Tests for Non-Cointegration based on the Frequency Domain

■❣♦r ❱✐✈❡✐r♦s ▼❡❧♦ ❙♦✉③❛

❚❡sts ❢♦r ◆♦♥✲❈♦✐♥t❡❣r❛t✐♦♥ ❜❛s❡❞ ♦♥ t❤❡

❋r❡q✉❡♥❝② ❉♦♠❛✐♥

■❣♦r ❱✐✈❡✐r♦s ▼❡❧♦ ❙♦✉③❛

❚❡sts ❢♦r ◆♦♥✲❈♦✐♥t❡❣r❛t✐♦♥ ❜❛s❡❞ ♦♥ t❤❡

❋r❡q✉❡♥❝② ❉♦♠❛✐♥

❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ ■♥st✐t✉t♦ ❞❡ ❈✐✲

ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡✲

r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✱ ♣❛r❛ ❛ ♦❜t❡♥çã♦

❞❡ ❚ít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ❊st❛tíst✐❝❛✱

♥❛ ➪r❡❛ ❞❡ ❙ér✐❡s ❚❡♠♣♦r❛✐s✳

❖r✐❡♥t❛❞♦r✿ ❱❛❧❞❡r✐♦ ❆♥s❡❧♠♦ ❘❡✐✲

s❡♥

❈♦✲❖r✐❡♥t❛❞♦r❛✿ ●❧❛✉r❛ ❞❛ ❈♦♥❝❡✐✲

çã♦ ❋r❛♥❝♦

■❣♦r ❱✐✈❡✐r♦s ▼❡❧♦ ❙♦✉③❛✱

❚❡sts ❢♦r ◆♦♥✲❈♦✐♥t❡❣r❛t✐♦♥ ❜❛s❡❞ ♦♥ t❤❡ ❋r❡✲ q✉❡♥❝② ❉♦♠❛✐♥

✺✽ ♣á❣✐♥❛s

❚❡s❡ ✭❉♦✉t♦r❛❞♦✮ ✲ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✳ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛✳

✶✳ ❈♦✐♥t❡❣r❛çã♦ ❋r❛❝✐♦♥ár✐❛ ✷✳ ❉♦♠í♥✐♦ ❞❛ ❋r❡q✉ê♥❝✐❛ ✸✳ ❊st✐♠❛❞♦r ❙❡♠✐♣❛r❛♠étr✐❝♦

✹✳ ❉❡t❡r♠✐♥❛♥t❡ ❞❛ ▼❛tr✐③ ❞❡ ❉❡♥s✐❞❛❞❡ ❊s♣❡❝tr❛❧

■✳ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✳ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✳ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛✳

❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛✿

Pr♦❢✳ ❉r✳ ●➳♦r❣② ❚❡r❞✐❦ Pr♦❢✳ ❉r✳ ▼árt♦♥ ■s♣á♥②

✭❯♥✐✈❡rs✐t② ♦❢ ❉❡❜r❡❝❡♥✮ ✭❯♥✐✈❡rs✐t② ♦❢ ❉❡❜r❡❝❡♥✮

Pr♦❢✳ ❉r✳ ❋❧á✈✐♦ ❆✉❣✉st♦ ❩❡✐❣❡❧♠❛♥♥ Pr♦❢✳ ❉r✳ ❇r✉♥♦ ❞❡ P❛✉❧❛ ❘♦❝❤❛

✭❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ❙✉❧✮ ✭❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✮

Pr♦❢✳ ❉r✳ ❱❛❧❞❡r✐♦ ❆♥s❡❧♠♦ ❘❡✐s❡♥ Pr♦❢❛✳ ❉r❛✳ ●❧❛✉r❛ ❞❛ ❈♦♥❝❡✐çã♦ ❋r❛♥❝♦

❆❣r❛❞❡❝✐♠❡♥t♦s

❉✐✜❝✐❧♠❡♥t❡ ❛❧❣✉é♠ ❝❛♠✐♥❤❛ s♦③✐♥❤♦✳ ❈♦♠✐❣♦ ♥✉♥❝❛ ❢♦✐ ❞✐❢❡r❡♥t❡✳ ❙❡♠♣r❡ t✐✈❡ ❛♦ ♠❡✉ ❧❛❞♦ ✐♥ú♠❡r❛s ♣❡ss♦❛s q✉❡ ♠❡ ❛❥✉❞❛r❛♠ ❡♠ ❞✐✈❡rs❛s ❞✐♠❡♥sõ❡s✳ P♦rt❛♥t♦✱ ❝♦♠❡ç♦ ♣♦r ❛❣r❛❞❡❝❡r à ♠✐♥❤❛ ❡s♣♦s❛ ❡ ♠✐♥❤❛s ✜❧❤❛s✳ ●✐s❡❧❧❡ s❡♠♣r❡ ❡st❡✈❡ ❛♦ ♠❡✉ ❧❛❞♦ ♠❡ ❞❛♥❞♦ t♦❞♦ ❛♣♦✐♦ ♥❡❝❡ssár✐♦ ❡ ♦s r❡s✉❧t❛❞♦s q✉❡ ♦❜t✐✈❡ ❞❡✈♦ ❣r❛♥❞❡ ♣❛rt❡ ❛ ❡❧❛✳ ➚s ♠✐♥❤❛s ✜❧❤❛s ❊❧❡❛♥♦r ❡ ❙♦✜s❛ ♣❡ç♦ ❞❡s❝✉❧♣❛s ♣♦r ♥ã♦ t❡r s✐❞♦ ♠❛✐s ♣r❡s❡♥t❡ ❡♠❜♦r❛ s❡♠♣r❡ t❡♥❤❛ ♠❡ ❡s❢♦rç❛❞♦ ♣❛r❛ ♥ã♦ ❞❡✐①❛r ❧❛❝✉♥❛s✳ ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♣❛✐s✱ ▼❛r✐③❛ ❡ ▼❛r❝♦♥❡✱ q✉❡ ♥✉♥❝❛ ♠❡❞✐r❛♠ ❡s❢♦rç♦s ♣❛r❛ ♠❡ ❛❥✉❞❛r ❡ ❛♦s ♠❡✉s s♦❣r♦s ▼❛r❧❡♥❡ ❡ ❆❧❢r❡❞♦ ❜❡♠ ❝♦♠♦ ♠✐♥❤❛ ✐r♠ã ❏❛♥❛í♥❛ ❡ ❛♦ ❉❛♥✐❡❧✳ ❆❧❣✉♥s ❛♠✐❣♦s t❛♠❜é♠ ❢♦r❛♠ ✐♠♣♦rt❛♥t❡s ♥❡ss❛ tr❛❥❡tór✐❛✳ ❊♠ ❡s♣❡❝✐❛❧ ♠❡✉s ❝♦❧❡❣❛s ❞❡ ❞♦✉t♦r❛❞♦ ❋r❛♥❦ ❡ ■✈❛✐r✳ ❆❣r❛❞❡ç♦ à ♥♦ss❛ q✉❡r✐❞❛ ●❡r❛❧❞❛ ❊✉sé❜✐❛✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❉♦♥❛ ●❡r❛❧❞❛✱ ♣♦r t♦❞♦ ❡s❢♦rç♦ ❡ ❞❡❞✐❝❛çã♦ ❝♦♠ ❛s ♠✐♥❤❛s ✜❧❤❛s ♥❡st❡ ♣❡rí♦❞♦✳ ❙❡♠ ❛ s✉❛ ❛❥✉❞❛✱ ●ê❣❡✱ ❡✉ t❛♠❜é♠ ♥ã♦ t❡r✐❛ ❝♦♥s❡❣✐❞♦ ✐r ♠✉✐t♦ ❧♦♥❣❡✳

❘❡s✉♠♦

❊st❛ t❡s❡ s❡ ♣r♦♣õ❡ ❛ ❡st✉❞❛r ❛ ❝♦✐♥t❡❣r❛çã♦ ❢r❛❝✐♦♥ár✐❛ ♥♦ ❞♦♠í♥✐♦ ❞❛ ❢r❡q✉ê♥✲ ❝✐❛✳ ❆q✉✐ ✐♥✈❡st✐❣❛♠✲s❡ ❛s r❡str✐çõ❡s q✉❡ ❛ ❛✉sê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ ❝♦✐♥t❡❣r❛çã♦ ✐♠♣õ❡ s♦❜r❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ ❞❡ ❞❡♥s✐❞❛❞❡ ❡s♣❡❝tr❛❧ ❞❡ ✉♠ ✈❡t♦r ❞❡ sér✐❡s ❜✐✈❛r✐❛❞♦✱ ✐♥t❡❣r❛❞♦ ❞❡ ♦r❞❡♠ ✶✱ q✉❛♥❞♦ ❛✈❛❧✐❛❞♦ ♥❛ ♣r✐♠❡✐r❛ ❞✐❢❡r❡♥ç❛✳ P❡r♠✐t❡✲s❡✱ ❛q✉✐✱ q✉❡ ♦s ❡rr♦s ❞❛ r❡❧❛çã♦ ❞❡ ❝♦✐♥t❡❣r❛çã♦ s❡❥❛♠ ❢r❛❝✐♦♥❛❧♠❡♥t❡ ✐♥t❡❣r❛❞♦s✳ ◆❡st❡ ❡st✉❞♦ é ♠♦str❛❞♦ q✉❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ ❞❡ ❞❡♥s✐✲ ❞❛❞❡ ❡s♣❡❝tr❛❧ é ✉♠❛ ❢✉♥çã♦ ♣♦tê♥❝✐❛ ❞♦ ♣❛râ♠❡tr♦ q✉❡ ♠❡♥s✉r❛ ❛ r❡❞✉çã♦ ♥❛ ♦r❞❡♠ ❞❡ ✐♥t❡❣r❛çã♦ ❞♦ ❡rr♦ ✭❞❡♥♦t❛❞♦ ♣♦r b✮ ♣❛r❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❢r❡q✉ê♥❝✐❛s

❞❡ ❋♦✉r✐❡r ♣ró①✐♠❛s ❞❛ ♦r✐❣❡♠✳ ❆ ♣❛rt✐r ❞✐st♦✱ ❞✉❛s ♣r♦♣♦st❛s ♣❛r❛ ❛ ❡st✐♠❛çã♦ ❞♦ ♣❛râ♠❡tr♦ ❞❡ ❝♦✐♥t❡❣r❛çã♦ b sã♦ s✉❣❡r✐❞❛s✳ ❚❡st❡s s♦❜ ❛ ❤✐♣ót❡s❡ ♥✉❧❛ ❞❡ ♥ã♦

❝♦✐♥t❡❣r❛çã♦ sã♦ ❞❡r✐✈❛❞♦s ❛ ♣❛rt✐r ❞♦s ❡st✐♠❛❞♦r❡s ❛♣r❡s❡♥t❛❞♦s ❡ s✉❛s ♣r♦♣r✐✲ ❡❞❛❞❡s ❛ss✐♥tót✐❝❛s ❞✐s❝✉t✐❞❛s✳ ❊st✉❞♦s ❝♦♠ ❛♠♦str❛s ✜♥✐t❛s ❢♦r❛♠ r❡❛❧✐③❛❞♦s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❛✈❛❧✐❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❡♠♣ír✐❝♦ ❞♦s ❡st✐♠❛❞♦r❡s ❡ ❞♦s t❡st❡s ♣r♦♣♦st♦s ❛tr❛✈és ❞♦ ❝❛❧❝✉❧♦ ❞♦ ✈í❝✐♦✱ ❞♦ ❡rr♦ q✉❛❞rát✐❝♦ ♠é❞✐♦✱ ❞♦s ♥í✈❡✐s ❞❡ s✐❣♥✐✜❝â♥❝✐❛ ❡ ❞♦ ♣♦❞❡r✳ ❖s r❡s✉❧t❛❞♦s s✉❣❡r❡♠ q✉❡ ♦s t❡st❡s ♣♦ss✉❡♠ ♥í✈❡✐s ❞❡ s✐❣♥✐✜❝â♥❝✐❛ ❡♠♣ír✐❝♦s ♣ró①✐♠♦s ❛♦s ♥í✈❡✐s ♥♦♠✐♥❛✐s✳ ❆❧é♠ ❞✐st♦✱ ♦ ♣♦❞❡r ❞♦s t❡st❡s ❛♣r❡s❡♥t❛ ✉♠ ❞❡s❡♠♣❡♥❤♦ s✐♠✐❧❛r q✉❛♥❞♦ ❝♦♠♣❛r❛❞♦ ❝♦♠ ♦ ❞❡s❡♠♣❡♥❤♦ ❞❡ ♦✉tr♦s t❡st❡s ❝❧áss✐❝♦s ♥❛ ❧✐t❡r❛t✉r❛ ❞❡ ❝♦✐♥t❡❣r❛çã♦✳

❆❜str❛❝t

❚❤✐s t❤❡s✐s ♣r♦♣♦s❡s t♦ st✉❞② t❤❡ ❢r❛❝t✐♦♥❛❧ ❝♦✐♥t❡❣r❛t✐♦♥ ✐♥ t❤❡ ❢r❡q✉❡♥❝② ❞♦✲ ♠❛✐♥✳ ❍❡r❡ ✐s ✐♥✈❡st✐❣❛t❡❞ t❤❡ r❡str✐❝t✐♦♥s t❤❛t t❤❡ ❛❜s❡♥❝❡ ♦r t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❝♦✐♥t❡❣r❛t✐♦♥ ✐♠♣♦s❡s ♦♥ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ♠❛tr✐① ♦❢ ❛ ✈❡❝t♦r ♦❢ ❜✐✈❛r✐❛t❡ s❡r✐❡s✱ ✐♥t❡❣r❛t❡❞ ♦❢ ♦r❞❡r ✶✱ ✇❤❡♥ ❡✈❛❧✉❛t❡❞ ❛t t❤❡ ✜rst ❞✐✛❡r✲ ❡♥❝❡✳ ❚❤❡ ❡rr♦rs ♦❢ t❤❡ ❝♦✐♥t❡❣r❛t✐♦♥ r❡❧❛t✐♦♥s❤✐♣ ❛r❡ ❛❧❧♦✇❡❞ t♦ ❜❡ ❢r❛❝t✐♦♥❛❧❧② ✐♥t❡❣r❛t❡❞✳ ■♥ t❤✐s st✉❞② ✐t ✐s s❤♦✇♥ t❤❛t t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ t❤❡ s♣❡❝tr❛❧ ❞❡♥✲ s✐t② ♠❛tr✐① ✐s ❛ ♣♦✇❡r ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♣❛r❛♠❡t❡r t❤❛t ♠❡❛s✉r❡s r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ♦r❞❡r ♦❢ ✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ ❡rr♦r s❡r✐❡s ✭❞❡♥♦t❡❞ ❤❡r❡ ❜② b✮ ❢♦r ❛ s❡t ♦❢ ❋♦✉r✐❡r

❢r❡q✉❡♥❝✐❡s ❝❧♦s❡ t♦ t❤❡ ♦r✐❣✐♥✳ ❋r♦♠ t❤✐s✱ t✇♦ ♣r♦♣♦s❛❧s ❢♦r t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ❝♦✐♥t❡❣r❛t✐♥❣ ♣❛r❛♠❡t❡r b ❛r❡ s✉❣❣❡st❡❞✳ ❚❡sts ✉♥❞❡r t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s ♦❢ ♥♦♥✲

❝♦✐♥t❡❣r❛t✐♦♥ ❛r❡ ❞❡r✐✈❡❞ ❢r♦♠ t❤❡s❡ ❡st✐♠❛t♦rs ❛♥❞ t❤❡✐r ❛s②♠♣t♦t✐❝ ♣r♦♣❡rt✐❡s ❛r❡ ❞✐s❝✉ss❡❞✳ ❆ ✜♥✐t❡ s❛♠♣❧❡ ✐♥✈❡st✐❣❛t✐♦♥ ✇❛s ❝♦♥❞✉❝t❡❞ ✐♥ ♦r❞❡r t♦ ❡✈❛❧✉❛t❡ t❤❡ ❡♠♣✐r✐❝❛❧ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ❡st✐♠❛t♦rs ❛♥❞ t❡sts ❜② ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ❜✐❛s✱ t❤❡ ♠❡❛♥ sq✉❛r❡ ❡rr♦r✱ t❤❡ s✐❣♥✐✜❝❛♥❝❡ ❧❡✈❡❧s ❛♥❞ t❤❡ ♣♦✇❡r✳ ❚❤❡ r❡s✉❧ts s✉❣❣❡st t❤❛t t❡sts ❤❛✈❡ ❡♠♣✐r✐❝❛❧ s✐❣♥✐✜❝❛♥❝❡ ❧❡✈❡❧s ❝❧♦s❡ t♦ ♥♦♠✐♥❛❧ ❧❡✈❡❧s✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ♣♦✇❡r ♦❢ t❤❡ t❡sts s❤♦✇s ❛ s✐♠✐❧❛r ♣❡r❢♦r♠❛♥❝❡ ❝♦♠♣❛r❡❞ ✇✐t❤ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ♦t❤❡r ❝❧❛ss✐❝❛❧ t❡sts ✐♥ ❝♦✐♥t❡❣r❛t✐♦♥ ❧✐t❡r❛t✉r❡✳

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✷✳✶ P♦✇❡r ♦❢ ❆❉✱ ▼❆❉ ❛♥❞ ▲❉❘ ❛t ✺✪ s✐❣♥✐✜❝❛♥❝❡ ❧❡✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✷ ❊♠♣✐r✐❝❛❧ ❞❡♥s✐t✐❡s ♦❢ st❛♥❞❛r❞✐③❡❞ st❛t✐st✐❝s ❢♦r ❆❉✱ ▼❆❉ ❛♥❞

▲❉❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✸ ▲♦❣❛r✐t❤♠ ♦❢ st♦❝❦ ✈❛❧✉❡s✱ ❉♦✇ ❏♦♥❡s ✭s♦❧✐❞✮ ❛♥❞ ❋❚❙❊ ✶✵✵ ✭❞♦tt❡❞✮ ✸✼

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✷✳✶ ❙✐♠✉❧❛t❡❞ ✈❛❧✉❡s ❢♦r ❝♦rr❡❧❛t✐♦♥s ✉s❡❞ ✐♥ t❤❡ ❛s②♠♣t♦t✐❝ ✈❛r✐❛♥❝❡ ♦❢ ❆❉ ❛♥❞ ▼❆❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷ ❊st✐♠❛t❡s✱ s✐③❡ ❛♥❞ ♣♦✇❡r ❢♦r t❤❡ ❆❉ ♠❡t❤♦❞ ❛t ✺✪ s✐❣♥✐✜❝❛♥❝❡

❧❡✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✸ ❊st✐♠❛t❡s✱ s✐③❡ ❛♥❞ ♣♦✇❡r ❢♦r t❤❡ ▼❆❉ ♠❡t❤♦❞ ❛t ✺✪ s✐❣♥✐✜❝❛♥❝❡

❧❡✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✹ ❈r✐t✐❝❛❧ ✈❛❧✉❡s ❢♦r ▲❉❘ t₋like st❛t✐st✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✷✳✺ ❊st✐♠❛t❡s✱ s✐③❡ ❛♥❞ ♣♦✇❡r ❢♦r t❤❡ ▲❉❘ ♠❡t❤♦❞ ❛t ✺✪ s✐❣♥✐✜❝❛♥❝❡ ❧❡✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✻ ❙✐③❡ ❛♥❞ ♣♦✇❡r ❝♦♠♣❛r✐s♦♥ ❛t ✺✪ s✐❣♥✐✜❝❛♥❝❡ ❧❡✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✼ ❘♦❜✉st♥❡ss ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♠❡t❤♦❞s t♦ β ✈❛r✐❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✷✳✽ ❱❛❧✉❡s ♦❢ ✉♥✐t r♦♦t t❡st st❛t✐st✐❝ ❛♥❞ ❝r✐t✐❝❛❧ ✈❛❧✉❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✾ ❊st✐♠❛t❡s ❢♦rb❛♥❞ t❡st st❛t✐st✐❝ ❢♦r ♥♦♥✲❝♦✐♥t❡❣r❛t✐♦♥ ❜❡t✇❡❡♥ ❉❏

❛♥❞ ❋❚❙❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✸✳✶ ❊st✐♠❛t❡s✱ s✐③❡ ❛♥❞ ♣♦✇❡r ✉♥❞❡r ♦✉t❧✐❡rs ♣r❡s❡♥❝❡ ✈s ♦✉t❧✐❡rs ❢r❡❡ ✳ ✺✵ ✸✳✷ ❊st✐♠❛t❡s✱ s✐③❡ ❛♥❞ ♣♦✇❡r ✉s✐♥❣ r♦❜✉st ♣❡r✐♦❞♦❣r❛♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶

✷ ❚❡sts ❢♦r ♥♦♥✲❝♦✐♥t❡❣r❛t✐♦♥ ❜❛s❡❞ ♦♥ t❤❡ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥ ✹

✷✳✶ ■♥tr♦❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✷ ❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ♠❛tr✐① ❢♦r ❛ ❜✐✈❛r✐❛t❡ s❡r✐❡s ✾

✷✳✸ ❊st✐♠❛t✐♥❣b ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✷✳✸✳✶ ❚❤❡ ❧♦❣❣❡❞ ❞❡t❡r♠✐♥❛♥t r❡❣r❡ss✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✸✳✷ ❚❤❡ ❆✈❡r❛❣❡❞ ❉❡t❡r♠✐♥❛♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✷✳✹ ▼♦♥t❡ ❈❛r❧♦ ❙t✉❞② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✹✳✶ ❊♠♣✐r✐❝❛❧ r❡s✉❧ts ❢♦r t❡st✐♥❣ ♥♦♥✲❝♦✐♥t❡❣r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✷✳✹✳✷ ❘♦❜✉st♥❡ss t♦ ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ ❝♦✐♥t❡❣r❛t✐♦♥ ❞❡❝❧✐✈✐t②(β) ✲ ❆ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ ❧♦❣ ❝♦❤❡r❡♥❝❡ r❡❣r❡ss✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✺ ❆♣♣❧✐❝❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✷✳✻ ❈♦♥❝❧✉s✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

❆♣♣❡♥❞✐❝❡s ✸✾

✳✶ ❚❡❝❤♥✐❝❛❧ ▲❡♠♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✳✷ Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

✳✸ Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✳✹ Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✸ ❆❞❞✐t✐♦♥❛❧ ❘❡s✉❧ts✿ ❘♦❜✉st♥❡ss t♦ ♦✉t❧✐❡rs ✹✺

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r ❝♦✐♥t❡❣r❛çã♦ ❢r❛❝✐♦♥ár✐❛ s♦❜ ❞✐❢❡r❡♥t❡s ❝♦♥✲ t❡①t♦s ❡ ♣r♦♣♦r ♠ét♦❞♦s ❞❡ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❞❡ ✐♥t❡r❡ss❡ ❡ t❡st❡s ❞❡ ♥ã♦ ❝♦✐t❡❣r❛çã♦ ❜❛s❡❛❞♦s ♥♦ ❞♦♠í♥✐♦ ❞❛ ❢r❡q✉ê♥❝✐❛✳ ❖ ❝♦♥❝❡✐t♦ ❞❡ ❝♦✐♥t❡❣r❛çã♦✱ ✐♥tr♦❞✉③✐❞♦ ♣♦r ●r❛♥❣❡r ✭✶✾✽✶✮✱ t♦r♥♦✉✲s❡ ✉♠❛ ❞❛s té❝♥✐❝❛s ♠❛✐s ♣♦♣✉❧❛r❡s ❡♥✲ tr❡ ♦s ❡❝♦♥♦♠❡tr✐st❛s ✉♠❛ ✈❡③ q✉❡ ♣❡r♠✐t❡ ✈❡r✐✜❝❛r s❡ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ sér✐❡s ♥ã♦ ❡st❛❝✐♦♥ár✐❛s✱ ✐♥t❡❣r❛❞❛s ❞❡ ♠❡s♠❛ ♦r❞❡♠✱ ♣r♦❞✉③ ❡rr♦s ❝✉❥❛ ♦r❞❡♠ ❞❡ ✐♥t❡❣r❛çã♦ é r❡❞✉③✐❞❛✳

◆❡st❡ ❡st✉❞♦✱ ✐♥✈❡st✐❣❛♠✲s❡ ❛s r❡str✐çõ❡s q✉❡ ❛ ❛✉sê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ ❝♦✐♥t❡❣r❛✲ çã♦ ✐♠♣õ❡ s♦❜r❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ ❞❡ ❞❡♥s✐❞❛❞❡ ❡s♣❡❝tr❛❧ ❞❡ ✉♠ ✈❡t♦r ❞❡ sér✐❡s✳ P❛r❛ ❛ ♣❡sq✉✐s❛ ♣r♦♣♦st❛ sã♦ ❝♦♥s✐❞❡r❛❞♦s ✈❡t♦r❡s ❜✐✈❛r✐❛❞♦s ❡ ✐♥t❡❣r❛❞♦s ❞❡ ♦r❞❡♠ ✶ ❡ ❛✈❛❧✐❛❞♦s ♥❛ ♣r✐♠❡✐r❛ ❞✐❢❡r❡♥ç❛ ❝♦♠ ♦s ❡rr♦s ❞❛ r❡❧❛çã♦ ❞❡ ❝♦✐♥t❡✲ ❣r❛çã♦ ❢r❛❝✐♦♥❛❧♠❡♥t❡ ✐♥t❡❣r❛❞♦s✳ ❖ ♣♦♥t♦ ❢✉♥❞❛♠❡♥t❛❧ s♦❜ ♦ q✉❛❧ ❡st❡ tr❛❜❛❧❤♦ s❡ ❜❛s❡✐❛ é ♦ ❢❛t♦ ❞❡ q✉❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ ❞❡ ❞❡♥s✐❞❛❞❡ ❡s♣❡❝tr❛❧ é ✉♠❛ ❢✉♥çã♦ ♣♦tê♥❝✐❛ ❞♦ ♣❛râ♠❡tr♦ q✉❡ r❡❞✉③ ❛ ♦r❞❡♠ ❞❡ ✐♥t❡❣r❛çã♦ ❞♦ ❡rr♦✱ ❞❡♥♦t❛❞♦ ♣♦r b✱ ♣❛r❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❢r❡q✉ê♥❝✐❛s ❞❡ ❋♦✉r✐❡r ♣ró①✐♠❛s ❞❛ ♦r✐❣❡♠✳

◆❡ss❡ ❝♦♥t❡①t♦✱ ♥♦ ❈❛♣ít✉❧♦ ✷ é ❛♣r❡s❡♥t❛❞♦ ♦ ❛rt✐❣♦ ❚❡sts ❢♦r ♥♦♥✲❝♦✐♥t❡❣r❛t✐♦♥ ❜❛s❡❞ ♦♥ t❤❡ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥ q✉❡ é ❛ ♣❛rt❡ ❝❡♥tr❛❧ ❞❡st❛ ♣❡sq✉✐s❛✳ ❈♦♠ ❜❛s❡ ♥❛ t❡♦r✐❛ ❞♦ ❞♦♠í♥✐♦ ❞❛ ❢r❡q✉ê♥❝✐❛✱ ❛s ♣r♦♣r✐❡❞❛❞❡s ♠❛t❡♠át✐❝❛s ❡ ❡st❛tíst✐❝❛s ❞♦s ♣r♦❝❡ss♦s ❝♦✐♥t❡❣r❛❞♦s sã♦ ❞✐s❝✉t✐❞❛s✳ ❊♠ ❛❞✐çã♦✱ ❞✉❛s ♣r♦♣♦st❛s ♣❛r❛ ❛ ❡st✐♠❛çã♦ ❞♦ ♣❛râ♠❡tr♦ ❞❡ ❝♦✐♥t❡❣r❛çã♦b sã♦ s✉❣❡r✐❞❛s✿ ❛ ♣r✐♠❡✐r❛✱ ❜❛s❡❛❞❛ ❡♠

●❡✇❡❦❡ ❛♥❞ P♦rt❡r✲❍✉❞❛❦ ✭✶✾✽✸✮✱ ♣r♦♣õ❡ ✉♠❛ r❡❣r❡ssã♦ ❞♦ ❧♦❣❛r✐t♠♦ ❞♦ ❞❡t❡r✲ ♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ ❡s♣❡❝tr❛❧ ❞♦ ♣r♦❝❡ss♦ ❜✐✈❛r✐❛❞♦ ❡♠ ❡st✉❞♦✳ ❈♦♠♦ s❡❣✉♥❞❛ ♣r♦♣♦st❛✱ s✉❣❡r❡✲s❡ ✉♠ ❡st✐♠❛❞♦r s❡♠✐✲♣❛r❛♠étr✐❝♦ ❞♦ ❞❡t❡r♠✐♥❛♥t❡ ♠é❞✐♦ ❜❛✲ s❡❛❞♦ ♥❛ ♣r♦♣♦st❛ ❞❡ ❘♦❜✐♥s♦♥ ✭✶✾✾✹✮✳

❖ ❛rt✐❣♦ t❛♠❜é♠ ♣r♦♣õ❡ t❡st❡s s♦❜ ❛ ❤✐♣ót❡s❡ ♥✉❧❛ ❞❡ ♥ã♦ ❝♦✐♥t❡❣r❛çã♦✱ ♦ q✉❛✐s sã♦ ❞❡r✐✈❛❞♦s à ♣❛rt✐r ❞♦s ❡st✐♠❛❞♦r❡s s✉❣❡r✐❞♦s ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❛ss✐♥tó✲ t✐❝❛s ❞❡ss❡s t❡st❡s sã♦ ❞❡r✐✈❛❞❛s✳ ❊st✉❞♦s ❝♦♠ ❛♠♦str❛s ✜♥✐t❛s ❢♦r❛♠ r❡❛❧✐③❛❞♦s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❛✈❛❧✐❛r✱ ❡♠♣✐r✐❝❛♠❡♥t❡✱ ♦ ❞❡s❡♠♣❡♥❤♦ ❞♦s ❡st✐♠❛❞♦r❡s ❡ ❞♦s t❡st❡s ♣r♦♣♦st♦s ♣♦r ♠❡✐♦ ❞♦ ❝❛❧❝✉❧♦ ❞♦ ✈í❝✐♦ ✱❞♦ ❡rr♦ q✉❛❞rát✐❝♦ ♠é❞✐♦✱ ❞♦s ♥í✲ ✈❡✐s ❞❡ s✐❣♥✐✜❝â♥❝✐❛ ❡ ❞♦ ♣♦❞❡r✳ ❖s r❡s✉❧t❛❞♦s ❞♦ ♣♦❞❡r ❞♦s t❡st❡s ❡✈✐❞❡♥❝✐❛r❛♠ ✉♠ ❞❡s❡♠♣❡♥❤♦ s✐♠✐❧❛r ❝♦♠♣❛r❛❞♦ ❝♦♠ ♦✉tr♦s t❡st❡s ❝❧áss✐❝♦s ♥❛ ❧✐t❡r❛t✉r❛ ❞❡ ❝♦✐♥t❡❣r❛çã♦ ❞✐s❝✉t✐❞♦s ❡♠ ❉✐tt♠❛♥♥ ✭✷✵✵✵✮✳

❆ ❛✈❛❧✐❛çã♦ ❡♠♣ír✐❝❛ ❡①t❡♥❞❡✲s❡ ♣♦r ♠❡✐♦ ❞❡ ❝♦♠♣❛r❛çã♦ ❝♦♠ ❛ ♠❡t♦❞♦❧♦❣✐❛ ❛♣r❡s❡♥t❛❞❛ ❡♠ ❱❡❧❛s❝♦ ✭✷✵✵✸✮✳ ❊ss❡ ❛✉t♦r s✉❣❡r❡ ✉♠ ♠ét♦❞♦ ❛❧t❡r♥❛t✐✈♦ ♣❛r❛ ❡st✐♠❛çã♦ ❞♦ ♣❛râ♠❡tr♦ b✳ ❱❛❧❡ r❡ss❛❧t❛r q✉❡ ❱❡❧❛s❝♦ ✭✷✵✵✸✮ ❛♣r❡s❡♥t❛✱ ❞✐❢❡✲

r❡♥t❡♠❡♥t❡ ❞♦ ❡st✉❞♦ ♣r♦♣♦st♦✱ ❛s ♣r♦♣r✐❡❞❛❞❡s ❛ss✐♥tót✐❝❛s ❞♦ ❡st✐♠❛❞♦r s♦❜ ❛ ❤✐♣ót❡s❡ ❞❡ ❝♦✐♥t❡❣r❛çã♦✳ ▼✉✐t♦ ❡♠❜♦r❛ ♦ ❡st✐♠❛❞♦r s✉❣❡r✐❞♦ ❡♠ ❱❡❧❛s❝♦ ✭✷✵✵✸✮ ♣❡r♠✐t❛ q✉❡ ❛ ♦r❞❡♠ ❞❡ ✐♥t❡❣r❛çã♦ ❞♦ ✈❡t♦r s❡❥❛ s✉♣❡r✐♦r ❛ ✶✱ ♦s r❡s✉❧t❛❞♦s ❞❛s s✐♠✉❧❛çõ❡s ♠♦str❛♠ q✉❡ ♦ ♠ét♦❞♦ ❞❡ ❱❡❧❛s❝♦ ✭✷✵✵✸✮ ♥ã♦ é r♦❜✉st♦ ❛ ❞✐❢❡r❡♥t❡s ♣❛r❛♠❡tr✐③❛çõ❡s ❞♦ ✈❡t♦r ❝♦✐♥t❡❣r❛çã♦✱ ❞❡♥♦t❛❞♦ ❛q✉✐ ♣♦r β✱ ❛♦ ♣❛ss♦ q✉❡ ❛s

♣r♦♣♦st❛s s✉❣❡r✐❞❛s ♥❡st❛ t❡s❡ ♠♦str❛r❛♠✲s❡ r♦❜✉st❛s à ✈❛r✐❛çõ❡s ❡♠ β✳

❈♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ✐❧✉str❛r ❛ ❛♣❧✐❝❛çã♦ ❞♦s t❡st❡s ♣r♦♣♦st♦s✱ ♦ ❛rt✐❣♦ ❛♣r❡s❡♥t❛ ❛♥á❧✐s❡ ❞❡ sér✐❡s r❡❛✐s✳ ◆❡ss❡ ❝♦♥t❡①t♦✱ ❛ ❤✐♣ót❡s❡ ❞❡ ♥ã♦ ❝♦✐♥t❡❣r❛çã♦ é t❡st❛❞❛ ❡♥tr❡ ❛s sér✐❡s ❞♦ í♥❞✐❝❡ ❉♦✇ ❏♦♥❡s ❞❛ ❜♦❧s❛ ❞❡ ✈❛❧♦r❡s ❞❡ ◆♦✈❛ ■♦rq✉❡ ❡ ♦ í♥❞✐❝❡ ❋✐♥❛♥❝✐❛❧ ❚✐♠❡s ❙t♦❝❦ ❊①❝❤❛♥❣❡ ✶✵✵ ❞❛ ❜♦❧s❛ ❞❡ ▲♦♥❞r❡s✳ P❛r❛ t❛❧ ✐♥t❡r❡ss❡✱ ❢♦r❛♠ ❝♦❧❡t❛❞❛s ♦❜s❡r✈❛çõ❡s ♠❡♥s❛✐s ❝♦♠♣r❡❡♥❞✐❞❛s ❡♥tr❡ ❥❛♥❡✐r♦ ❞❡ ✶✾✽✺ ❡ ♠❛✐♦ ❞❡ ✷✵✶✹✳

❖ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ s✉❣❡r❡ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ♣❡r✐♦❞♦❣r❛♠❛s r♦❜✉st♦s ♥♦s t❡st❡ q✉❛♥❞♦ ❛s sér✐❡s ♣♦ss✉❡♠ ♦✉t❧✐❡rs✳ ❖ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s t❡st❡s r♦❜✉st♦s é ✈❡r✐✲ ✜❝❛❞♦ ♣♦r ♠❡✐♦ ❞❡ ❡♥s❛✐♦s ❡♠♣ír✐❝♦s✳ P♦r ✜♠✱ ♦ ❈❛♣ít✉❧♦ ✹ ❝♦♥❝❧✉✐ ♦ tr❛❜❛❧❤♦ ❝♦♠ ❛s s✉❣❡stõ❡s ❞❡ ♣❡sq✉✐s❛s ❢✉t✉r❛s✳

❈❛♣ít✉❧♦ ✷

❚❡sts ❢♦r ♥♦♥✲❝♦✐♥t❡❣r❛t✐♦♥ ❜❛s❡❞

♦♥ t❤❡ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥

■❣♦r ❱✐✈❡✐r♦s ▼❡❧♦ ❙♦✉③❛a_{✱ ❱❛❧❞❡r✐♦ ❆♥s❡❧♠♦ ❘❡✐s❡♥}b_{✱ ●❧❛✉r❛ ❞❛}

❈♦♥❝❡✐çã♦ ❋r❛♥❝♦c

a_{❉❊❈❊●✱ ❯❋❖P ❛♥❞ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛✱ ❯❋▼●}

b_{❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛✱ ❯❋❊❙ ❛♥❞ PP●❊❆✱ ❯❋❊❙}

c_{❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛✱ ❯❋▼●}

❆❜str❛❝t ❚❤❡ ❛✐♠ ♦❢ t❤✐s ♣❛♣❡r ✐s t♦ ♣r♦♣♦s❡ ♠❡t❤♦❞s t♦ t❡st t❤❡ ♥✉❧❧ ❤②✲ ♣♦t❤❡s✐s ♦❢ ♥♦♥✲❝♦✐♥t❡❣r❛t✐♦♥ ✐♥ ❜✐✈❛r✐❛t❡ s❡r✐❡s ❜❛s❡❞ ♦♥ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ♠❛tr✐① ❢♦r t❤❡ ❢r❡q✉❡♥❝✐❡s ❝❧♦s❡ t♦ t❤❡ ♦r✐❣✐♥✳ ❚✇♦ ❞✐✛❡r❡♥t st❛✲ t✐st✐❝s ❛r❡ ♣r♦♣♦s❡❞✿ t❤❡ ✜rst ♦♥❡ ✐s ❜❛s❡❞ ♦♥ ❛ r❡❣r❡ss✐♦♥ ♦❢ ❧♦❣❣❡❞ ❞❡t❡r♠✐♥❛♥t ♦♥ ❛ s❡t ♦❢ ❧♦❣❣❡❞ ❋♦✉r✐❡r ❢r❡q✉❡♥❝✐❡s ❛♥❞ t❤❡ s❡❝♦♥❞ st❛t✐st✐❝ ✐s t❤❡ s❡♠✐♣❛r❛✲ ♠❡tr✐❝ ❛✈❡r❛❣❡❞ ❞❡t❡r♠✐♥❛♥t ❡st✐♠❛t♦r✳ ■♥ t❤❡ st✉❞②✱ s❡r✐❡s ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡

I(1) ❛♥❞ t❤❡ ♦r❞❡r ♦❢ ✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ ❡rr♦r s❡r✐❡s ✐s I(1₋b)✱ b_∈[0,1]✱ t❤❛t ✐s✱ t❤❡ ♣❛r❛♠❡t❡r b ❞❡t❡r♠✐♥❡s t❤❡ r❡❞✉❝t✐♦♥ ✐♥ t❤❡ ♦r❞❡r ♦❢ ✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ ❡rr♦r

s❡r✐❡s✳ ❇❡s✐❞❡s✱ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ♠❛tr✐① ❢♦r t❤❡ ✜rst ❞✐✛❡✲ r❡♥❝❡ s❡r✐❡s ✐s ❛ ♣♦✇❡r ❢✉♥❝t✐♦♥ ♦❢b✳ ❆♥ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ♠❡t❤♦❞s ♣r♦♣♦s❡❞ ❤❡r❡

♦✈❡r t❤❡ st❛♥❞❛r❞ ♠❡t❤♦❞s ✐s t❤❛t t❤❡② ❛❧❧♦✇ t♦ ❦♥♦✇ t❤❡ ♦r❞❡r ♦❢ ✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ ❡rr♦r s❡r✐❡s ✇✐t❤♦✉t ❡st✐♠❛t✐♥❣ ❛ r❡❣r❡ss✐♦♥ ❡q✉❛t✐♦♥✳ ▼❡t❤♦❞s ❞✐s❝✉ss❡❞ ❤❡r❡ ♣♦ss❡ss ❝♦rr❡❝t s✐③❡ ❛♥❞ ❣♦♦❞ ♣♦✇❡r ❢♦r ♠♦❞❡r❛t❡ s❛♠♣❧❡ s✐③❡s ✇❤❡♥ ❝♦♠♣❛r❡❞ ✇✐t❤ ♦t❤❡r ♣r♦♣♦s❛❧s✳

❑❡②✇♦r❞s✿ ❋r❛❝t✐♦♥❛❧ ❝♦✐♥t❡❣r❛t✐♦♥❀ ❉❡t❡r♠✐♥❛♥t ♦❢ s♣❡❝tr❛❧ ❞❡♥s✐t② ♠❛tr✐①✱ ❙❡♠✐♣❛r❛♠❡tr✐❝ ❡st✐♠❛t♦r

✷✳✶ ■♥tr♦❞✉❝t✐♦♥

❚♦ st✉❞② t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❛♠♦♥❣ ❡❝♦♥♦♠✐❝ ✈❛r✐❛❜❧❡s✱ t❤❡ ❝♦♥❝❡♣t ♦❢ ❝♦✐♥t❡❣r❛✲ t✐♦♥✱ ✐♥tr♦❞✉❝❡❞ ❜② ●r❛♥❣❡r ✭✶✾✽✶✮✱ ❤❛s ❜❡❡♥ ✇✐❞❡❧② ❡♠♣❧♦②❡❞✱ ♠❛✐♥❧② ❞✉❡ t♦ t❤❡ s♣✉r✐♦✉s r❡❣r❡ss✐♦♥ ♣r♦❜❧❡♠✳ ❚❤❡ ❜❛s✐❝ ✐❞❡❛ ♦❢ ❝♦✐♥t❡❣r❛t✐♦♥ ❝♦♥s✐sts ✐♥ t❤❡ ❢❛❝t t❤❛t ❛h_×1✈❡❝t♦r s❡r✐❡sX_t✱_{t = 1,2, ...}✱ ✇❤❡r❡ ❡❛❝❤ ❝♦♠♣♦♥❡♥t ✐s ♥♦♥✲st❛t✐♦♥❛r②✱ ❝❛♥ ♣r♦❞✉❝❡ s♦♠❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ✐ts ❝♦♦r❞✐♥❛t❡s t❤❛t ❤❛s ❛ ❧♦✇❡r ♦r❞❡r ♦❢ ✐♥t❡❣r❛t✐♦♥✳ ❆❢t❡r t❤❡ s❡♠✐♥❛❧ ✇♦r❦ ♦❢ ●r❛♥❣❡r ✭✶✾✽✶✮✱ s❡✈❡r❛❧ st✉❞✐❡s ❛❜♦✉t t❤✐s t♦♣✐❝ ❤❛✈❡ ❜❡❡♥ ❞❡✈❡❧♦♣❡❞✳ ■♥ t❤❡ ❝❧❛ss✐❝ ❝♦♥t❡①t✱ t❤❡ ♠♦st ✉s❡❞ t❡sts ❢♦r ❝♦✐♥t❡❣r❛t✐♦♥ ❛r❡ t❤❡ ❊♥❣❧❡ ❛♥❞ ●r❛♥❣❡r ✭✶✾✽✼✮ t❡st ✭❊●✮✱ t❤❡ P❤✐❧❧✐♣s ❛♥❞ ❖✉✲ ❧✐❛r✐s ✭✶✾✽✽✮ t❡st ❛♥❞ t❤❡ ❏♦❤❛♥s❡♥ ✭✶✾✾✶✮ ♣r♦❝❡❞✉r❡✳ ❇❡s✐❞❡s✱ t❡sts t♦ ✈❡r✐❢② t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ✉♥✐t r♦♦t ❛r❡ ♥❡❝❡ss❛r② t♦ ✉s❡ ❛♣♣r♦♣r✐❛t❡ ♣r♦❝❡❞✉r❡s ❢♦r ♠♦❞❡❧✐♥❣ t❤❡ ❞❛t❛✳

❉❡s♣✐t❡ ✐ts ✇✐❞❡s♣r❡❛❞ ✉s❡✱ t❤❡ ❝❧❛ss✐❝❛❧ s❡t ✉♣ ♦❢ ❝♦✐♥t❡❣r❛t✐♦♥ ✐s ❧❛t❡❧② ❜❡✐♥❣ ❝♦♥s✐❞❡r❡❞ q✉✐t❡ r❡str✐❝t✐✈❡ ❢♦r ♠❛♥② r❡❛❧ ♣r♦❜❧❡♠s✳ ❆s ❛♥ ❛❧t❡r♥❛t✐✈❡✱ ❢r❛❝t✐♦✲ ♥❛❧ ❝♦✐♥t❡❣r❛t✐♦♥ ❤❛s ❡♠❡r❣❡❞ ❛s ❛ ♠♦r❡ ❛❞❡q✉❛t❡ ♠❡t❤♦❞♦❧♦❣② ❛♥❞ ❡①❛♠♣❧❡s

t♦ r❡❛❧ ♣r♦❜❧❡♠s ❝❛♥ ❜❡ s❡❡♥ ✐♥ ❈❤❡✉♥❣ ❛♥❞ ▲❛✐ ✭✶✾✾✸✮✱ ❇❛✐❧❧✐❡ ❛♥❞ ❇♦❧❧❡rs❧❡✈ ✭✶✾✾✹✮✱ ❉✐tt♠❛♥♥ ✭✷✵✵✶✮✱ ▼❝❍❛❧❡ ❛♥❞ P❡❡❧ ✭✷✵✶✵✮ ❛♥❞ ❈✉❡st❛s ❡t ❛❧✳ ✭✷✵✶✹✮✳ ❉✐✛❡r❡♥t ❛♣♣r♦❛❝❤❡s ❤❛✈❡ ❜❡❡♥ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ t❤❡ ❡st✐♠❛t✐♦♥ ❛♥❞ ❝♦♥str✉❝t✐♦♥ ♦❢ ❤②♣♦t❤❡s✐s t❡sts ❝♦♥❝❡r♥✐♥❣ ❢r❛❝t✐♦♥❛❧ ♣r♦❝❡ss❡s✳ ❙❡❡✱ ❢♦r ❡①❛♠♣❧❡✱ ❘♦❜✐♥s♦♥ ✭✶✾✾✹✮✱ ❘♦❜✐♥s♦♥ ❛♥❞ ▼❛r✐♥✉❝❝✐ ✭✷✵✵✶✮✱ ▼❛r✐♥✉❝❝✐ ❛♥❞ ❘♦❜✐♥s♦♥ ✭✷✵✵✶✮✱ ❘♦❜✐♥✲ s♦♥ ❛♥❞ ❨❛❥✐♠❛ ✭✷✵✵✷✮ ❛♥❞ ❱❡❧❛s❝♦ ✭✷✵✵✸✮✳

❚❤❡ ✜rst st❡♣ ✐♥ ❝♦✐♥t❡❣r❛t✐♦♥ ❛♥❛❧②s✐s ✐s t♦ ✈❡r✐❢② t❤❡ ♦r❞❡r ♦❢ ✐♥t❡❣r❛t✐♦♥ ♦❢ s❡r✐❡s Xi,t✱ t= 1,2, ...,✱ i= 1, ..., h✱ t❤❛t ❝♦♠♣♦s❡s t❤❡ ✈❡❝t♦rXt✳ ❆ s❡r✐❡s Xi,t ✐s

s❛✐❞ ✐♥t❡❣r❛t❡❞ ♦❢ ♦r❞❡r d✱ d _{∈ ℜ}✱ ❞❡♥♦t❡❞ ❜② Xi,t ∼ I(d)✱ ✐❢ d ✐s t❤❡ ♠✐♥✐♠✉♠

♥✉♠❜❡r ♦❢ ❞✐✛❡r❡♥❝❡s r❡q✉✐r❡❞ t♦ ♦❜t❛✐♥ ❛ ♣r♦❝❡ss t❤❛t ❛❞♠✐ts ❛♥ ❆✉t♦r❡❣r❡ss✐✈❡ ▼♦✈✐♥❣ ❆✈❡r❛❣❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭❆❘▼❆✮✳ ■♥ t❤✐s ❝♦♥t❡①t✱ ♣❛r❛♠❡t❡r d ♠❡❛s✉r❡s

t❤❡ ♠❡♠♦r② ♦❢ t❤❡ s❡r✐❡s✳ ❯s✐♥❣ t❤❡ ❆❘▼❆ r❡♣r❡s❡♥t❛t✐♦♥✱ s❡r✐❡s Xi,t ❝❛♥ ❜❡

✇r✐tt❡♥ ❛s✿

Xi,t = (1−B)−dei,t ✭✷✳✶✳✶✮

✇❤❡r❡ei,t =θq(B)φp−1(B)ui,t ✇✐t❤ui,t ❜❡✐♥❣ ❛ ✇❤✐t❡ ♥♦✐s❡ ♣r♦❝❡ss ✇✐t❤ ③❡r♦ ♠❡❛♥

❛♥❞ ❝♦♥st❛♥t ✈❛r✐❛♥❝❡ σ2

u✱ θq(B) ❛♥❞ φp(B) ❛r❡ ♣♦❧②♥♦♠✐❛❧s ✐♥ B ✇✐t❤ ♦r❞❡r q

❛♥❞ p✱ r❡s♣❡❝t✐✈❡❧②✱ ✇✐t❤ ❛❧❧ r♦♦ts ♦✉ts✐❞❡ ♦❢ t❤❡ ✉♥✐t ❝✐r❝❧❡ ✭s❡❡ ❍♦s❦✐♥❣ ✭✶✾✽✶✮✮✳

B ✐s t❤❡ ❜❛❝❦s❤✐❢t ♦♣❡r❛t♦r✱ t❤❛t ✐s✱ Bτ_X

i,t = Xi,t−τ ∀ τ ∈ N✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡

s❡r✐❡sXi,t ✐s s❛✐❞ t♦ ❜❡ ❛♥ ❆✉t♦r❡❣r❡ss✐✈❡ ❋r❛❝t✐♦♥❛❧❧② ■♥t❡❣r❛t❡❞ ▼♦✈✐♥❣ ❆✈❡r❛❣❡

♣r♦❝❡ss✱ ❞❡♥♦t❡❞ ❜② ❆❘❋■▼❆ ✭p, d, q✮✳ ❉✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ d ❛tt❛❝❤ ❞✐✛❡r❡♥t

♣r♦♣❡rt✐❡s t♦ t❤❡ s❡r✐❡s Xi,t✳ ❆ ♣r♦❝❡ss ✇✐t❤ d ≤ −0.5 ✐s st❛t✐♦♥❛r② ❜✉t ♥♦t

✐♥✈❡rt✐❜❧❡✳ ❲❤❡♥ d _∈ (₋0.5,0.5)✱ Xi,t ✐s ❜♦t❤ st❛t✐♦♥❛r② ❛♥❞ ✐♥✈❡rt✐❜❧❡✳ ❲❤❡♥

✐♥ t❤❡ s❡♥s❡ t❤❛t ✐♥♥♦✈❛t✐♦♥s ❞♦ ♥♦t ❤❛✈❡ ❧♦♥❣✲r✉♥ ✐♠♣❛❝t ♦♥ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ♣r♦❝❡ss✳ ❋♦r ✈❛❧✉❡s ♦❢ d_≥1✱ t❤❡ ♠❡❛♥✲r❡✈❡rs✐♦♥ ♣r♦♣❡rt② ✐s ♥♦ ❧♦♥❣❡r ✈❛❧✐❞ ✭❢♦r

❞❡t❛✐❧s s❡❡ ❈❤❡✉♥❣ ❛♥❞ ▲❛✐ ✭✶✾✾✸✮✮✳

■❢Xi,t✐s ❛ st❛t✐♦♥❛r② ♣r♦❝❡ss✱ ✐t ❤❛s ❛ s♣❡❝tr❛❧ ❞❡♥s✐t② ❢✉♥❝t✐♦♥✶✱fX(λ)✱ ✇❤✐❝❤

❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s✿

fX(λ) = fe(λ)

1₋e−iλ−2d ✭✷✳✶✳✷✮

✇❤❡r❡ fe(λ) ✐s t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ♦❢ ❛ st❛t✐♦♥❛r② ❆❘▼❆ ♣r♦❝❡ss et ✇✐t❤ λ ∈

[0,2π) ❛♥❞ d _{∈ ℜ}✳ ❲❤❡♥ t❤❡ s❡r✐❡s Xi,t ✐s ♥♦♥✲st❛t✐♦♥❛r②✱ ✐✳❡✳✱ d ≥ 0.5✱ t❤❡

❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ✐♥ ❊q✉❛t✐♦♥ ✷✳✶✳✷ ✐s ✉s✉❛❧❧② ❞❡♥♦t❡❞ ♣s❡✉❞♦✲s♣❡❝tr❛❧ ❞❡♥s✐t② ✭s❡❡✱ ❢♦r ❡①❛♠♣❧❡✱ ❱❡❧❛s❝♦ ✭✷✵✵✸✮✮✳

❆ ❣❡♥❡r❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❢r❛❝t✐♦♥❛❧ ❝♦✐♥t❡❣r❛t✐♦♥ ✇❛s ❣✐✈❡♥ ❜② ❘♦❜✐♥s♦♥ ❛♥❞ ▼❛r✐♥✉❝❝✐ ✭✶✾✾✽✮✱ ❛❧❧♦✇✐♥❣ ❛ ❞✐✛❡r❡♥t ♦r❞❡r ♦❢ ✐♥t❡❣r❛t✐♦♥ ❢♦r Xi,t✱ t❤❛t ✐s✱Xi,t ∼

I(di)✱ di >0✱ ∀ i✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❢r❛❝t✐♦♥❛❧ ❝♦✐♥t❡❣r❛t✐♦♥ ❢♦r ❛h×1✈❡❝t♦r Xt ✐s

❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿

❉❡✜♥✐t✐♦♥ ✶✳ ▲❡t X_t✱ _{t = 1,2, ...}✱ ❜❡ ❛ _h×1 ✈❡❝t♦r s❡r✐❡s ✇❤♦s❡ ✐✲t❤ ❡❧❡♠❡♥t

Xi,t ∼ I(di), di > 0, i = 1, ..., h✳ Xt ✐s ❝❛❧❧❡❞ ❢r❛❝t✐♦♥❛❧❧② ❝♦✐♥t❡❣r❛t❡❞✱ ❞❡♥♦t❡❞

❜② X_{t ∼ F CI(d1, ..., dh, dε)}✱ ✐❢ t❤❡r❡ ❡①✐sts ❛ _h×1 ✈❡❝t♦r _{β 6= 0} s✉❝❤ t❤❛t _{εt =}

βTX_t∼I(dε)✱ ✇❤❡r❡ _{0< dε <min1≤i≤hdi}✳

❚❤❡ ❛❜♦✈❡ ❞❡✜♥✐t✐♦♥ ✐s ✈❛❧✐❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢di =dj ❢♦r s♦♠❡i6=j✱i, j = 1, ..., h✳

❚❤❡ ✈❡❝t♦r β ✐s ❝❛❧❧❡❞ ❝♦✐♥t❡❣r❛t✐♦♥ ✈❡❝t♦r✳ ■♥ t❤❡ ❝❛s❡ t❤❛td1 =...=dh =d✱ ✐t

✐s ✉s✉❛❧ t♦ ✇r✐t❡ X_{t ∼ CI(d, b)} ✇❤❡r❡ _{b =d−dε}✳ ❲❤❡♥ _{b = 0} t❤❡ ✈❡❝t♦r X_t ✐s ♥♦♥✲❝♦✐♥t❡❣r❛t❡❞✳ ■♥ t❤✐s s❡♥s❡✱ ♣❛r❛♠❡t❡r b ♠❡❛s✉r❡s t❤❡ r❡❞✉❝t✐♦♥ ✐♥ t❤❡ ♦r❞❡r

✶_{❚❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ♦❢ ❛♥ st❛t✐♦♥❛r② ♣r♦❝❡ssX}

t ✐s t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ t❤❡ ❛✉t♦❝♦✈❛✲

r✐❛♥❝❡ ❢✉♥❝t✐♦♥✱γX(τ) =E{(Xt+τ−µX) (Xt−µX)}✱ t❤❛t ✐s✱fX(λ) =21π

∞ P

τ=−∞

γX(τ)eiτ λ✳

♦❢ ✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ ❡rr♦r s❡r✐❡sεt✳

❚♦ t❡st t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s ♦❢ ❢r❛❝t✐♦♥❛❧ ♥♦♥✲❝♦✐♥t❡❣r❛t✐♦♥✱ ❛ ❣❡♥❡r❛❧ ❛♣♣r♦❛❝❤ ✐s t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ♦r❞❡r ♦❢ ✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ r❡s✐❞✉❛❧ s❡r✐❡s εˆt = ˆβ

T

X_t✱ ❛❢t❡r ❡st✐♠❛t✐♥❣ ✈❡❝t♦r β ✭s❡❡ ❉✐tt♠❛♥♥ ✭✷✵✵✵✮✮✳

❱❛r✐♦✉s ❡st✐♠❛t♦rs ♦❢dε❝❛♥ ❜❡ ✉s❡❞ ✐♥ ❤②♣♦t❤❡s✐s t❡sts ❢♦r ❢r❛❝t✐♦♥❛❧❧② ❝♦✐♥t❡✲

❣r❛t❡❞ ♣r♦❝❡ss❡s ✭s❡❡✱ ❢♦r ❡①❛♠♣❧❡✱ ❉✐tt♠❛♥♥ ✭✷✵✵✵✮ ❛♥❞ ❙❛♥t❛♥❞❡r ❡t ❛❧✳ ✭✷✵✵✸✮✮✳ ❆♥♦t❤❡r ❛♣♣r♦❛❝❤ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❱❡❧❛s❝♦ ✭✷✵✵✸✮ ✇❤♦ ♣r♦♣♦s❡❞ ❛ ♠❡t❤♦❞ t♦ ❡s✲ t✐♠❛t❡ ❛♥❞ t❡st t❤❡ ♣❛r❛♠❡t❡r b ✉♥❞❡r t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s ♦❢ ❝♦✐♥t❡❣r❛t✐♦♥✳

❚❤✉s✱ t❤❡ ♠❛✐♥ ♦❜❥❡❝t✐✈❡s ♦❢ t❤✐s ✇♦r❦ ❛r❡ t♦ ♣r♦♣♦s❡ ♥❡✇ ♠❡t❤♦❞s ❢♦r ❡s✲ t✐♠❛t✐♥❣ t❤❡ ♣❛r❛♠❡t❡r b ❛♥❞✱ ❛❧s♦✱ ❛ t❡st ♦❢ ♥♦♥✲❝♦✐♥t❡❣r❛t✐♦♥ ❜❛s❡❞ ♦♥ t❤❡

❞❡t❡r♠✐♥❛♥t ♦❢ t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ♠❛tr✐① ♦❢ t❤❡ ✈❡❝t♦r (∆X1,t,∆X2,t)✱ ✇❤❡r❡

∆ ✐s t❤❡ ✜rst ❞✐✛❡r❡♥❝❡ ♦♣❡r❛t♦r✱ t❤❛t ✐s✱ ∆ = (1₋B)✳ ❍❡r❡✱ s♣❡❝✐❛❧ ❛tt❡♥t✐♦♥

✐s ♣❛✐❞ t♦ t❤❡ ❝❛s❡ ✇❤❡r❡ d = 1✱ ❛❧t❤♦✉❣❤ t❤❡ ♣r♦❝❡❞✉r❡s ❝❛♥ ❛❧s♦ ❜❡ ❛❞❛♣t❡❞

t♦ ♦t❤❡r ❝❛s❡s s✉❝❤ ❛s d ₆= 1✳ ■♥ t❤✐s s✐t✉❛t✐♦♥ ❛♥ ❛♣♣r♦♣r✐❛t❡ ❡st✐♠❛t♦r ♦❢ d ✐s

r❡q✉✐r❡❞✳

❙♦♠❡ t❤❡♦r❡t✐❝❛❧ r❡s✉❧ts ❛r❡ ❡st❛❜❧✐s❤❡❞ ❢♦r t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞s ❛♥❞ ❛♥ ❡♠♣✐r✐❝❛❧ ▼♦♥t❡ ❈❛r❧♦ st✉❞② ✐s ❝♦♥❞✉❝t t♦ ❡✈❛❧✉❛t❡ t❤❡✐r ♣❡r❢♦r♠❛♥❝❡ ❢♦r s♠❛❧❧ s❛♠♣❧❡ s✐③❡s✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ❝❧❛ss✐❝❛❧ ❝♦✐♥t❡❣r❛t✐♦♥ ♠❡t❤♦❞s ❛r❡ ❛❧s♦ ❝♦♥s✐❞❡r❡❞ ✐♥ ❡♠♣✐r✐❝❛❧ st✉❞✐❡s ❢♦r ❝♦♠♣❛r✐s♦♥ ♣✉r♣♦s❡s✳

❚❤❡ ♣❛♣❡r ✐s str✉❝t✉r❡❞ ❛s ❢♦❧❧♦✇s✳ ■♥ ❙❡❝t✐♦♥ ✷ s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞❡✲ t❡r♠✐♥❛♥t ♦❢ t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ♠❛tr✐① ❢♦r ❝♦✐♥t❡❣r❛t❡❞ ❛♥❞ ♥♦♥✲❝♦✐♥t❡❣r❛t❡❞ s❡r✐❡s ✐♥ t❤❡ ✜rst ❞✐✛❡r❡♥❝❡ ❛r❡ ❛♥❛❧②s❡❞✳ ❙❡❝t✐♦♥ ✸ ♣r❡s❡♥ts t❤❡ ❧♦❣ ❞❡t❡r♠✐♥❛♥t r❡❣r❡ss✐♦♥ ❡st✐♠❛t♦r✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ❛✈❡r❛❣❡❞ ❞❡t❡r♠✐♥❛♥t ❡st✐♠❛t♦r ❛♥❞ ✐ts ♠♦❞✐✜❝❛t✐♦♥ ❛r❡ ❞✐s❝✉ss❡❞✳ ❆ ▼♦♥t❡ ❈❛r❧♦ st✉❞② t♦ ❛♥❛❧②s❡ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ♣r♦♣♦s❛❧s s✉❣❣❡st❡❞ ❤❡r❡ ✐♥ t❡r♠s ♦❢ ❜✐❛s✱ s✐③❡ ❛♥❞ ♣♦✇❡r ✐s ♣r❡s❡♥t❡❞ ✐♥ ❙❡❝t✐♦♥

✹✳ ❚❤✐s s❡❝t✐♦♥ ❛❧s♦ ❝♦♠♣❛r❡s ♠❡t❤♦❞s ♣r♦♣♦s❡❞ ❤❡r❡ ✇✐t❤ r❡s✐❞✉❛❧ ❜❛s❡❞ t❡sts ♣r❡s❡♥t❡❞ ✐♥ ❉✐tt♠❛♥♥ ✭✷✵✵✵✮ ❛♥❞ t♦ t❤❡ ▲♦❣ ❈♦❤❡r❡♥❝② ❘❡❣r❡ss✐♦♥ ♠❡t❤♦❞ ✐♥ ❱❡❧❛s❝♦ ✭✷✵✵✸✮✳ ❙❡❝t✐♦♥ ✺ s❤♦✇s ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞♦❧♦❣✐❡s t♦ ❛ r❡❛❧ t✐♠❡ s❡r✐❡s ❛♥❞ ❙❡❝t✐♦♥ ✻ ❝♦♥❝❧✉❞❡s t❤❡ ✇♦r❦✳

✷✳✷ ❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ♠❛✲

tr✐① ❢♦r ❛ ❜✐✈❛r✐❛t❡ s❡r✐❡s

❚❤✐s s❡❝t✐♦♥ ♣r❡s❡♥ts t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ♠❛tr✐① ❢♦r t❤❡ ✈❡❝t♦r(∆X1,t,∆X2,t)✇❤❡r❡ ❜♦t❤ ❝♦♠♣♦♥❡♥ts ❛r❡I(1)❛♥❞ s❛t✐s✜❡s

t❤❡ ❧✐♥❡❛r r❡❧❛t✐♦♥s❤✐♣ X1,t = βX2,t +εt ❢♦r s♦♠❡ β 6= 0✳ ❚❤❡ ❡rr♦r t❡r♠ εt ✐s

❛ss✉♠❡❞ t♦ ❜❡ I(1₋b)✱ ✇✐t❤0_≤b_≤1✱ t❤❛t ✐s✱ t❤❡ ♦r❞❡r ♦❢ ✐♥t❡❣r❛t✐♦♥ ❝❛♥ t❛❦❡ ♥♦♥✐♥t❡❣❡r ✈❛❧✉❡s✳

■❢(X1,t, X2,t)✐s ❝♦✐♥t❡❣r❛t❡❞✱ ✐✳❡✳✱b ∈(0,1]✱ t❤❛♥ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ t❤❡ s♣❡❝✲

tr❛❧ ❞❡♥s✐t② ♠❛tr✐① ♦❢(∆X1,t,∆X2,t)✐s ❛ ♣♦✇❡r ❢✉♥❝t✐♦♥ ♦❢ b✳ ▲❡t t❤❡ ♦❜s❡r✈❛❜❧❡

❜✐✈❛r✐❛t❡ t✐♠❡ s❡r✐❡s (X1,t, X2,t) ❜❡ ❢♦r♠❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠✿

X1,t =β1Tt+w1,t

X2,t =β2Tt+w2,t

✭✷✳✷✳✶✮

❢♦r t = 1,2, ...✱ β1 = 06 ❛♥❞ β2 6= 0✳ ❚❤❡ s❡r✐❡s Tt ✐s ❛ ❝♦♠♠♦♥ ✉♥♦❜s❡r✈❛❜❧❡

st♦❝❤❛st✐❝ tr❡♥❞ s✉❝❤ t❤❛t✿

Tt = (1−B)−1ηt ✭✷✳✷✳✷✮

❛♥❞ t❤❡ ✐♥♥♦✈❛t✐♦♥s ηt ❛r❡ ❛ st❛t✐♦♥❛r② ❆❘▼❆ ♣r♦❝❡ss ✇✐t❤ ③❡r♦ ♠❡❛♥ s✉❝❤ t❤❛t

∞

P

τ=−∞|

γη(τ)| < ∞ ✇❤❡r❡ γη(τ) ✐s t❤❡ ❛✉t♦❝♦✈❛r✐❛♥❝❡ ♦❢ ♦r❞❡r τ✳ ❚❤❡ ♣❛✐r ♦❢

✐♥♥♦✈❛t✐♦♥s (w1,t, w2,t) ❢♦❧❧♦✇s t❤❡ ♣r♦❝❡ss❡s✿

w1,t = (1−B)−(1−b1)e1,t

w2,t = (1−B)−(1−b2)e2,t

✭✷✳✷✳✸✮

✇❤❡r❡b1 ∈[0,1]❛♥❞ b2 ∈[0,1]✳ ❚❤❡ ✈❡❝t♦r(e1,t, e2,t)❢♦❧❧♦✇s ❛ ③❡r♦ ♠❡❛♥ ❆❘▼❆

♣r♦❝❡ss ✇✐t❤ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✷ _{Σ =}

   

∞

P

τ=−∞

γe1(τ) 0

0 P∞

τ=−∞

γe2(τ)

  

✱ s✉❝❤ t❤❛t

∞

P

τ=−∞|

γe1(τ)|<∞✱

∞

P

τ=−∞|

γe2(τ)|<∞❛♥❞ ✐t ✐s ✉♥❝♦rr❡❧❛t❡❞ ✇✐t❤ηt✳ ❚❤❡ s②st❡♠

❞❡s❝r✐❜❡❞ ✐♥ ❊q✉❛t✐♦♥ ✷✳✷✳✶ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✿

X1,t=βX2,t+εt ✭✷✳✷✳✹✮

✇❤❡r❡ β =β1/β2✱ β1 6= 0✱ β2 6= 0 ❛♥❞ εt=w1,t−(β1/β2)w2,t ✐s ❛ ♥♦♥✲♦❜s❡r✈❛❜❧❡

❡rr♦r t❡r♠ s✉❝❤ t❤❛t εt∼I(1−b) ✇✐t❤ b= min (b1, b2)✳

❋♦❧❧♦✇✐♥❣ ❉❡✜♥✐t✐♦♥ ✶✱ t❤❡ ✈❡❝t♦r (X1,t, X2,t) ✇✐❧❧ ❜❡ ♥♦♥✲❝♦✐♥t❡❣r❛t❡❞ ✐❢ ❛♥❞

♦♥❧② ✐❢b = 0✳ ◆♦t❡ t❤❛t ✐♥ ❊q✉❛t✐♦♥ ✷✳✷✳✹✱ t❤❡ ✐♥♣✉t s❡r✐❡s X2,t ❛♥❞ ❡rr♦r t❡r♠εt

❛r❡ ❝♦rr❡❧❛t❡❞✳ ❚♦ ✐♠♣♦s❡ ♦rt❤♦❣♦♥❛❧✐t② ❜❡t✇❡❡♥ X2,t ❛♥❞ εt ✐t ✐s ♥❡❝❡ss❛r② t❤❛t

σ2

u2 = 0 ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t X2,t =Tt✳ ❚❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ♠❛tr✐① ♦❢ t❤❡ ✈❡❝t♦r

✷_{❲✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱Σ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❞✐❛❣♦♥❛❧ ✐♥ ♦r❞❡r t♦ ❛✈♦✐❞ t❤❡ ❝r♦ss s♣❡❝tr✉♠} t❡r♠s ❜❡t✇❡❡♥e1,t ❛♥❞e2,t ❛♥❞ t♦ ♠❛❦❡ t❤❡ ❝❛❧❝✉❧❛t✐♦♥s ❡❛s✐❡r

(∆X1,t,∆X2,t) ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ✭s❡❡ Pr✐❡st❧❡② ✭✶✾✽✶✮ ♣✳✻✺✽✲✻✺✾✮✿

F_{(λ) =}

∞

P

τ=−∞

1 2π

  

E_{{∆X1,t+τ∆X1,t}} E_{{∆X1,t+τ∆X2,t}} E_{{∆X2,t+τ∆X1,t}} E_{{∆X2,t+τ∆X2,t}}

  e−iλτ

=

 

 f∆X1(λ) f∆X1∆X2(λ)

f∆X2∆X1(λ) f∆X2(λ)

  ,

✭✷✳✷✳✺✮

✇❤❡r❡ f∆X1(λ) ❛♥❞ f∆X2(λ) ❛r❡ t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t✐❡s ♦❢ ∆X1,t ❛♥❞ ∆X2,t✱ r❡s✲

♣❡❝t✐✈❡❧② ❛♥❞f∆X1∆X2(λ)❛♥❞f∆X2∆X1(λ)❛r❡ t❤❡ ❝r♦ss✲s♣❡❝tr✉♠ ❜❡t✇❡❡♥∆X1,t

❛♥❞ ∆X2,t✳ ❚❤❡ ♠❛tr✐① F(λ) ✐s ❍❡r♠✐t✐❛♥ ✇❤✐❝❤ ♠❡❛♥s t❤❛t f∆X1∆X2(λ) =

f∆X2∆X1(λ)✱ ✇❤❡r❡ t❤❡ ♦✈❡r ❧✐♥❡ ♠❡❛♥s t❤❡ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡✳

❯s✐♥❣ t❤❡ st❛♥❞❛r❞ s♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s ♦❢ ♠✉❧t✐✈❛r✐❛t❡ t✐♠❡ s❡r✐❡s✱ F_(λ) ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s ✭s❡❡ Pr✐❡st❧❡② ✭✶✾✽✶✮✮✿

F_{(λ) =}

  β

2

1fη(λ) +|1−e−iλ| 2b1

fe1(λ) β1β2fη(λ)

β2β1fη(λ) β22fη(λ) +|1−e−iλ| 2b2

fe2(λ)

 

. ✭✷✳✷✳✻✮

❚❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ♠❛tr✐① F_(λ)✐s✿

D(λ) = _|1₋e−iλ_|2b1

β₂2fe1(λ)fη(λ) +|1−e

−iλ

|2b2

β₁2fe2(λ)fη(λ)+

|1₋e−iλ_|2(b1+b2)fe1(λ)fe2(λ). ✭✷✳✷✳✼✮

❆ss✉♠✐♥❣ ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t② t❤❛t b1 ≤ b2✱ ✇❤✐❝❤ ♠❛❦❡s b = b1✱ ❛♥❞

✉s✐♥❣ t❤❡ ❢❛❝t t❤❛t✱

|1₋e−iλ_|2b∗ = (2₋2 cosλ)b∗,

❛♥❞✿

lim

λ→0+

(2₋2 cosλ)b∗

λ2b∗ = 1,

✇❤✐❝❤ ♠❡❛♥s t❤❛t ❢♦r b∗ _{∈ ℜ✱ |1−e}−iλ_|2b∗ _=O(λ2b∗

)✱ t❤❡ ❞❡t❡r♠✐♥❛♥t D(λ) ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s✿

D(λ) = _|1₋e−iλ_|2bG(λ)

G(0)G(0) +O(λ

2b2_{) +O(λ}2(b+b2)_), ✭✷✳✷✳✽✮

✇❤❡r❡ G(λ) ✐s ❛ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥ ❞✉❡ t♦ t❤❡ st❛t✐♦♥❛r✐t② ♦❢ t❤❡ ♣r♦❝❡ss❡s e1,t✱

e2,t ❛♥❞ ηt s✉❝❤ t❤❛t✿

lim

λ→0+

G(λ)

G(0) = 1.

❋r♦♠ t❤✐s✱ t❤❡ ❞❡t❡r♠✐♥❛♥tD(λ)❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛s✿

D(λ)_{∼ |}1₋e−iλ_|2bG(λ)

G(0)G(0) as λ−→0

+_{, ✭✷✳✷✳✾✮}

✇❤❡r❡ t❤❡ s②♠❜♦❧ ”_∼ ”♠❡❛♥s t❤❛t r❛t✐♦ ♦❢ ❧❡❢t ❛♥❞ r✐❣❤t✲❤❛♥❞ s✐❞❡s t❡♥❞s t♦ ❛ ❝♦♥st❛♥t 0< C < _∞❛sλ _→0+_{✳ ❋r♦♠ t❤❡ ❊q✉❛t✐♦♥ ✷✳✷✳✾✱ D(λ)❞❡♣❡♥❞s ♦♥ t❤❡}

r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ♦r❞❡r ♦❢ ✐♥t❡❣r❛t✐♦♥ b ✐♠♣♦s❡❞ ❜② ❝♦✐♥t❡❣r❛t✐♦♥✳ ❙✐♠✐❧❛r r❡s✉❧ts

❛r❡ ❛❧s♦ ❞❡s❝r✐❜❡❞ ❜② ◆✐❡❧s❡♥ ✭✷✵✵✹✮✳ ❚❤❡r❡❢♦r❡✱ ✐❢ (X1,t, X2,t) ✐s ❝♦✐♥t❡❣r❛t❡❞✱

t❤❛t ✐s✱ 0 < b_≤1✱ D(λ) _→0 ❛s λ _−→0+_{✳ ■t ♠❡❛♥s t❤❛t F(λ) ✐s ❛ ♠❛tr✐① ✇✐t❤}

✐♥❝♦♠♣❧❡t❡ r❛♥❦ ❛tλ= 0 ✭s❡❡ P❤✐❧❧✐♣s ❛♥❞ ❖✉❧✐❛r✐s ✭✶✾✽✽✮✮✳ ■♥ t❤❡ ❝❛s❡ ♦❢b= 0✱

t❤❛t ✐s✱ (X1,t, X2,t)✐s ♥♦♥✲❝♦✐♥t❡❣r❛t❡❞✱ D(λ)→C ❛sλ −→0+ ❛♥❞ F(λ)❤❛s ❢✉❧❧

r❛♥❦ ❛t λ= 0✳

❚❤❡r❡❢♦r❡✱ ♥❡✇ ♠❡t❤♦❞s t♦ ❡st✐♠❛t❡ ♣❛r❛♠❡t❡r b ❛♥❞ t❡st t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s

♦❢ ♥♦♥✲❝♦✐♥t❡❣r❛t✐♦♥ ❛r❡ ♣r♦♣♦s❡❞ ❜② ❛♥❛❧②s✐♥❣ t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ D(λ)✐♥

❛ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ③❡r♦ ❢r❡q✉❡♥❝②✳

✷✳✸ ❊st✐♠❛t✐♥❣

b

❙t❛♥❞❛r❞ ❡st✐♠❛t✐♦♥ ♠❡t❤♦❞s ❢♦r t❤❡ ♠❡♠♦r② ♣❛r❛♠❡t❡r d✱ ✇❡❧❧ ❞✐s❝✉ss❡❞ ✐♥

t❤❡ ❧✐t❡r❛t✉r❡ ♦❢ ❧♦♥❣ ♠❡♠♦r② ♣r♦❝❡ss❡s✱ ❝❛♥ ❜❡ ✉s❡❞ ❛s ❛❧t❡r♥❛t✐✈❡ ♣r♦❝❡❞✉r❡s t♦ ♦❜t❛✐♥ ❡st✐♠❛t❡s ♦❢ b✳ ❚❤❡s❡ ♣r♦❝❡❞✉r❡s ❛r❡ ❛❞❞r❡ss❡❞ ❤❡r❡ ✉s✐♥❣ t❤❡ ❢❛❝t

t❤❛t D(λ) _∼ O(λ2b_{)✳ ❚❤❡ ✜rst ♣r♦♣♦s❛❧ ✐s s✐♠✐❧❛r t♦ t❤❡ ❛♣♣r♦❛❝❤ ♦❢ ●❡✇❡❦❡}

❛♥❞ P♦rt❡r✲❍✉❞❛❦ ✭✶✾✽✸✮✱ ✇❤❡r❡ t❤❡ ❧♦❣❣❡❞ ♣❡r✐♦❞♦❣r❛♠ ✐s r❡❣r❡ss❡❞ ♦♥ ❧♦❣❣❡❞ ❋♦✉r✐❡r ❢r❡q✉❡♥❝✐❡s✳ ❚❤❡ s❡❝♦♥❞ ♦♥❡ ✐s ❜❛s❡❞ ♦♥ ❘♦❜✐♥s♦♥ ✭✶✾✾✹✮ s❡♠✐♣❛r❛♠❡tr✐❝ ❛✈❡r❛❣❡❞ ♣❡r✐♦❞♦❣r❛♠ ❡st✐♠❛t♦r ♦❢ d✱ ✇❤❡r❡ ❛ ❧♦❣❣❡❞ r❛t✐♦ ♦❢ t❤❡ ♣❡r✐♦❞♦❣r❛♠ ✐s

❡✈❛❧✉❛t❡❞ ✐♥ ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ③❡r♦ ❢r❡q✉❡♥❝②✳

✷✳✸✳✶ ❚❤❡ ❧♦❣❣❡❞ ❞❡t❡r♠✐♥❛♥t r❡❣r❡ss✐♦♥

❙✐♠✐❧❛r t♦ t❤❡ ❡st✐♠❛t♦r ♦❢ d ♣r♦♣♦s❡❞ ❜② ●❡✇❡❦❡ ❛♥❞ P♦rt❡r✲❍✉❞❛❦ ✭✶✾✽✸✮

✭●P❍✮✱ ❛♥ ❡st✐♠❛t❡ ♦❢b❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❢r♦♠ ❛♥ ❛♣♣r♦①✐♠❛t❡❞ r❡❣r❡ss✐♦♥ ❡q✉❛✲

t✐♦♥ ♦❢lnD(λ)_∼2 ln_|1₋e−iλ_{|✇❤❡♥λ−→0}+_{✳ ❇② t❛❦✐♥❣ t❤❡ ❧♦❣ ✐♥ t❤❡ ❊q✉❛t✐♦♥}

✷✳✷✳✾ ②✐❡❧❞s✿

lnD(λ)_∼lnG(0) + lnG(λ)

G(0) +bln|1−e

−iλ

|2 as λ_−→0+. ✭✷✳✸✳✶✮

❋♦r ❛ ♣❛✐r ♦❢ s❡r✐❡s (∆X1,t,∆X2,t) ✇✐t❤ ❛ s❛♠♣❧❡ ♦❢ s✐③❡ n✱ ✐❡✱ t = 1, ..., n ✱

t❤❡ ✜rst st❡♣ ✐♥ ♦r❞❡r t♦ ✐♠♣❧❡♠❡♥t t❤❡ ❛❜♦✈❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧ ✐s t♦ ❡st✐♠❛t❡ t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ♠❛tr✐①✱ F_(λ)✱ ✐♥ ✷✳✷✳✻✳ ▲❡t _{λj = 2πj/n}✱ _{j = l, l+ (2r+ 1), l+}

2(2r+ 1), ..., m₋(2r+ 1), m✱ ✇❤❡r❡r, l_∈N∗✱ ✇✐t❤_{r < l < m}❛♥❞ _{m < n}✳ ❍❡♥❝❡✱

t❤❡ ❡st✐♠❛t❡ ♦❢ F_(λj) ✐s ❣✐✈❡♥ ❜②

b

F_{r(λj) =} 1 (2r+ 1)

    

j_P+r

v=j−r

In,∆X1(λv)

j_P+r

v=j−r

In,∆X1∆X2(λv)

j_P+r

v=j−r

In,∆X2∆X1(λv)

j_P+r

v=j−r

In,∆X2(λv)

   

, ✭✷✳✸✳✷✮

✇❤❡r❡ ❡❛❝❤ ❞✐❛❣♦♥❛❧ t❡r♠ ♦❢Fb_r(λj)✐s t❤❡ ❛✈❡r❛❣❡ ♦❢_{2r+ 1}❞✐st✐♥❝t ♣❡r✐♦❞♦❣r❛♠s

❝❡♥t❡r❡❞ ❛t ❢r❡q✉❡♥❝② j ❣✐✈❡♥ ❜②✿

In,∆Xi(λj) =

1 2πn n X t=1

Xi,te−iλjt

2 , ✭✷✳✸✳✸✮

❢♦ri= 1,2✳ ❚❤❡ ♦✛✲❞✐❛❣♦♥❛❧ t❡r♠s ♦❢Fb_r(λj)❛r❡ ❛❧s♦ ❛♥ ❛✈❡r❛❣❡ ♦❢_2r+1❞✐st✐♥❝t ❝r♦ss✲♣❡r✐♦❞♦❣r❛♠s ❝❡♥t❡r❡❞ ❛t ❢r❡q✉❡♥❝② j t❤❛t ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜②✿

In,∆Xs∆Xp(λj) =

n

X

t=1

∆Xs,te−iλjt n

X

t=1

∆Xp,teiλjt

!

/2πn ✭✷✳✸✳✹✮

✇❤❡r❡ p, s= 1,2✱ p=₆ s✳ ❚❤❡ ♥❛t✉r❛❧ ❡st✐♠❛t❡ ♦❢ D(λj) ✐♥ ❊q✉❛t✐♦♥ ✷✳✸✳✶ ✐s t❤❡

❞❡t❡r♠✐♥❛♥t ♦❢ Fb_r(λj) ❞❡♥♦t❡❞ ❤❡r❡ ❜② _Db_r(λj)✳ ❊q✉❛t✐♦♥ ✾✳✺✳✶✷ ❢r♦♠ Pr✐❡st❧❡②

✭✶✾✽✶✮✱ ♣✳✻✾✼✱ st❛t❡s t❤❛t covIn,∆Xs₁∆Xp₁(λj), In,∆Xs₂∆Xp₂(λk)

→ 0 ❛sn _{→ ∞}✱

✇❤❡r❡ s1, p1, s2, p2 = 1,2✱ ❛♥❞ λj = 2πj/n✱ λk = 2πk/n✱ j, k = 1, ..., n ✇✐t❤

j ₆=k✳ ❙✐♥❝❡ t❤❡ q✉❛♥t✐t✐❡s ✐♥Fb_r(λj)❛r❡ ❝❛❧❝✉❧❛t❡❞ ✇✐t❤ ♥♦♥✲♦✈❡r❧❛♣♣✐♥❣ ❋♦✉r✐❡r ❢r❡q✉❡♥❝✐❡s✱ t❤❡② s❛t✐s❢② t❤❡ ❝♦♥❞✐t✐♦♥s ♣r❡s❡♥t❡❞ ❜② Pr✐❡st❧❡② ✭✶✾✽✶✮ ✐♥ ♦r❞❡r t♦ ❜❡ ❛s②♠♣t♦t✐❝❛❧❧② ✉♥❝♦rr❡❧❛t❡❞ ❛♥❞✱ ❛s ❛ r❡s✉❧t✱ cov[D(λj), D(λk)] → 0 ❛s

n _{→ ∞}✳ ■❢ t❤❡ ♣r♦❝❡ss (∆X1,t,∆X2,t) ✐s ●❛✉ss✐❛♥✱ t❤❡♥ cov[D(λj), D(λk)] = 0

∀n✳

❯s✐♥❣ t❤❡ ❢❛❝t t❤❛t lnhDbr(λj)

D(λj)

i

−lnDbr(λj) = −lnD(λj) ❛♥❞ r❡♣❧❛❝✐♥❣D(λj)

❜② t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ ❊q✉❛t✐♦♥ ✷✳✸✳✶✱ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❣r❡ss✐♦♥ ❡q✉❛t✐♦♥ ✐s ♦❜✲ t❛✐♥❡❞✿

lnDbr(λj) = lnG(0) + ln

G(λj)

G(0) +c(r) +bln|1−e

−iλj_|2₊

(

ln

" b

Dr(λj)

D(λj)

#

−c(r)

)

✭✷✳✸✳✺✮ ✇❤❡r❡ c(r) =En_lnhDbr(λj)

D(λj)

io

✳

❚❤❡r❡❢♦r❡✱ t❤❡ ♦r❞✐♥❛r② ❧❡❛st sq✉❛r❡s ❡st✐♠❛t♦r ♦❢b✱ˆbLDR✱ ✐s✿

ˆ_{bLDR =}

m

X

j=l

˜

Zj 2

!−1 _m

X

j=l

˜

Zj(lnDbr(λj)), ✭✷✳✸✳✻✮

✇❤❡r❡ Zj = ln (2−2 cosλj) ❛♥❞ Z˜j = Zj −Z¯✱ Z¯ ✐s t❤❡ ♠❡❛♥ ♦❢ Zj✳ ■♥ ♦r❞❡r

t♦ ♦❜t❛✐♥ s♦♠❡ ❛s②♠♣t♦t✐❝ r❡s✉❧ts ♦❢ ˆ_{bLDR✱ ✉♥❞❡r t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s ♦❢ ♥♦♥✲} ❝♦✐♥t❡❣r❛t✐♦♥✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss✉♠♣t✐♦♥s ❛r❡ ✐♥tr♦❞✉❝❡❞✿

❆ss✉♠♣t✐♦♥ ✶✳ ❚❤❡ ✈❡❝t♦r ♦❢ ✐♥♥♦✈❛t✐♦♥s (∆X1,t,∆X2,t) ❢♦❧❧♦✇s ❛ ●❛✉ss✐❛♥

✇❤✐t❡ ♥♦✐s❡ ♣r♦❝❡ss ✇✐t❤ ③❡r♦ ♠❡❛♥ ❛♥❞ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① Σ✳

❆ss✉♠♣t✐♦♥ ✷✳ ▲❡t m=g(n) s✉❝❤ t❤❛t g(_nn)+ 1

g(n) +

lnn

g(n) →0 ❛s n → ∞✳

❘❡♠❛r❦ ✶✳ ❯♥❞❡r ❆ss✉♠♣t✐♦♥ ✶✱ t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ♦❢ (∆X1,t,∆X2,t)✱ F(λj)✱

✐s ❝♦♥st❛♥t ❛❝r♦ss ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ λj✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✈❛❧✉❡ ♦❢ F(λj) ✐s

✐♥❞❡♣❡♥❞❡♥t ♦❢ j✱ j = 1, ..., m✱ ❛♥❞✱ t❤❡r❡❢♦r❡✱ D(λj) = Λ✱ ✇❤❡r❡ Λ ✐s ❛ ♣♦s✐t✐✈❡

❝♦♥st❛♥t✳ ▼♦r❡♦✈❡r✱ ✐❢ ❆ss✉♠♣t✐♦♥ ✶ ❤♦❧❞s✱ t❤❛♥ t❤❡ s②st❡♠ ❞❡s❝r✐❜❡❞ ❜② ❊q✉❛t✐♦♥ ✷✳✷✳✶ ✐s ♥❡❝❡ss❛r✐❧② ♥♦♥✲❝♦✐♥t❡❣r❛t❡❞✱ t❤❛t ✐s✱ b =b1 =b2 = 0✳

❋♦❧❧♦✇✐♥❣ ●♦♦❞♠❛♥ ✭✶✾✻✸✮ ❛♥❞ ✉♥❞❡r ❆ss✉♠♣t✐♦♥ ✶✱ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ q✉❛♥t✐t②D _{= ln}h_{4 (2r+ 1)}2_Db_r(λj)/Λ)i❤❛s t❤❡ s❛♠❡ ♣r♦♣❡rt✐❡s ♦❢_ln(χ2

(4r+2)χ2(4r))✱

t❤❛t ✐s✱ D ₌d _ln(χ2

(4r+2)χ2(4r)) ✇❤❡r❡ χ2(4r+2) ❛♥❞ χ2(4r) ❛r❡ ❝❤✐✲sq✉❛r❡❞ r❛♥❞♦♠ ✈❛✲

r✐❛❜❧❡s ✇✐t❤ (4r+ 2) ❛♥❞ 4r ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ s②♠❜♦❧ =d ♠❡❛♥s ❡q✉❛❧✐t② ✐♥ ❞✐str✐❜✉t✐♦♥✳ ■♥ ❛❞❞✐t✐♦♥✱ s✐♥❝❡ t❤❡ ✈❡❝t♦r (∆X1,t, X2,t) ✐s ❛

✇❤✐t❡ ♥♦✐s❡✱ G(λ)✐♥ ❊q✉❛t✐♦♥ ✷✳✷✳✽ ✐s ❝♦♥st❛♥t✱ t❤❛t ✐s✱ G(λ) = G(0)✳

Pr♦♣♦s✐t✐♦♥ ✶✳ ▲❡t t❤❡ ❜✐✈❛r✐❛t❡ t✐♠❡ s❡r✐❡s (X1,t, X2,t)s❛t✐s❢②✐♥❣ t❤❡ ❊q✉❛t✐♦♥

✷✳✷✳✹✳ ■❢ ❆ss✉♠♣t✐♦♥s ✶ ❛♥❞ ✷ ❤♦❧❞✱ t❤❡♥✿

✶✳ Ehˆ_bLDRi_{= 0}❀

✷✳ Vhˆ_bLDRi ₌ ψ(1)(2r+1)+_m ψ(1)(2r) P

j=l

˜

Zj2

❀

✸✳ Vhˆ_bLDRi _→0 ❛s _{m→ ∞}✱

✇❤❡r❡ ψ(1)_{(z)✐s t❤❡ P♦❧②❣❛♠♠❛ ❢✉♥❝t✐♦♥ ♦❢ ♦r❞❡r ✶✱ t❤❛t ✐s✱ ψ}(1)_{(z) =} d2_{ln Γ(z)}

dz2 ✳

❆ss✉♠♣t✐♦♥ ✶ ✐s q✉✐t❡ str♦♥❣ ❜✉t ♥❡❝❡ss❛r② t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ st❛t✐st✐❝ ˆ_{bLDR✳ ❆s ♣♦✐♥t❡❞ ♦✉t✱ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ ♥♦♥✲❝♦✐♥t❡❣r❛t✐♦♥ ❛♥❞} ❢♦r ❛ ✜①❡❞m ❛♥❞ r✱ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ˆbLDR ✇✐❧❧ ❜❡ t❤❡ s❛♠❡ ♦❢ ❛ ✇❡✐❣❤t❡❞ s✉♠

♦❢ {Wj}m_j=l ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✇❤❡r❡ ❡❛❝❤ Wj d

= ln(χ2

(4r+2)χ2(4r)) ❛♥❞

t❤❡ ✇❡✐❣❤ts ❛r❡ _Z˜_j/ Pm

j=l

˜

Zj 2

!

✳ ❖♥❝❡ m _{→ ∞} t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥ ❝❛♥ ❜❡

st❛t❡❞✿

Pr♦♣♦s✐t✐♦♥ ✷✳ ▲❡t ❆ss✉♠♣t✐♦♥s ✶ ❛♥❞ ✷ ❤♦❧❞✳ ❚❤❡♥✱ ❢♦r ❛ ✜①❡❞ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r

r

❛s m _{→ ∞}✱ ✇❤❡r❡ ˆbLDR ❛♥❞ V

h

ˆ

bLDR

i

❛r❡ ❣✐✈❡♥ ❜② ❊q✉❛t✐♦♥s ✷✳✸✳✻ ❛♥❞ ✳✷✳✼✱ r❡s♣❡❝t✐✈❡❧②✳

❚❤❡ ♣r♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥s ✶ ❛♥❞ ✷ ❝❛♥ ❜❡ ✈✐❡✇❡❞ ✐♥ ❆♣♣❡♥❞✐①❡s ✳✷ ❛♥❞ ✳✸✱ r❡s♣❡❝t✐✈❡❧②✳ ❆ss✉♠♣t✐♦♥ ✶ ❝❛♥ ❜❡ r❡❧❛①❡❞ ❛❧❧♦✇✐♥❣ t❤❡ t❡r♠sD(λj)❛♥❞lnG(λj)

t♦ ✈❛r② ❛❝r♦ss ❞✐✛❡r❡♥t ❢r❡q✉❡♥❝✐❡s✳ ■♥ t❤❛t ❝❛s❡✱ t❤❡ ❡st✐♠❛t♦r ˆ_{bLDR ✐s st✐❧❧} ❝♦♥s✐st❡♥t s✐♥❝❡ ❢♦r s✉✣❝✐❡♥t❧② ❝❧♦s❡ ❢r❡q✉❡♥❝✐❡s λj, λk✱ j 6=k✱ D(λj)≈D(λk) ❛s

n _{→ ∞}❛♥❞ lnG(λj)→lnG(0) ❛s m→ ∞✳

✷✳✸✳✷ ❚❤❡ ❆✈❡r❛❣❡❞ ❉❡t❡r♠✐♥❛♥t

❇❛s❡❞ ♦♥ t❤❡ s❡♠✐♣❛r❛♠❡tr✐❝ ❛✈❡r❛❣❡❞ ♣❡r✐♦❞♦❣r❛♠ ❡st✐♠❛t♦r ♦❢ d ♣r♦♣♦s❡❞ ❜②

❘♦❜✐♥s♦♥ ✭✶✾✾✹✮✱ ❛♥ ❛❧t❡r♥❛t✐✈❡ t♦ ❡st✐♠❛t❡ b ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❞✉❡ t♦ t❤❡ ❢❛❝t

t❤❛t t❤❡ D(λ)✐♥ ❊q✉❛t✐♦♥ ✷✳✷✳✾ ✐s ❛ r❡❣✉❧❛r❧② ✈❛r②✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ✐♥❞❡① 2b✱ t❤❛t

✐s✿ ✸

lim

λ→0+

D(qλ)

D(λ) =q

2b_{, ✭✷✳✸✳✼✮}

✇❤❡r❡ q ✐s ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t✳ ❇❛s❡❞ ♦♥ ❊q✉❛t✐♦♥ ✷✳✸✳✼✱

b ∼= (2 lnq)−1lnD(qλ)

D(λ) . ✭✷✳✸✳✽✮

❚❤❡ ❡st✐♠❛t❡ ♦❢b✱ s❛②ˆbAD✱ ✐s ❝♦♠♣✉t❡❞ ❜② r❡♣❧❛❝✐♥❣D(·)❜② ✐ts ❡st✐♠❛t❡ ❛♥❞

✸_{❋♦❧❧♦✇✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ❣✐✈❡♥ ❜② ❇✐♥❣❤❛♠ ❡t ❛❧✳ ✭✶✾✽✼✮✱ ❛ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥ H :}

[a∗

,∞) → (0,∞)✱ ∀a∗

∈ ℜ✱ ✐s s❛✐❞ t♦ ❜❡ r❡❣✉❧❛r❧② ✈❛r②✐♥❣ ♦❢ ✐♥❞❡① ϑ✱ ϑ ∈ ℜ✱ ✐❢ H s❛t✐s✲

✜❡s✿

lim

y→∞

H(αy)

H(y) =α

ϑ

∀α >0. ■♥ t❤❡ ♣r❡s❡♥t ❝❛s❡✱ ❧❡ty= 1/λ✳ ❚❤✉s✱ ❢♦r ❛ ♣♦s✐t✐✈❡q✿

lim

λ→0+ D(qλ)

D(λ) = limy→∞

H(qy)

H(y) =q

2b

t❤✐s ✐s ❞✐s❝✉ss❡❞ ❛s ❢♦❧❧♦✇s✳ ❋♦r ❛ s❛♠♣❧❡ ♦❢ (∆X1,t,∆X2,t)✱ t = 1, ..., n✱ ❧❡t ♥♦✇

t❤❡ ❡st✐♠❛t❡ ♦❢ F_(λj)✱_{j = 1, ..., m}✱ ❜❡ ❣✐✈❡♥ ❜②✿

c

F_{(λj) =}

 

I∆X1∆X1(λj) I∆X1∆X2(λj)

I∆X2∆X1(λj) I∆X2∆X2(λj)

 

. ✭✷✳✸✳✾✮

❚❤❡ ❡st✐♠❛t❡ ♦❢ D(λ) ❛♥❞ D(qλ) ❛r❡ t❤❡♥ ♦❜t❛✐♥❡❞ ❛s ❢♦❧❧♦✇s✿

b

D(λm) =

m X j=1 c

F_(λj)

❛♥❞ Db(qλm) = m X j=1 c

F_(qλj)

, ✭✷✳✸✳✶✵✮

✇❤❡r❡kA_k❞❡♥♦t❡s t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ t❤❡ ♠❛tr✐① A❛♥❞ _ms❛t✐s✜❡s ❆ss✉♠♣t✐♦♥ ✷✳ ❚❤❡r❡❢♦r❡✱Db(qλm)→Db(0)❛♥❞Db(λm)→Db(0)✳ ❚❤❡ ♣❛r❛♠❡t❡rb✐s ❡st✐♠❛t❡❞

❜②✿

bbAD = (2 lnq)−1ln

b

D(qλm)

b

D(λm)

. ✭✷✳✸✳✶✶✮

❚❤❡ st❛t✐st✐❝ ✐♥ ✷✳✸✳✶✶ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ t❤❡ ❆✈❡r❛❣❡ ❉❡t❡r♠✐♥❛♥t ✭❆❉✮ ❡st✐♠❛✲ t♦r✳ ❯♥❞❡r ❆ss✉♠♣t✐♦♥ ✶✱ t❤❛t ✐s✱ (∆X1,t,∆X2,t)❢♦❧❧♦✇s ❛ ●❛✉ss✐❛♥ ✇❤✐t❡ ♥♦✐s❡

♣r♦❝❡ss✱bbAD ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s✿

bbAD =

n

lnhDb(qλm)/Λ

i

−lnhDb(λm)/Λ

io

(2 lnq)−1. ✭✷✳✸✳✶✷✮

❙✐♥❝❡λj ✐s t❤❡ ❋♦✉r✐❡r ❢r❡q✉❡♥❝②✱ t❤❡ s❡t ♦❢ ✈❛r✐❛❜❧❡sFc(λj)❛r❡ ✐♥❞❡♣❡♥❞❡♥t❧②

❞✐str✐❜✉t❡❞ ❛♥❞ ❡❛❝❤ ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛s ❛ 2_×2 ❝♦♠♣❧❡① ❲❤✐s❤❛rt ♠❛tr✐①✱ t❤❛t ✐s✱ Fc_(λj)∼W₂c_(1,f_(λ)) ✭❙❡❡ ❇r✐❧❧✐♥❣❡r ✭✶✾✽✶✮ ♣♣✳ ✸✵✺✮✳

■♥ t❤❡ ❝❛s❡ ✇❤❡r❡q✐s ♥♦t ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ♥✉♠❜❡r✱ t❤❡ q✉❛♥t✐t②2πjq/n✐s ♥♦

❧♦♥❣❡r ❛ ❋♦✉r✐❡r ❢r❡q✉❡♥❝② ❛♥❞✱ ✐♥ ♦r❞❡r t♦ ❣✉❛r❛♥t❡❡ ❛s②♠♣t♦t✐❝ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢