ ❞❡ ♦❜t❡r✱ s✐st❡♠❛t✐❝❛♠❡♥t❡✱ r❡♥❞✐♠❡♥t♦s s✉♣❡r✐♦r❡s à ♠é❞✐❛ ❞♦ ♠❡r❝❛❞♦ ❬✶✼❪✱ ❡✱ ❛ss✐♠✱ H = 0,5✳ ❊st✉❞♦s ❡♠♣ír✐❝♦s✱ ♥♦ ❡♥t❛♥t♦✱ ♠♦str❛♠
r❡s✉❧t❛❞♦s q✉❡ ❡♥❢r❛q✉❡❝❡♠ ❡ss❛ ❤✐♣ót❡s❡✱ ❡♠ q✉❡ H <0,5❬✶✻✱ ✶✼✱ ✽✷❪✳
P♦r ❡①❡♠♣❧♦✱ às ✈és♣❡r❛s ❞❡ ✉♠❛ q✉❡❜r❛ ♥❛ ❜♦❧s❛s ❞❡ ✈❛❧♦r❡s ✭❝r❛s❤✮ ♦✉ ❞❡ ✉♠❛ ❝r✐s❡ ❡❝♦♥ô♠✐❝❛ ❡♠ ❣r❛♥❞❡ ❡s❝❛❧❛✱ ♦s ❛❣❡♥t❡s q✉❡ ❝♦♠♣r❛♠ ❡ ✈❡♥❞❡♠ ❛t✐✈♦s ♣♦❞❡♠ s❡❣✉✐r ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ❝♦❧❡t✐✈♦ ❡♠ ♠❛ss❛ ✭♦ q✉❡✱ ❡♠ ♣❛rt❡✱ ❛❥✉❞❛ ❛ ❡①♣❧✐❝❛r ❛ ♣r❡s❡♥ç❛ ❞❡ ❞❡♣❡♥❞ê♥❝✐❛ s❡r✐❛❧ ♥♦ ♣❡rí♦❞♦ q✉❡ ❛♥t❡❝❡❞❡ ✉♠❛ ❝r✐s❡✮✳ ❊♠ ❢❡♥ô♠❡♥♦s ❞❡ r✉♣t✉r❛✱ ❉✳ ❙♦r♥❡tt❡ ♦❜s❡r✈♦✉ ✉♠ ♣❛❞rã♦ ❧♦❣✲♣❡r✐ó❞✐❝♦ ♥❛ ❢♦r♠❛ xt ∼ cos lnt ❬✶✶✶❪✱ ❡♠ q✉❡ t é ♦
t❡♠♣♦ ❡ xt é ✉♠❛ ✈❛r✐á✈❡❧ ❞♦ s✐st❡♠❛✳ ❊ ❛ss✐♠✱ ❡✈✐❞ê♥❝✐❛s ❞❡ ❧♦❣✲♣❡r✐♦❞✐❝✐❞❛❞❡ ❢♦r❛♠
❡♥❝♦♥tr❛❞❛s ❡♠ í♥❞✐❝❡s ❞❡ ❜♦❧s❛s ❞❡ ✈❛❧♦r❡s ❬✶✶✵✱ ✶✶✷❪✱ ❡♠ t❛①❛ ❞❡ ❝â♠❜✐♦ ❬✼✾❪ ❡ ♥♦ í♥❞✐❝❡ ❉♦✇ ❏♦♥❡s ❬✼✽❪✳
❛❧❣♦r✐t♠♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❞✐s♣♦♥í✈❡❧ ♣❛r❛ ❣❡rá✲❧♦ é tã♦ ❣r❛♥❞❡ q✉❛♥t♦ ❛♦ ♣ró♣r✐♦ str✐♥❣✳ ❆ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ t❛♠❛♥❤♦ ❞❡ ✉♠ str✐♥❣ ❡ ♦ ♠❡♥♦r ❛❧❣♦r✐t♠♦ ♣♦ssí✈❡❧ r❡♣r❡s❡♥t❛ ♦ s❡✉ ❣r❛✉ ❞❡ ❝♦♠♣r❡ss✐❜✐❧✐❞❛❞❡✳ ❉❡ss❡ ♠♦❞♦✱ ✉♠ str✐♥❣ ❞❡ ❜❛✐①❛ ❝♦♠♣❧❡①✐❞❛❞❡ é ❛❧t❛✲ ♠❡♥t❡ ❝♦♠♣r❡ssí✈❡❧✱ ❡♥q✉❛♥t♦ ✉♠ str✐♥❣ ❞❡ ❞í❣✐t♦s ❜✐♥ár✐♦s ❛❧❡❛tór✐♦s é ✐♥❝♦♠♣r❡ssí✈❡❧✳ ❊ss❛ ❛❜♦r❞❛❣❡♠ ♣❡r♠✐t❡✱ ♣♦r ❡①❡♠♣❧♦✱ ❞❡s❝r❡✈❡r ❡ ❝❧❛ss✐✜❝❛r ♦s ♠❡r❝❛❞♦s ❝♦♠ ❜❛s❡ ♥♦ ❛❧❣♦r✐t♠♦ ❞❡ ❝♦♠♣r❡ssã♦ ❞❡ ❞❛❞♦s ❞❡ ▲❡♠♣❡❧✲❩✐✈ ❬✶✻✱ ✹✶✱ ✹✷✱ ✹✸❪✳
❉❛❞❛ ❛ ❛❜r❛♥❣ê♥❝✐❛ ❞♦ t❡♠❛✱ ❡st❡ tr❛❜❛❧❤♦ s❡ r❡str✐♥❣❡ ❛♦s ❛s♣❡❝t♦s ❞❛ ✐♥❢❡rê♥❝✐❛ ❡st❛tíst✐❝❛ ✈✐❛ ❢✉♥çõ❡s ❝❛r❛❝t❡ríst✐❝❛s✱ ❞❡✐①❛♥❞♦ à ♠❛r❣❡♠ ❞✐✈❡rs♦s ❛ss✉♥t♦s ❝♦♠♦ ❝♦♠✲ ♣❧❡①✐❞❛❞❡✱ ❝r✐t✐❝❛❧✐❞❛❞❡✱ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡✱ ❝❛♦s ❡ ❧♦❣♣❡r✐♦❞✐❝✐❞❛❞❡✳
✶✳✸ ❖❜❥❡t✐✈♦s
◆♦s ❡st✉❞♦s ❛♥t❡r✐♦r❡s✱ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦ ♣r♦❝❡ss♦S∆′ t♣❛r❛ ❛ ❣❛✉ss✐❛♥❛ ❢♦✐ ❛✈❛❧✐❛❞❛ ❝♦♠
❜❛s❡ ♥♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❢✉♥çã♦ w∆t(q) ✭❊q✳ ✭✶✳✽✮✮✳ ❈♦♥s✐❞❡r❛♥❞♦ q✉❡ ❛ q✉❛s❡ ❡st❛✲
❜✐❧✐❞❛❞❡ r❡♠❡t❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❛♦ r❡❣✐♠❡ ❞❡ ▲é✈②✱ q✉❡ ❛ ❞✐str✐❜✉✐çã♦ ♣❛r❛ ∆t →1 s❡
❛ss❡♠❡❧❤❛ ❛ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡stá✈❡❧ ✭♣❡❧♦ ♠❡♥♦s ♥❛ r❡❣✐ã♦ ♠♦❞❛❧ ❞❛ ❞✐str✐❜✉✐çã♦✮✱ ❡ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦ ♣r♦❝❡ss♦ ❡♠♣ír✐❝♦ ❡ ♦ ❤✐♣♦tét✐❝♦ ♣♦❞❡ s❡r ♠❡❞✐❞❛ ❝♦♠ ❜❛s❡ ♥❛s ❢✉♥çõ❡s ❝❛r❛❝t❡ríst✐❝❛s ❡♠♣ír✐❝❛ ❡ ❤✐♣♦tét✐❝❛ ❬✷✻✱ ✷✼✱ ✾✺✱ ✾✼✱ ✶✶✼✱ ✶✷✺✱ ✶✷✵❪✱ ❡st❡ tr❛❜❛❧❤♦ ♣r♦♣õ❡ ✉♠ ❡st✉❞♦ ❞❛ ❞✐♥â♠✐❝❛ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦s ♣r♦❝❡ss♦s s♦❜ ❛ ♣❡rs♣❡❝t✐✈❛ ❞❛s ❞✐str✐❜✉✐çõ❡s q✉❛s❡ ❡stá✈❡✐s✳ ❖✉ s❡❥❛✱ ♥♦ ❝❛s♦ s✐♠étr✐❝♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❧✉❣❛r ❞❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ✭✶✳✼✮✱ ♣r♦♣õ❡✲s❡ q✉❡ ♦ ♣r♦❝❡ss♦ S∆′ t s❡❥❛ ❞❡s❝r✐t♦ ♣❡❧❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ♥❛ ❢♦r♠❛
φS′
∆t(q;
θs)≈φS
∆t(q;θs) =
eiqS∆t= exp(−γ
∆t∆tqα∆t).
❆ss✐♠✱ ❤á ❡st❛❜✐❧✐❞❛❞❡ s❡ α∆t =α ❢♦r ❝♦♥st❛♥t❡ ♣❛r❛ t♦❞♦ ∆t ❡ γ∆t = ∆tγ✳ P♦ré♠✱ ♥❛s
❝♦♥❞✐çõ❡s ❞♦ t❡♦r❡♠❛ ❧✐♠✐t❡ ❝❡♥tr❛❧✱ s❡ ♥ã♦ ❤♦✉✈❡r ❡st❛❜✐❧✐❞❛❞❡✱ ❡s♣❡r❛✲s❡ q✉❡ α∆t → 2
à ♠❡❞✐❞❛ q✉❡ ∆t ❛✉♠❡♥t❛✳ ❊✱ s❡ α∆t ≈ α ❡♠ ❛❧❣✉♠ ✐♥t❡r✈❛❧♦ ∆t0 ≤ ∆t ≤ ∆t1✱ ❡♥tã♦ ❤á q✉❛s❡ ❡st❛❜✐❧✐❞❛❞❡✳ ❊✱ ❛✐♥❞❛✱ ♦ ❡❢❡✐t♦ ❞❛ ❞❡♣❡♥❞ê♥❝✐❛ t❡♠♣♦r❛❧ ♥♦ ♣❛râ♠❡tr♦ ❞❡ ❡s✲ ❝❛❧❛ ♣♦❞❡ s❡r ❛✈❛❧✐❛❞♦ ❝♦♠ ❜❛s❡ ♥♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ γ∆t ✈❡rs✉s ∆t✳ ❊ss❛ ❛❜♦r❞❛❣❡♠
r❡q✉❡r ❡st✐♠❛çã♦ ❞❡α∆t❡ ❞❡γ∆t♣❛r❛ ❝❛❞❛∆t❞❡s❡❥❛❞♦✳ ❖ ♠ét♦❞♦ ❞❛ ♠á①✐♠❛ ✈❡r♦ss✐♠✐✲
❧❤❛♥ç❛ ✭▼▼❱✮ ❢♦r♥❡❝❡ ❡st✐♠❛t✐✈❛s ❝♦♠ ❜♦❛s ♣r♦♣r✐❡❞❛❞❡s ❡st❛tíst✐❝❛s ❝♦♠♦ ❝♦♥s✐stê♥❝✐❛✱ ❡✜❝✐ê♥❝✐❛ ❡ ♥♦r♠❛❧✐❞❛❞❡ ❛ss✐♥tót✐❝❛ ❞❛s ❞✐str✐❜✉✐çõ❡s ❛♠♦str❛✐s ❬✽✾✱ ✾✽✱ ✶✵✶❪✳ P♦ré♠✱ ♦ ❢❛t♦ ❞❡ ❛ ❢✉♥çã♦ ❞❡ ❞❡♥s✐❞❛❞❡ ✭❋❉✮ ❞❛ ❞✐str✐❜✉✐çã♦ ❡stá✈❡❧ ♥ã♦ ♣♦ss✉✐r ❢♦r♠❛ ❢❡❝❤❛❞❛ ♣❛r❛ α 6= 1 ❡ 2 ❬✾✼✱ ✾✾❪ ♠♦t✐✈♦✉ ❛ ❜✉s❝❛ ♣♦r ❞✐❢❡r❡♥t❡s ♦✉tr♦s ♠ét♦❞♦s ❞❡ ❡st✐♠❛çã♦
❬✷✶✱ ✼✺✱ ✶✵✸✱ ✽✷✱ ✽✼✱ ✾✼❪✱ ❡♠❜♦r❛ ❡❧❡s s❡❥❛♠ ♠❡♥♦s ❡✜❝✐❡♥t❡s ❞♦ q✉❡ ♦ ▼▼❱✳ ❆♦ ❝♦♥trár✐♦ ❞❛ ❋❉✱ ❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ✭❋❈✮ ❞❛ ❞✐str✐❜✉✐çã♦ ❡stá✈❡❧ ♣♦ss✉✐ ❢♦r♠❛ ❢❡❝❤❛❞❛✳ P♦r ❝❛✉s❛ ❞❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ❛ ❋❉ ❡ ❛ ❋❈✱ ❡s♣❡r❛✲s❡ q✉❡ s❡❥❛ ♣♦ssí✈❡❧ ♦❜t❡r ❡st✐♠❛t✐✈❛s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✭▼❱✮ ❝♦♠ ❜❛s❡ ❡♠ ❢✉♥çõ❡s ❝❛r❛❝t❡ríst✐❝❛s ❬✶✷✺❪✳ ❆s ❡q✉❛çõ❡s ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✖ q✉❡ ❢♦r♠❛♠ ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ♣❛r❛ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛s ❡st✐♠❛✲ t✐✈❛s ❞❡ ▼❱ ✖ s❡ r❡❧❛❝✐♦♥❛♠ ❝♦♠ ❛ ❞✐✈❡r❣ê♥❝✐❛ ❞❡ ❑✉❧❧❜❛❝❦✲▲❡✐❜❧❡r ❡♥tr❡ ❛ ❞✐str✐❜✉✐çã♦ ❡♠♣ír✐❝❛ ❡ ❛ ❤✐♣♦tét✐❝❛ ❬✷✻✱ ✷✼❪✳
❆ss✐♠✱ ♦ ♣r✐♠❡✐r♦ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❞❡s❡♥✈♦❧✈❡r ✉♠❛ ❡q✉❛çã♦ ❞❡ ✈❡r♦ss✐♠✐✲ ❧❤❛♥ç❛ ❝♦♠ ❜❛s❡ ❡♠ ❢✉♥çõ❡s ❝❛r❛❝t❡ríst✐❝❛s✱ ❝♦♥s✐❞❡r❛♥❞♦✲s❡ ❛s ❞✐str✐❜✉✐çõ❡s ✭❛♣r♦①✐♠❛✲ ❞❛♠❡♥t❡✮ ❡stá✈❡✐s s✐♠étr✐❝❛s ❡ ❛s ❛ss✐♠étr✐❝❛s✳
❈♦♠♦ ❡ss❛ ✐♥❢❡rê♥❝✐❛ ❡st❛tíst✐❝❛ ❞❡♣❡♥❞❡ ❞❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❡♠♣ír✐❝❛ ✭❋❈❊✮✱ ♦ s❡❣✉♥❞♦ ♦❜❥❡t✐✈♦ tr❛t❛ ❞♦ ❡st✉❞♦ ❞♦ tr✉♥❝❛♠❡♥t♦ ♥❛t✉r❛❧ ❞❡ss❛ ❢✉♥çã♦✳ P♦r ❝❛✉s❛ ❞❛ ✜♥✐t✉❞❡ ❞♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛✱ ❡st❛t✐st✐❝❛♠❡♥t❡✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❤✐♣ót❡s❡φS∆t(q;θs) = 0 ♥ã♦ ♣♦❞❡r✐❛ s❡r r❡❥❡✐t❛❞❛ ❝❛s♦ s✉❛ ❡st✐♠❛t✐✈❛ ✭φˆ(q)✮ s❡ ❡♥❝♦♥tr❡ ♥♦ ♥í✈❡❧ ❞❡ r✉í❞♦✳
❈♦♥s✐❞❡r❛♥❞♦ q✉❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ s❡r✐❛❧ ♣r♦❞✉③ q✉❛s❡ ❡st❛❜✐❧✐❞❛❞❡ ♥♦ ♣r♦❝❡ss♦ S∆′ t✱ ♦
t❡r❝❡✐r♦ ♦❜❥❡t✐✈♦ é ♣r♦♣♦r ✉♠ ♥♦✈♦ t❡st❡ ❞❡ ❤✐♣ót❡s❡s ♣❛r❛ ❛ ❞❡t❡❝çã♦ ❞❡ ❞❡♣❡♥❞ê♥❝✐❛ ♥ã♦ ❧✐♥❡❛r ❬✽✵❪✳ ❖ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦ ❡ ❛ ❢✉♥çã♦ ❞❡ ❛✉t♦❝♦rr❡❧❛çã♦ ♥ã♦ sã♦ ❝♦♥✲ s✐st❡♥t❡s ♣❛r❛ ♦s ❝❛s♦s ❡♠ q✉❡ ❤á ❞❡♣❡♥❞ê♥❝✐❛ ♥ã♦ ❧✐♥❡❛r ♥♦s ❞❛❞♦s✳ P❛r❛ ❞✐str✐❜✉✐çõ❡s q✉❡ ♥ã♦ ♣♦ss✉❡♠ ♠♦♠❡♥t♦ ✜♥✐t♦ ♦✉ q✉❡ ❛♣r❡s❡♥t❛♠ ❞❡♣❡♥❞ê♥❝✐❛ ♥ã♦ ❧✐♥❡❛r✱ ❡s♣❡r❛✲s❡ q✉❡ ✉♠ t❡st❡ ❡❧❛❜♦r❛❞♦ ❝♦♠ ❜❛s❡ ♥❛ ❞✐✈❡r❣ê♥❝✐❛ ❞❡ ❑✉❧❧❜❛❝❦✲▲❡✐❜❧❡r ❢♦r♥❡ç❛ r❡s✉❧t❛❞♦s ❝♦♥s✐st❡♥t❡s ❬✽✵✱ ✽✾✱ ✾✽✱ ✶✵✶❪✳
✶✳✹ ❉❛❞♦s
P❛r❛ ❛s ✐❧✉str❛çõ❡s ❛♣r❡s❡♥t❛❞❛s ♥❡st❡ tr❛❜❛❧❤♦✱ ❝♦♥s✐❞❡r❛♠♦s ❛ sér✐❡ t❡♠♣♦r❛❧ ❞♦ í♥❞✐❝❡ ❞✐ár✐♦ ❞❛ ❇♦❧s❛ ❞❡ ❱❛❧♦r❡s ❞❡ ❙ã♦ P❛✉❧♦ ✭■❇♦✈❡s♣❛✮✱ ❛ sér✐❡ ✐♥tr❛❞❛② ✭♠✐♥✉t♦ ❛ ♠✐♥✉t♦✮ ❞♦ í♥❞✐❝❡ ❉♦✇ ❏♦♥❡s ■♥❞✉str✐❛❧ ❆✈❡r❛❣❡ ✭❉❏■❆✮ ❞❛ ❇♦❧s❛ ❞❡ ❱❛❧♦r❡s ❞❡ ◆♦✈❛ ■♦rq✉❡ ✭◆❨❙❊✮✱ ❛s sér✐❡s ✐♥tr❛❞❛② ✭♠✐♥✉t♦ ❛ ♠✐♥✉t♦✮ ❞♦s ♣r❡ç♦s ❞❛s ❛çõ❡s ❞❡ ❛❧❣✉♠❛s ❡♠♣r❡s❛s ♥❡❣♦❝✐❛❞❛s ♥❛ ◆❨❙❊✱ ❡ ❛s ❞❛s t❛①❛s ❞✐ár✐❛s ❞❡ ❝â♠❜✐♦ ❞❡ ❛❧❣✉♠❛s ♠♦❡❞❛s ✭❚❛❜✳ ✶✳✷✮ ❢r❡♥t❡ ❛♦ ❞ó❧❛r ❛♠❡r✐❝❛♥♦✳ ❆ s❡❣✉✐r✱ ❞❡s❝r❡✈❡♠♦s ❜r❡✈❡♠❡♥t❡ ❡ss❛s sér✐❡s ✜♥❛♥❝❡✐r❛s✳
❋✐❣✉r❛ ✶✳✶✿ ❖❜s❡r✈❛çõ❡s ❞✐ár✐❛s ❞♦ ❧♦❣❛r✐t♠♦ ♥❛t✉r❛❧ ❞♦ ■❇♦✈❡s♣❛✱ lnWt ✭♣❛✐♥❡❧ s✉♣❡r✐♦r✮✱ ❡ s❡✉s
r❡t♦r♥♦s Xt ✭♣❛✐♥❡❧ ✐♥❢❡r✐♦r✮✱ ❞❡ ✷ ❞❡ ❥❛♥❡✐r♦ ❞❡ ✶✾✻✽ ❛ ✷✾ ❞❡ ❢❡✈❡r❡✐r♦ ❞❡ ✷✵✶✷✳ ❖ ✐♥st❛♥t❡ t = 6.500
❝♦rr❡s♣♦♥❞❡ ❛ ✵✹✴✵✼✴✶✾✾✹✱ três ❞✐❛s ❛♣ós ♦ ❞✐❛ ❡♠ q✉❡ ♦ P❧❛♥♦ ❘❡❛❧ ❡♥tr♦✉ ❡♠ ✈✐❣♦r✳
✶✳✹✳✶ ❖ ■❇♦✈❡s♣❛
❝á❧❝✉❧♦ s❡ ♠❛♥t❡✈❡ ❛ ♠❡s♠❛ ❞❡s❞❡ s✉❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❡♠ ✶✾✻✽✳ ❖ ♣❛✐♥❡❧ s✉♣❡r✐♦r ❞❛ ❋✐❣✳ ✶✳✶ ♠♦str❛ ❛ sér✐❡ ❤✐stór✐❝❛ ❞♦ ❧♦❣❛r✐t♠♦ ❞❛ ♣♦♥t✉❛çã♦ ❞❡ ❢❡❝❤❛♠❡♥t♦ ❞♦ ■❜♦✈❡s♣❛ ❞❡ ✷ ❞❡ ❥❛♥❡✐r♦ ❞❡ ✶✾✻✽ ❛ ✷✾ ❞❡ ❢❡✈❡r❡✐r♦ ❞❡ ✷✵✶✷✱ ♣❡r❢❛③❡♥❞♦ ♦ t♦t❛❧ ❞❡ ✶✵✳✽✼✵ ♦❜s❡r✈❛çõ❡s✳ ❈♦♥s✐❞❡r❛♥❞♦ q✉❡Wt r❡♣r❡s❡♥t❛ ❛ ♣♦♥t✉❛çã♦ ❞♦ ■❇♦✈❡s♣❛ ❛♦ ✜♥❛❧ ❞♦ ❞✐❛t ✭✐❣♥♦r❛♥❞♦✲s❡
❢❡r✐❛❞♦s ❡ ✜♥❛✐s ❞❡ s❡♠❛♥❛✮✱ ❞❡✜♥❡✲s❡ ♦ r❡t♦r♥♦ ❧♦❣❛rít♠✐❝♦ ❝♦♠♦
Rt= ln(Wt)−ln(Wt−1), ✭✶✳✾✮ ❡ ♦ r❡t♦r♥♦ ❝❡♥tr❛❞♦ ♥❛ ♠é❞✐❛ ❤✐stór✐❝❛ ❞♦s r❡t♦r♥♦s é ❞❛❞♦ ♣♦r
Xt =Rt−µ, ✭✶✳✶✵✮
❡♠ q✉❡ µ=hRti✳ ❖ ♣❛✐♥❡❧ ✐♥❢❡r✐♦r ❞❛ ❋✐❣✳ ✶✳✶ ♠♦str❛ ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❛ sér✐❡ ❞♦s
r❡t♦r♥♦s Xt✳
❋✐❣✉r❛ ✶✳✷✿ ❖❜s❡r✈❛çõ❡s ♠✐♥✉t♦ ❛ ♠✐♥✉t♦ ❞♦ ❧♦❣❛r✐t♠♦ ♥❛t✉r❛❧ ❞♦ ❉❏■❆✱lnWt✭♣❛✐♥❡❧ s✉♣❡r✐♦r✮✱ ❡ s❡✉s
r❡t♦r♥♦sXt✭♣❛✐♥❡❧ ✐♥❢❡r✐♦r✮✱ ❞❡ ✶✺❤✵✾ ❞♦ ❞✐❛ ✶✽ ❞❡ s❡t❡♠❜r♦ ❞❡ ✷✵✵✾ ❛ ✶✵❤✵✾ ❞♦ ❞✐❛ ✷✺ ❞❡ ♠❛✐♦ ❞❡ ✷✵✶✵✳
❖ ✢❛s❤ ❝r❛s❤ ♦❝♦rr❡✉ ❡♠ ✻ ❞❡ ♠❛✐♦ ❞❡ ✷✵✶✵ ✭60.491≤t≤60.881✮✳
✶✳✹✳✷ ❖ ❮♥❞✐❝❡ ❉❏■❆
❖ ♣❛✐♥❡❧ s✉♣❡r✐♦r ❞❛ ❋✐❣✳ ✶✳✷ ♠♦str❛ ❛ ❡✈♦❧✉çã♦ ♠✐♥✉t♦ ❛ ♠✐♥✉t♦ ❞♦ ❧♦❣❛r✐t♠♦ ♥❛t✉r❛❧ ❞♦ í♥❞✐❝❡ ❉❏■❆ ✭❉♦✇ ❏♦♥❡s ■♥❞✉str✐❛❧ ❆✈❡r❛❣❡✮ ❞❛ ❜♦❧s❛ ❞❡ ✈❛❧♦r❡s ❞❡ ◆♦✈❛ ■♦rq✉❡✱ ❛ ♣❛rt✐r ❞❡ ✶✺❤✵✾ ❞♦ ❞✐❛ ✶✽ ❞❡ s❡t❡♠❜r♦ ❞❡ ✷✵✵✾ ❛té ✶✵❤✵✾ ❞♦ ❞✐❛ ✷✺ ❞❡ ♠❛✐♦ ❞❡ ✷✵✶✵✱ ♣❡r❢❛③❡♥❞♦ ♦ t♦t❛❧ ❞❡ ✻✺✳✺✸✺ ♦❜s❡r✈❛çõ❡s✳ ◆❡ss❛ sér✐❡ t❡♠♣♦r❛❧✱ ✉♠ ❡♣✐só❞✐♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✢❛s❤ ❝r❛s❤ ❬✼✽❪ ♠❛r❝♦✉ ♦ ❞✐❛ ✻ ❞❡ ♠❛✐♦ ❞❡ ✷✵✶✵ ✭♥❛ ❋✐❣✳ ✶✳✷ ❡ss❡ ❞✐❛ ❝♦rr❡s♣♦♥❞❡ ❛♦ ✐♥t❡r✈❛❧♦
60.491 ≤ t ≤ 60.881✮✳ ◆❡ss❛ q✉✐♥t❛✲❢❡✐r❛ ♥❡❣r❛✱ r❡♣❡♥t✐♥❛♠❡♥t❡✱ ♦ í♥❞✐❝❡ s♦❢r❡✉ ✉♠❛
q✉❡❞❛ ❛❜r✉♣t❛ ❞❡ 998.5 ♣♦♥t♦s✳ ❆ q✉❡❞❛ ♦❝♦rr❡✉ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❡♥tr❡ ✶✹❤✹✵ ❡ ✶✺❤✵✵✱ ❡
♥❡ss❡ ♣❡rí♦❞♦ ♦ ♣r❡ç♦ ❞❛ ❛çã♦ ❞❛ ❡♠♣r❡s❛ ❞❡ ❝♦♥s✉❧t♦r✐❛ ❆❝❝❡♥t✉r❡✱ ♣♦r ❡①❡♠♣❧♦✱ ❞❡s♣❡♥✲ ❝♦✉ ❞❡ ❯❙✩ ✻✵✱✵✵ ♣❛r❛ ❯❙✩ ✵✱✵✶✳ ❊ss❛ q✉❡❜r❛ ❢♦✐ ♣r♦✈♦❝❛❞❛ ♣♦r ✉♠❛ ♦r❞❡♠ ❞❡ ✈❡♥❞❛ ❞❡ ❝♦♥tr❛t♦s ❢✉t✉r♦s ❢❡✐t❛ ♣♦r ✉♠ ♦♣❡r❛❞♦r q✉❡ ✉t✐❧✐③♦✉ ✉♠❛ ♣❧❛t❛❢♦r♠❛ ❛✉t♦♠❛t✐③❛❞❛ ♣❛r❛ s✉❛s ♥❡❣♦❝✐❛çõ❡s✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ ór❣ã♦ r❡❣✉❧❛❞♦r ❙❡❝✉r✐t✐❡s ✫ ❊①❝❤❛♥❣❡ ❈♦♠♠✐ss✐♦♥✱ ❡ss❛ ♦r❞❡♠ ❛✉t♦♠❛t✐③❛❞❛ ✈❡♥❞❡✉✱ ❡♠ ❛♣❡♥❛s ✷✵ ♠✐♥✉t♦s✱ ✼✺ ♠✐❧ ❝♦♥tr❛t♦s ❢✉t✉r♦s ❊✲♠✐♥✐ ❞♦ ❙✫P ✺✵✵✱ ❝♦♠ ✈❛❧♦r ❡st✐♠❛❞♦ ❡♠ ❯❙✩ ✹✱✶ ♠✐❧❤õ❡s✳ ❆ r❛♣✐❞❡③ ❞❛ ❡①❡❝✉çã♦ ❞❡ss❛ ♦r❞❡♠ ♣r♦✈♦✉ ✉♠ ❝❤♦q✉❡ ♥♦ ♠❡r❝❛❞♦✱ ❡ ♦ ❞❡❝❧í♥✐♦ q✉❡ s❡ s❡❣✉✐✉ ♥♦s í♥❞✐❝❡s ❞❡ ❢✉t✉r♦s ❛❧❛r♠♦✉ ♦s ❞❡♠❛✐s ♦♣❡r❛❞♦r❡s✳ ❆ ❢✉❣❛ ♠❛ss✐✈❛ ❞❡ss❡s ♦♣❡r❛❞♦r❡s ♣r♦❞✉③✐✉ ❛ q✉❡❞❛ ❡♠ ♣♦✉❝♦s ♠✐♥✉t♦s ✭❥á q✉❡ ❛ ♦r❞❡♠ ❞❡ ♣r♦t❡çã♦ ❝♦♥tr❛ ♣❡r❞❛s ♥❛ ♥❡❣♦❝✐❛çã♦ ❞❡ ❢✉t✉r♦s t❛♠❜é♠ é ❛✉t♦♠❛t✐③❛❞❛✮✳
