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■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

❆❜♦✉❜❛❝❛r ❆♠✐r✐ ∗✱ ❈❤r✐st♦♣❤❡ ❈r❛♠❜❡s † ❛♥❞ ❇❛❜❛ ❚❤✐❛♠∗

❏♦✉r♥é❡s ❞❡ ❙t❛t✐st✐q✉❡ ❋♦♥❝t✐♦♥♥❡❧❧❡✱ ▼♦♥t♣❡❧❧✐❡r

✷✽ ❏✉✐♥ ✷✵✶✷

∗✳ ❯♥✐✈❡rs✐té ▲✐❧❧❡ ✸ †✳ ❯♥✐✈❡rs✐té ▼♦♥t♣❡❧❧✐❡r ✷

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

✶

■♥tr♦❞✉❝t✐♦♥

✷

❆s②♠♣t♦t✐❝ r❡s✉❧ts

✸

❆♣♣❧✐❝❛t✐♦♥

✹

❈♦♥❝❧✉s✐♦♥

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

✶

■♥tr♦❞✉❝t✐♦♥

✷

❆s②♠♣t♦t✐❝ r❡s✉❧ts

✸

❆♣♣❧✐❝❛t✐♦♥

✹

❈♦♥❝❧✉s✐♦♥

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

▼♦t✐✈❛t✐♦♥s

➨ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ♣r♦❝❡ss (Xk, k ≥ 1) , ♦❜s❡r✈❡❞ ✉♥t✐❧ ❛ ❣✐✈❡♥ ✐♥st❛♥t n ❋♦r❡❝❛st Xn+1✳ ❚❤❡ s❛♠♣❧❡ s✐③❡ ✐s ♥♦t ✜①❡❞ ✐♥ ❛❞✈❛♥❝❡✱ ❜✉t t❤❡ ❞❛t❛❜❛s❡ ✐s ❝♦♥t✐♥✉♦✉s❧② ✉♣❞❛t❡❞ ✉♣❞❛t❡ t❤❡ ❢♦r❡❝❛st ❛t ❡❛❝❤ t✐♠❡✳ ❘❡❝✉rs✐✈❡ ❛♣♣r♦❛❝❤ ✿ ❡✳❣✳ ❡①♣♦♥❡♥t✐❛❧ s♠♦♦t❤✐♥❣ t❡❝❤♥✐q✉❡s✳ ❚❤❡ ♣r❡❞✐❝t♦r ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ r❡❝✉rs✐✈❡❧② ❜② ✿ ✭✶✮ ■t ❝❛♥ ❜❡ ✉♣❞❛t❡❞ ✇✐t❤ ❡❛❝❤ ♥❡✇ ♦❜s❡r✈❛t✐♦♥ ❛❞❞❡❞ t♦ t❤❡ ❞❛t❛❜❛s❡✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

▼♦t✐✈❛t✐♦♥s

➨ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ♣r♦❝❡ss (Xk, k ≥ 1) , ♦❜s❡r✈❡❞ ✉♥t✐❧ ❛ ❣✐✈❡♥ ✐♥st❛♥t n ❋♦r❡❝❛st Xn+1✳ ➨ ❚❤❡ s❛♠♣❧❡ s✐③❡ ✐s ♥♦t ✜①❡❞ ✐♥ ❛❞✈❛♥❝❡✱ ❜✉t t❤❡ ❞❛t❛❜❛s❡ ✐s ❝♦♥t✐♥✉♦✉s❧② ✉♣❞❛t❡❞ ✉♣❞❛t❡ t❤❡ ❢♦r❡❝❛st ❛t ❡❛❝❤ t✐♠❡✳ ❘❡❝✉rs✐✈❡ ❛♣♣r♦❛❝❤ ✿ ❡✳❣✳ ❡①♣♦♥❡♥t✐❛❧ s♠♦♦t❤✐♥❣ t❡❝❤♥✐q✉❡s✳ ❚❤❡ ♣r❡❞✐❝t♦r ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ r❡❝✉rs✐✈❡❧② ❜② ✿ ✭✶✮ ■t ❝❛♥ ❜❡ ✉♣❞❛t❡❞ ✇✐t❤ ❡❛❝❤ ♥❡✇ ♦❜s❡r✈❛t✐♦♥ ❛❞❞❡❞ t♦ t❤❡ ❞❛t❛❜❛s❡✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

▼♦t✐✈❛t✐♦♥s

➨ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ♣r♦❝❡ss (Xk, k ≥ 1) , ♦❜s❡r✈❡❞ ✉♥t✐❧ ❛ ❣✐✈❡♥ ✐♥st❛♥t n ❋♦r❡❝❛st Xn+1✳ ➨ ❚❤❡ s❛♠♣❧❡ s✐③❡ ✐s ♥♦t ✜①❡❞ ✐♥ ❛❞✈❛♥❝❡✱ ❜✉t t❤❡ ❞❛t❛❜❛s❡ ✐s ❝♦♥t✐♥✉♦✉s❧② ✉♣❞❛t❡❞ ✉♣❞❛t❡ t❤❡ ❢♦r❡❝❛st ❛t ❡❛❝❤ t✐♠❡✳ ➨ ❘❡❝✉rs✐✈❡ ❛♣♣r♦❛❝❤ ✿ ❡✳❣✳ ❡①♣♦♥❡♥t✐❛❧ s♠♦♦t❤✐♥❣ t❡❝❤♥✐q✉❡s✳ ❚❤❡ ♣r❡❞✐❝t♦r ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ r❡❝✉rs✐✈❡❧② ❜② ✿

• Xn+1 = γ

Xn + (1 − γ)Xn+1 (0 < γ < 1). ✭✶✮ ■t ❝❛♥ ❜❡ ✉♣❞❛t❡❞ ✇✐t❤ ❡❛❝❤ ♥❡✇ ♦❜s❡r✈❛t✐♦♥ ❛❞❞❡❞ t♦ t❤❡ ❞❛t❛❜❛s❡✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

▼♦t✐✈❛t✐♦♥s

➨ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ♣r♦❝❡ss (Xk, k ≥ 1) , ♦❜s❡r✈❡❞ ✉♥t✐❧ ❛ ❣✐✈❡♥ ✐♥st❛♥t n ❋♦r❡❝❛st Xn+1✳ ➨ ❚❤❡ s❛♠♣❧❡ s✐③❡ ✐s ♥♦t ✜①❡❞ ✐♥ ❛❞✈❛♥❝❡✱ ❜✉t t❤❡ ❞❛t❛❜❛s❡ ✐s ❝♦♥t✐♥✉♦✉s❧② ✉♣❞❛t❡❞ ✉♣❞❛t❡ t❤❡ ❢♦r❡❝❛st ❛t ❡❛❝❤ t✐♠❡✳ ➨ ❘❡❝✉rs✐✈❡ ❛♣♣r♦❛❝❤ ✿ ❡✳❣✳ ❡①♣♦♥❡♥t✐❛❧ s♠♦♦t❤✐♥❣ t❡❝❤♥✐q✉❡s✳ ❚❤❡ ♣r❡❞✐❝t♦r ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ r❡❝✉rs✐✈❡❧② ❜② ✿

• Xn+1 = γ

Xn + (1 − γ)Xn+1 (0 < γ < 1). ✭✶✮ ■t ❝❛♥ ❜❡ ✉♣❞❛t❡❞ ✇✐t❤ ❡❛❝❤ ♥❡✇ ♦❜s❡r✈❛t✐♦♥ ❛❞❞❡❞ t♦ t❤❡ ❞❛t❛❜❛s❡✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

▼♦t✐✈❛t✐♦♥s

➨ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ♣r♦❝❡ss (Xk, k ≥ 1) , ♦❜s❡r✈❡❞ ✉♥t✐❧ ❛ ❣✐✈❡♥ ✐♥st❛♥t n ❋♦r❡❝❛st Xn+1✳ ➨ ❚❤❡ s❛♠♣❧❡ s✐③❡ ✐s ♥♦t ✜①❡❞ ✐♥ ❛❞✈❛♥❝❡✱ ❜✉t t❤❡ ❞❛t❛❜❛s❡ ✐s ❝♦♥t✐♥✉♦✉s❧② ✉♣❞❛t❡❞ ✉♣❞❛t❡ t❤❡ ❢♦r❡❝❛st ❛t ❡❛❝❤ t✐♠❡✳ ➨ ❘❡❝✉rs✐✈❡ ❛♣♣r♦❛❝❤ ✿ ❡✳❣✳ ❡①♣♦♥❡♥t✐❛❧ s♠♦♦t❤✐♥❣ t❡❝❤♥✐q✉❡s✳ ❚❤❡ ♣r❡❞✐❝t♦r ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ r❡❝✉rs✐✈❡❧② ❜② ✿

• Xn+1 = γ

Xn + (1 − γ)Xn+1 (0 < γ < 1). ✭✶✮ ➨ ■t ❝❛♥ ❜❡ ✉♣❞❛t❡❞ ✇✐t❤ ❡❛❝❤ ♥❡✇ ♦❜s❡r✈❛t✐♦♥ ❛❞❞❡❞ t♦ t❤❡ ❞❛t❛❜❛s❡✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

▼♦t✐✈❛t✐♦♥s

➨ ❆ ❝♦♥s✐❞❡r❛❜❧❡ ❣❛✐♥ ♦❢ t✐♠❡ ♦❢ ❝♦♠♣✉t❛t✐♦♥✳ ❲❡ ❛r❡ ♥♦t r❡q✉✐r❡❞ t♦ st♦r❡ ❡①t❡♥s✐✈❡ ❞❛t❛ ✐♥ ♦r❞❡r t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ♣r❡❞✐❝t♦r✳ ❯♥❞❡r s♦♠❡ ❝♦♥❞✐t✐♦♥s✱ t❤❡ ♣r❡❞✐❝t✐♦♥ ♣r♦❜❧❡♠ ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ r❡❣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ ❛ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ♦♥ ❛♥ ❡①♣❧❛♥❛t♦r② ♦♥❡✳

• ♦❛❧ ✿ ❊st✐♠❛t✐♦♥ ♦❢ t❤❡ r❡❣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ❜② ♥♦♥♣❛r❛♠❡tr✐❝

❦❡r♥❡❧ ❛♣♣r♦❛❝❤ ✇✐t❤ ❦❡❡♣✐♥❣ t❤❡ ❛❞✈❛♥t❛❣❡s ♦❢ ❛ r❡❝✉rs✐✈❡ r❡❧❛t✐♦♥ ❛s ✭✶✮✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

▼♦t✐✈❛t✐♦♥s

➨ ❆ ❝♦♥s✐❞❡r❛❜❧❡ ❣❛✐♥ ♦❢ t✐♠❡ ♦❢ ❝♦♠♣✉t❛t✐♦♥✳ ➨ ❲❡ ❛r❡ ♥♦t r❡q✉✐r❡❞ t♦ st♦r❡ ❡①t❡♥s✐✈❡ ❞❛t❛ ✐♥ ♦r❞❡r t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ♣r❡❞✐❝t♦r✳ ❯♥❞❡r s♦♠❡ ❝♦♥❞✐t✐♦♥s✱ t❤❡ ♣r❡❞✐❝t✐♦♥ ♣r♦❜❧❡♠ ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ r❡❣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ ❛ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ♦♥ ❛♥ ❡①♣❧❛♥❛t♦r② ♦♥❡✳

• ♦❛❧ ✿ ❊st✐♠❛t✐♦♥ ♦❢ t❤❡ r❡❣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ❜② ♥♦♥♣❛r❛♠❡tr✐❝

❦❡r♥❡❧ ❛♣♣r♦❛❝❤ ✇✐t❤ ❦❡❡♣✐♥❣ t❤❡ ❛❞✈❛♥t❛❣❡s ♦❢ ❛ r❡❝✉rs✐✈❡ r❡❧❛t✐♦♥ ❛s ✭✶✮✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

▼♦t✐✈❛t✐♦♥s

➨ ❆ ❝♦♥s✐❞❡r❛❜❧❡ ❣❛✐♥ ♦❢ t✐♠❡ ♦❢ ❝♦♠♣✉t❛t✐♦♥✳ ➨ ❲❡ ❛r❡ ♥♦t r❡q✉✐r❡❞ t♦ st♦r❡ ❡①t❡♥s✐✈❡ ❞❛t❛ ✐♥ ♦r❞❡r t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ♣r❡❞✐❝t♦r✳ ➨ ❯♥❞❡r s♦♠❡ ❝♦♥❞✐t✐♦♥s✱ t❤❡ ♣r❡❞✐❝t✐♦♥ ♣r♦❜❧❡♠ ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ r❡❣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ ❛ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ♦♥ ❛♥ ❡①♣❧❛♥❛t♦r② ♦♥❡✳

• ♦❛❧ ✿ ❊st✐♠❛t✐♦♥ ♦❢ t❤❡ r❡❣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ❜② ♥♦♥♣❛r❛♠❡tr✐❝

❦❡r♥❡❧ ❛♣♣r♦❛❝❤ ✇✐t❤ ❦❡❡♣✐♥❣ t❤❡ ❛❞✈❛♥t❛❣❡s ♦❢ ❛ r❡❝✉rs✐✈❡ r❡❧❛t✐♦♥ ❛s ✭✶✮✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

▼♦t✐✈❛t✐♦♥s

➨ ❆ ❝♦♥s✐❞❡r❛❜❧❡ ❣❛✐♥ ♦❢ t✐♠❡ ♦❢ ❝♦♠♣✉t❛t✐♦♥✳ ➨ ❲❡ ❛r❡ ♥♦t r❡q✉✐r❡❞ t♦ st♦r❡ ❡①t❡♥s✐✈❡ ❞❛t❛ ✐♥ ♦r❞❡r t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ♣r❡❞✐❝t♦r✳ ➨ ❯♥❞❡r s♦♠❡ ❝♦♥❞✐t✐♦♥s✱ t❤❡ ♣r❡❞✐❝t✐♦♥ ♣r♦❜❧❡♠ ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ r❡❣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ ❛ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ♦♥ ❛♥ ❡①♣❧❛♥❛t♦r② ♦♥❡✳ ➨ ●♦❛❧ ✿ ❊st✐♠❛t✐♦♥ ♦❢ t❤❡ r❡❣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ❜② ♥♦♥♣❛r❛♠❡tr✐❝ ❦❡r♥❡❧ ❛♣♣r♦❛❝❤ ✇✐t❤ ❦❡❡♣✐♥❣ t❤❡ ❛❞✈❛♥t❛❣❡s ♦❢ ❛ r❡❝✉rs✐✈❡ r❡❧❛t✐♦♥ ❛s ✭✶✮✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❘❡❣r❡ss✐♦♥ ♠♦❞❡❧

➨ ❉❛t❛ ❝♦♥s✐❞❡r❡❞ ❛s ❝✉r✈❡s ✭❘❛♠s❛② ❛♥❞ ❙✐❧✈❡r♠❛♥✱ ✭✷✵✵✺✮✱ ❋❡rr❛t② ❛♥❞ ❱✐❡✉ ✭✷✵✵✻✮✱✳✳✳✮ ❙✐♠✉❧❛t❡❞ ❞❛t❛ ✿ ❙❛♠♣❧❡ ✿ ✐✳✐✳❞✳✱ ✇✐t❤ t❤❡ s❛♠❡ ❧❛✇ ❛s ✇✐t❤ ❛♥❞ ▼♦❞❡❧ ✿

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❘❡❣r❡ss✐♦♥ ♠♦❞❡❧

➨ ❉❛t❛ ❝♦♥s✐❞❡r❡❞ ❛s ❝✉r✈❡s ✭❘❛♠s❛② ❛♥❞ ❙✐❧✈❡r♠❛♥✱ ✭✷✵✵✺✮✱ ❋❡rr❛t② ❛♥❞ ❱✐❡✉ ✭✷✵✵✻✮✱✳✳✳✮ ➨ ❙✐♠✉❧❛t❡❞ ❞❛t❛ ✿

0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 1 2

❙❛♠♣❧❡ ✿ ✐✳✐✳❞✳✱ ✇✐t❤ t❤❡ s❛♠❡ ❧❛✇ ❛s ✇✐t❤ ❛♥❞ ▼♦❞❡❧ ✿

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❘❡❣r❡ss✐♦♥ ♠♦❞❡❧

➨ ❉❛t❛ ❝♦♥s✐❞❡r❡❞ ❛s ❝✉r✈❡s ✭❘❛♠s❛② ❛♥❞ ❙✐❧✈❡r♠❛♥✱ ✭✷✵✵✺✮✱ ❋❡rr❛t② ❛♥❞ ❱✐❡✉ ✭✷✵✵✻✮✱✳✳✳✮ ❙✐♠✉❧❛t❡❞ ❞❛t❛ ✿ ➨ ❙❛♠♣❧❡ ✿ (Xi, Yi)i=1,...,n ✐✳✐✳❞✳✱ ✇✐t❤ t❤❡ s❛♠❡ ❧❛✇ ❛s (X, Y ) ✇✐t❤ X ∈ (E, .) ❛♥❞ Y ∈ R ▼♦❞❡❧ ✿

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❘❡❣r❡ss✐♦♥ ♠♦❞❡❧

➨ ❉❛t❛ ❝♦♥s✐❞❡r❡❞ ❛s ❝✉r✈❡s ✭❘❛♠s❛② ❛♥❞ ❙✐❧✈❡r♠❛♥✱ ✭✷✵✵✺✮✱ ❋❡rr❛t② ❛♥❞ ❱✐❡✉ ✭✷✵✵✻✮✱✳✳✳✮ ❙✐♠✉❧❛t❡❞ ❞❛t❛ ✿ ➨ ❙❛♠♣❧❡ ✿ (Xi, Yi)i=1,...,n ✐✳✐✳❞✳✱ ✇✐t❤ t❤❡ s❛♠❡ ❧❛✇ ❛s (X, Y ) ✇✐t❤ X ∈ (E, .) ❛♥❞ Y ∈ R ➨ ▼♦❞❡❧ ✿ Yi = r(Xi) + εi ✇✐t❤ E(εi|Xi) = 0 ❛♥❞ E(ε2

i |Xi) = σ2 ε(Xi)

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

◆♦♥♣❛r❛♠❡tr✐❝ ❡st✐♠❛t✐♦♥ ♦❢ r

➨ ❑❡r♥❡❧ ❡st✐♠❛t♦r ✭❋❡rr❛t② ❛♥❞ ❱✐❡✉✮ ✿ rn(χ) =

n

• i=1

YiK χ − Xi h

• n
• i=1

K χ − Xi h

• K ✿ ❦❡r♥❡❧
• h ✿ ❜❛♥❞✇✐❞t❤

❈♦♥✈❡r❣❡♥❝❡ r❡s✉❧ts ✿ ❆❧♠♦st ❝♦♠♣❧❡t❡ ❝♦♥✈❡r❣❡♥❝❡ ✭❋❡rr❛t② ❛♥❞ ❱✐❡✉✱ ✷✵✵✻✮ ▼❡❛♥ sq✉❛r❡ ❝♦♥✈❡r❣❡♥❝❡ ❛♥❞ ❛s②♠♣t♦t✐❝ ♥♦r♠❛❧✐t② ✭❋❡rr❛t②✱ ▼❛s ❛♥❞ ❱✐❡✉✱ ✷✵✵✼✮

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

◆♦♥♣❛r❛♠❡tr✐❝ ❡st✐♠❛t✐♦♥ ♦❢ r

➨ ❑❡r♥❡❧ ❡st✐♠❛t♦r ✭❋❡rr❛t② ❛♥❞ ❱✐❡✉✮ ✿ rn(χ) =

n

• i=1

YiK χ − Xi h

• n
• i=1

K χ − Xi h

• K ✿ ❦❡r♥❡❧
• h ✿ ❜❛♥❞✇✐❞t❤

➨ ❈♦♥✈❡r❣❡♥❝❡ r❡s✉❧ts ✿

• ❆❧♠♦st ❝♦♠♣❧❡t❡ ❝♦♥✈❡r❣❡♥❝❡ ✭❋❡rr❛t② ❛♥❞ ❱✐❡✉✱ ✷✵✵✻✮
• ▼❡❛♥ sq✉❛r❡ ❝♦♥✈❡r❣❡♥❝❡ ❛♥❞ ❛s②♠♣t♦t✐❝ ♥♦r♠❛❧✐t② ✭❋❡rr❛t②✱

▼❛s ❛♥❞ ❱✐❡✉✱ ✷✵✵✼✮

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ r

➨ ❈❛s❡ ♦❢ ❛ ❝♦✈❛r✐❛t❡ X ✐♥ Rd ✭❆♠✐r✐✱ ✷✵✶✷✮ ✿ r[ℓ]

n (x) = n

• i=1

Yi hdℓ

i

K x − Xi hi

• n
• i=1

1 hdℓ

i

K x − Xi hi

• K ✿ ❦❡r♥❡❧
• hi ✿ ❜❛♥❞✇✐❞t❤
• ℓ ✿ ♣❛r❛♠❡t❡r ✐♥ [0; 1]

➨ r[ℓ]

n (x) ❝♦♥t❛✐♥s t❤❡ ♠♦st ✉s❡❞ r❡❝✉rs✐✈❡ ❡st✐♠❛t♦rs ✿

❆❤♠❛❞✲▲✐♥ (ℓ = 0) ❛♥❞ ❉❡✈r♦②❡✲❲❛❣♥❡r (ℓ = 1)✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ r

➨ ❈❛s❡ ♦❢ ❛ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡ ✿ r[ℓ]

n (χ) = n

• i=1

Yi Fχ(hi)ℓ K χ − Xi hi

• n
• i=1

1 Fχ(hi)ℓ K χ − Xi hi

• K ✿ ❦❡r♥❡❧
• hi ✿ ❜❛♥❞✇✐❞t❤
• ℓ ✿ ♣❛r❛♠❡t❡r ✐♥ [0; 1]
• Fχ ✿ ❝✉♠✉❧❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ χ − X ✿

Fχ(hi) = P (χ − X ≤ hi)

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ r

➨ ◆♦t❛t✐♦♥s ✿ r[ℓ]

n (χ) = ϕ[ℓ] n (χ)

f[ℓ]

n (χ)

✇✐t❤

ϕ[ℓ]

n (χ) =

1

n

• i=1

Fχ(hi)1−ℓ

n

• i=1

Yi F(hi)ℓ K χ − Xi hi

• ❛♥❞

f[ℓ]

n (χ) =

1

n

• i=1

Fχ(hi)1−ℓ

n

• i=1

1 F(hi)ℓ K χ − Xi hi

• ❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ r

➨ ❘❡❝✉rs✐✈❡ ✇r✐t✐♥❣ ✿

r[ℓ]

n+1(χ) =

n

i=1 Fχ (hi)1−ℓ

ϕ[ℓ]

n (χ) +

n+1

i=1 Fχ (hi)1−ℓ

Yn+1K[ℓ]

n+1(χ − Xn+1)

n

i=1 Fχ (hi)1−ℓ

f[ℓ]

n (χ) +

n+1

i=1 Fχ (hi)1−ℓ

K[ℓ]

n+1(χ − Xn+1)

✇❤❡r❡ K[ℓ]

n+1(.) =

1 Fχ(hn+1)ℓ n+1

i=1 Fχ(hi)1−ℓ K

• .

hn+1

• ❈✉♠✉❧❛t❡❞ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♣❧❡①✐t② ✿

✈s ✱ ✇❤❡r❡ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❛❞❞✐t✐♦♥❛❧ ♦❜s❡r✈❛t✐♦♥s✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ r

➨ ❘❡❝✉rs✐✈❡ ✇r✐t✐♥❣ ✿

r[ℓ]

n+1(χ) =

n

i=1 Fχ (hi)1−ℓ

ϕ[ℓ]

n (χ) +

n+1

i=1 Fχ (hi)1−ℓ

Yn+1K[ℓ]

n+1(χ − Xn+1)

n

i=1 Fχ (hi)1−ℓ

f[ℓ]

n (χ) +

n+1

i=1 Fχ (hi)1−ℓ

K[ℓ]

n+1(χ − Xn+1)

✇❤❡r❡ K[ℓ]

n+1(.) =

1 Fχ(hn+1)ℓ n+1

i=1 Fχ(hi)1−ℓ K

• .

hn+1

• ➨ ❈✉♠✉❧❛t❡❞ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♣❧❡①✐t② ✿ O(n + T) ✈s O(nT)✱

✇❤❡r❡ T ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❛❞❞✐t✐♦♥❛❧ ♦❜s❡r✈❛t✐♦♥s✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❆ss✉♠♣t✐♦♥s

➨ (H.1) ✿ ❚❤❡ ♦♣❡r❛t♦rs r ❛♥❞ σ2

ε ❛r❡ ❝♦♥t✐♥✉♦✉s ♦♥ ❛

♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ χ✱ ❛♥❞ Fχ(0) = 0✳ ▼♦r❡♦✈❡r✱ t❤❡ ❢✉♥❝t✐♦♥ ζ : t − → E (r(X) − r(χ)| X − χ = t) ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❞❡r✐✈❛❜❧❡ ❛t t = 0 ✿ ❚❤❡ ❦❡r♥❡❧ ✐s s✉♣♣♦rt❡❞ ♦♥ t❤❡ ❝♦♠♣❛❝t ❛♥❞ ❤❛s ❛ ❝♦♥t✐♥✉♦✉s ❞❡r✐✈❛t✐✈❡ ♦♥ ✳ ▼♦r❡♦✈❡r✱ ❛♥❞ ✿ ❋♦r ❛♥② ✱ ❛s ✿ ✭✐✮ ❛♥❞ ❛s ✭✐✐✮ ❋♦r ❛❧❧ ✱ ❛s ✭✐✐✐✮ ❋♦r ❛❧❧ ✱ ❛s

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❆ss✉♠♣t✐♦♥s

➨ (H.1) ✿ ❚❤❡ ♦♣❡r❛t♦rs r ❛♥❞ σ2

ε ❛r❡ ❝♦♥t✐♥✉♦✉s ♦♥ ❛

♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ χ✱ ❛♥❞ Fχ(0) = 0✳ ▼♦r❡♦✈❡r✱ t❤❡ ❢✉♥❝t✐♦♥ ζ : t − → E (r(X) − r(χ)| X − χ = t) ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❞❡r✐✈❛❜❧❡ ❛t t = 0 ➨ (H.2) ✿ ❚❤❡ ❦❡r♥❡❧ K ✐s s✉♣♣♦rt❡❞ ♦♥ t❤❡ ❝♦♠♣❛❝t [0; 1] ❛♥❞ ❤❛s ❛ ❝♦♥t✐♥✉♦✉s ❞❡r✐✈❛t✐✈❡ ♦♥ [0; 1]✳ ▼♦r❡♦✈❡r✱ K′(s) ≤ 0 ❛♥❞ K(1) > 0 ✿ ❋♦r ❛♥② ✱ ❛s ✿ ✭✐✮ ❛♥❞ ❛s ✭✐✐✮ ❋♦r ❛❧❧ ✱ ❛s ✭✐✐✐✮ ❋♦r ❛❧❧ ✱ ❛s

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❆ss✉♠♣t✐♦♥s

➨ (H.1) ✿ ❚❤❡ ♦♣❡r❛t♦rs r ❛♥❞ σ2

ε ❛r❡ ❝♦♥t✐♥✉♦✉s ♦♥ ❛

♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ χ✱ ❛♥❞ Fχ(0) = 0✳ ▼♦r❡♦✈❡r✱ t❤❡ ❢✉♥❝t✐♦♥ ζ : t − → E (r(X) − r(χ)| X − χ = t) ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❞❡r✐✈❛❜❧❡ ❛t t = 0 ➨ (H.2) ✿ ❚❤❡ ❦❡r♥❡❧ K ✐s s✉♣♣♦rt❡❞ ♦♥ t❤❡ ❝♦♠♣❛❝t [0; 1] ❛♥❞ ❤❛s ❛ ❝♦♥t✐♥✉♦✉s ❞❡r✐✈❛t✐✈❡ ♦♥ [0; 1]✳ ▼♦r❡♦✈❡r✱ K′(s) ≤ 0 ❛♥❞ K(1) > 0 ➨ (H.3) ✿ ❋♦r ❛♥② s ∈ [0; 1]✱ τh(s) := Fχ(hs)

Fχ(h) → τ0(s) < ∞ ❛s

h → 0 ✿ ✭✐✮ ❛♥❞ ❛s ✭✐✐✮ ❋♦r ❛❧❧ ✱ ❛s ✭✐✐✐✮ ❋♦r ❛❧❧ ✱ ❛s

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❆ss✉♠♣t✐♦♥s

➨ (H.1) ✿ ❚❤❡ ♦♣❡r❛t♦rs r ❛♥❞ σ2

ε ❛r❡ ❝♦♥t✐♥✉♦✉s ♦♥ ❛

♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ χ✱ ❛♥❞ Fχ(0) = 0✳ ▼♦r❡♦✈❡r✱ t❤❡ ❢✉♥❝t✐♦♥ ζ : t − → E (r(X) − r(χ)| X − χ = t) ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❞❡r✐✈❛❜❧❡ ❛t t = 0 ➨ (H.2) ✿ ❚❤❡ ❦❡r♥❡❧ K ✐s s✉♣♣♦rt❡❞ ♦♥ t❤❡ ❝♦♠♣❛❝t [0; 1] ❛♥❞ ❤❛s ❛ ❝♦♥t✐♥✉♦✉s ❞❡r✐✈❛t✐✈❡ ♦♥ [0; 1]✳ ▼♦r❡♦✈❡r✱ K′(s) ≤ 0 ❛♥❞ K(1) > 0 ➨ (H.3) ✿ ❋♦r ❛♥② s ∈ [0; 1]✱ τh(s) := Fχ(hs)

Fχ(h) → τ0(s) < ∞ ❛s

h → 0 ➨ (H.4) ✿ ✭✐✮ hn → 0 ❛♥❞ nFχ(hn) → ∞ ❛s n → ∞ ✭✐✐✮ ❋♦r ❛❧❧ ℓ ∈ [0; 1]✱ 1 n

n

• i=1

hi hn Fχ(hi) Fχ(hn) 1−ℓ → αℓ ❛s n → ∞ ✭✐✐✐✮ ❋♦r ❛❧❧ r ≤ 2✱ 1 n

n

• i=1

Fχ(hi) Fχ(hn) r → βr ❛s n → ∞

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

◆♦t❛t✐♦♥s

❆s ✐♥ ❋❡rr❛t②✱ ▼❛s ❛♥❞ ❱✐❡✉ ✭✷✵✵✼✮ ✿ M0 = K(1) − 1 (sK(s))′τ0(s) ds M1 = K(1) − 1 K′(s)τ0(s) ds M2 = K(1)2 − 1 (K(s)2)′τ0(s) ds

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❆s②♠♣t♦t✐❝ ♠❡❛♥ sq✉❛r❡ ❡rr♦r

❯♥❞❡r (H1) − (H4) ✿ E

• r[ℓ]

n (χ) − r(χ)

• =

αℓ β1−ℓ ξ′(0)M0 M1 hn (1 + o(1)) + O

• 1

nFχ(hn)

• V
• r[ℓ]

n (χ)

• =

β1−2ℓ β2

1−ℓ

M2 M2

1

σ2

ǫ (χ)

1 nFχ(hn) (1 + o(1)) ➨ ❇✐❛s ր ➨ ❱❛r✐❛♥❝❡ ց

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❆s②♠♣t♦t✐❝ ♠❡❛♥ sq✉❛r❡ ❡rr♦r

❈♦r♦❧❧❛r✐❡s ✿ ➨ ❆s②♠♣t♦t✐❝ ♠❡❛♥ sq✉❛r❡ ❡rr♦r ✿

E

• r[ℓ]

n (χ) − r(χ)

2 =

• β1−2ℓ

β2

1−ℓ

M2 M2

1

σ2

ǫ (χ)

1 nFχ(hn) + ξ′(0)2 α2

ℓ

β2

1−ℓ

M2 M2

1

h2

n

• (1 + o(1))

➨ P❛rt✐❝✉❧❛r r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ✿ ✐❢ Fχ(t) ∼ ctκ✱ ✇✐t❤ t❤❡ ❝❤♦✐❝❡ hn ∼ An−

1 κ+2 ✿

lim

n→∞ n

2 2+κ E

• r[ℓ]

n (χ) − r(χ)

2 =

• β1−2ℓ

β2

1−ℓ

M2σ2

ǫ (χ)

AcM2

1

+ α2

ℓ

β2

1−ℓ

ξ′(0)2M2

0 A2

M2

1

• ❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❆❞❞✐t✐♦♥❛❧ ❛ss✉♠♣t✐♦♥s

➨ (H.5) ✿ ❚❤❡r❡ ❡①✐st λ > 0 ❛♥❞ µ > 0 s✉❝❤ t❤❛t E

• eλ|Y |µ

< +∞ ✿ ✭✐✮ ✭✐✐✮ ❋♦r ❛❧❧ ✱ ✭✐✐✐✮

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❆❞❞✐t✐♦♥❛❧ ❛ss✉♠♣t✐♦♥s

➨ (H.5) ✿ ❚❤❡r❡ ❡①✐st λ > 0 ❛♥❞ µ > 0 s✉❝❤ t❤❛t E

• eλ|Y |µ

< +∞ ➨ (H.6) ✿ ✭✐✮ limn→+∞

ln Fχ(hn) ln n

< ∞ ✭✐✐✮ ❋♦r ❛❧❧ ν ≥ 0✱ limn→+∞

nFχ(hn) (ln n)1+ 2

µ (ln ln n)2(ν+1) = +∞

✭✐✐✐✮ limn→+∞ Fχ(hn)(ln n)2/µ = 0

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❙tr♦♥❣ ❛♥❞ ✇❡❛❦ ❝♦♥s✐st❡♥❝②

➨ ❆❧♠♦st s✉r❡ ❝♦♥✈❡r❣❡♥❝❡ ✿ ❯♥❞❡r (H1) − (H6)✱ ✐❢ limn→∞ nh2

n = 0 ✿

lim sup

n→+∞

nFχ(hn) ln ln n 1/2 r[ℓ]

n (χ) − r(χ)

• =
• 2β1−2ℓ

β2

1−ℓ

σ2

ε(χ)M2

1/2 a.s. ❆s②♠♣t♦t✐❝ ♥♦r♠❛❧✐t② ✿ ❯♥❞❡r ✱ ✐❢ ✿

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❙tr♦♥❣ ❛♥❞ ✇❡❛❦ ❝♦♥s✐st❡♥❝②

➨ ❆❧♠♦st s✉r❡ ❝♦♥✈❡r❣❡♥❝❡ ✿ ❯♥❞❡r (H1) − (H6)✱ ✐❢ limn→∞ nh2

n = 0 ✿

lim sup

n→+∞

nFχ(hn) ln ln n 1/2 r[ℓ]

n (χ) − r(χ)

• =
• 2β1−2ℓ

β2

1−ℓ

σ2

ε(χ)M2

1/2 a.s. ➨ ❆s②♠♣t♦t✐❝ ♥♦r♠❛❧✐t② ✿ ❯♥❞❡r (H1) − (H6)✱ ✐❢ limn→∞ hn

• nF(hn) = 0 ✿
• nF(hn)
• r[ℓ]

n (χ) − r(χ)

D → N

• 0,

β1−2ℓ β2

1−ℓ

M2 M2

1

σ2

ε(χ)

• .

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❈♦♥❞✐t✐♦♥s ♦❢ s✐♠✉❧❛t✐♦♥s

➨ ❙❛♠♣❧❡ s✐③❡ ✿ n = 100 ➨ ❙✐♠✉❧❛t✐♦♥ ♦❢ X1, . . . , Xn ✿ st❛♥❞❛r❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥s ♦♥ [0; 1] ➨ ❉✐s❝r❡t✐③❛t✐♦♥ ♣♦✐♥ts ✿ p = 100 ➨ ❖♣❡r❛t♦r r ✿ r(χ) = 1

0 χ(s)2 ds

➨ ❙✐♠✉❧❛t✐♦♥ ♦❢ t❤❡ ♥♦✐s❡ ✿ ❣❛✉ss✐❛♥ N(0; 0.1) ➨ ◆✉♠❜❡r ♦❢ r❡♣❡t✐t✐♦♥s ♦❢ s✐♠✉❧❛t✐♦♥s ✿ N = 500

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❈❤♦✐❝❡ ♦❢ ♣❛r❛♠❡t❡rs

➨ ❈❤♦✐❝❡ ♦❢ K ✿ q✉❛❞r❛t✐❝ ❦❡r♥❡❧ ✿ K(u) = (1 − u2)1 1[0;1](u) ❈❤♦✐❝❡ ♦❢ ✿

▼❙P❊ ❚❛❜❧❡ ✿ ▼❡❛♥ ❛♥❞ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ ♣r❡❞✐❝t✐♦♥ ❡rr♦r✱ ❝♦♠♣✉t❡❞ ♦♥ r❡♣❡❛t❡❞ s✐♠✉❧❛t✐♦♥s✱ ❢♦r ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ ✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❈❤♦✐❝❡ ♦❢ ♣❛r❛♠❡t❡rs

➨ ❈❤♦✐❝❡ ♦❢ K ✿ q✉❛❞r❛t✐❝ ❦❡r♥❡❧ ✿ K(u) = (1 − u2)1 1[0;1](u) ➨ ❈❤♦✐❝❡ ♦❢ ℓ ✿ ℓ = 0

ℓ 0.25 0.5 0.75 1 ▼❙P❊ 0.4054848 0.4054814 0.4054786 0.4054764 0.4054746 (1.372965) (1.372930) (1.372896) (1.372863) (1.372831) ❚❛❜❧❡ ✿ ▼❡❛♥ ❛♥❞ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ ♣r❡❞✐❝t✐♦♥ ❡rr♦r✱ ❝♦♠♣✉t❡❞ ♦♥ 500 r❡♣❡❛t❡❞ s✐♠✉❧❛t✐♦♥s✱ ❢♦r ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ ℓ✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❈❤♦✐❝❡ ♦❢ t❤❡ s❡♠✐ ♥♦r♠

➨ ❙❡♠✐✲♥♦r♠ ❜❛s❡❞ ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥ts ❛♥❛❧②s✐s ♦❢ t❤❡ ❝✉r✈❡s✳ ➨ ❙❡♠✐✲♥♦r♠ ❜❛s❡❞ ♦♥ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❝✉r✈❡s ✐♥ ❛ ❋♦✉r✐❡r ❜❛s✐s✳ ➨ ❙❡♠✐✲♥♦r♠ ❜❛s❡❞ ♦♥ ❛ ❝♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❝✉r✈❡s✳ ♥♦r♠ [PCA] [FOU] [DERIV ] ▼❙P❊ 0.3936 0.4506 0.4527 (1.5190) (1.5624) (1.5616)

❚❛❜❧❡ ✿ ▼❡❛♥ ❛♥❞ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ ♣r❡❞✐❝t✐♦♥ ❡rr♦r✱ ❝♦♠♣✉t❡❞ ♦♥ 500 r❡♣❡❛t❡❞ s✐♠✉❧❛t✐♦♥s✱ ❢♦r ❞✐✛❡r❡♥t ❝❤♦✐❝❡s ♦❢ ♥♦r♠s✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❈❤♦✐❝❡ ♦❢ t❤❡ ❜❛♥❞✇✐❞t❤

➨ ❈❤♦✐❝❡ ♦❢ hn ✿ hi = C maxi=1,...,n Xi − χ i−δ ✇✐t❤ ✿ C ∈ {1, 2, 10} ❛♥❞ δ ∈ 1 10, 1 8, 1 6, 1 5, 1 4, 1 3, 1 2, 1

• ❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❙q✉❛r❡ ♣r❡❞✐❝t✐♦♥ ❡rr♦rs

➨ MSE1 ⇒ ❘❡❝✉rs✐✈❡ ❡st✐♠❛t♦r ➨ MSE2 ⇒ ❋❡rr❛t② ❛♥❞ ❱✐❡✉✬s ❡st✐♠❛t♦r

C 1 δ 1/10 1/8 1/6 1/5 1/4 1/3 1/2 1 MSE1 0.3022 0.3025 0.3033 0.3037 0.3041 0.3045 0.3045 0.3158 (0.6887) (0.6893) (0.6904) (0.6913) (0.6922) (0.6932) (0.6955) (0.7196) MSE2 NaN NaN NaN NaN NaN NaN NaN NaN (NaN) (NaN) (NaN) (NaN) (NaN) (NaN) (NaN) (NaN) C 2 δ 1/10 1/8 1/6 1/5 1/4 1/3 1/2 1 MSE1 0.3718 0.3718 0.3718 0.3718 0.3719 0.3721 0.3733 0.3765 (1.0690) (1.0693) (1.0698) (1.0702) (1.0712) (1.0731) (1.0790) (1.0957) MSE2 0.3202 0.3202 0.3202 0.3202 0.3202 0.3202 0.3202 0.3202 (0.7454) (0.7454) (0.7454) (0.7454) (0.7454) (0.7454) (0.7454) (0.7454) C 10 δ 1/10 1/8 1/6 1/5 1/4 1/3 1/2 1 MSE1 0.4031 0.4031 0.4031 0.4031 0.4031 0.4031 0.4032 0.4035 (0.8946) (0.8946) (0.8946) (0.8946) (0.8947) (0.8947) (0.8950) (0.8968) MSE2 0.3721 0.3721 0.3721 0.3721 0.3721 0.3721 0.3721 0.3721 (1.6497) (1.6497) (1.6497) (1.6497) (1.6497) (1.6497) (1.6497) (1.6497) ❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❈♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡

❈✉♠✉❧❛t❡❞ ❝♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡s ❢♦r T ❛❞❞✐t✐♦♥❛❧ ♦❜s❡r✈❛t✐♦♥s ✿

T 1 50 100 200 500 r[ℓ]

n+1, . . . , r[ℓ] n+T ✭s❡❝♦♥❞s✮

0.125 0.484 0.859 1.563 3.656 rn+1, . . . , rn+T ✭❋❡rr❛t② ❛♥❞ ❱✐❡✉✮ ✭s❡❝♦♥❞s✮ 0.047 1.922 5.594 21.938 152.719 ❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❘❡❛❧ ❞❛t❛s❡t ✿ ❊❧ ◆✐˜ ♥♦ ♣❤❡♥♦♠❡♥♦♥

➨ ▼♦♥t❤❧② s❡❛ s✉r❢❛❝❡ t❡♠♣❡r❛t✉r❡ ❢r♦♠ ❏❛♥✉❛r②✱ ✶✾✽✷ ✉♣ t♦ ❉❡❝❡♠❜❡r✱ ✷✵✶✶ ✭360 ♠♦♥t❤s✮✳ ➨ 30 ②❡❛r❧② ❝✉r✈❡s X1, . . . , X30 ❢r♦♠ ✶✾✽✷ t♦ ✷✵✶✶✱ ❞✐s❝r❡t✐③❡❞ ✐♥t♦ p = 12 ♣♦✐♥ts✳ ➨ ❚❤❡ ♦❜s❡r✈❛t✐♦♥ ♦❢ t❤❡ ✈❛r✐❛❜❧❡ ♦❢ ✐♥t❡r❡st ❛ ❝❡rt❛✐♥ ♠♦♥t❤ j ♦❢ t❤❡ ②❡❛r i ✐s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ s❡❛ t❡♠♣❡r❛t✉r❡ Xi+1 t❤❡ ♠♦♥t❤ j ✭✐✳❡ ❢♦r j = 1, . . . , 12 ❛♥❞ ❢♦r i = 1, . . . , 29, Y [j]

i

= Xi+1(j)).

2 4 6 8 10 12 20 22 24 26 28 months temperature

❋✐❣✉r❡ ✿ ❊❧ ◆✐♥♦ ②❡❛r❧② ❝✉r✈❡s t❡♠♣❡r❛t✉r❡s ❢r♦♠ ✶✾✽✷ ✉♣ t♦ ✷✵✶✶✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❘❡❛❧ ❞❛t❛s❡t ✿ ❊❧ ◆✐˜ ♥♦ ♣❤❡♥♦♠❡♥♦♥

➨ ❲❡ ✇❛♥t t♦ ♣r❡❞✐❝t t❤❡ ✈❛❧✉❡s ♦❢ Y [1]

29 , . . . , Y [12] 29

✭ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❝✉r✈❡ X30✮✳ ➨ ▼❡❛♥ sq✉❛r❡❞ ♣r❡❞✐❝t✐♦♥ MSPE = 1

12

12

j=1

• Y [j]

29 − Y [j] 29

2 ❘❡❝✉rs✐✈❡ ❡st✐♠❛t♦r ⇒ 0.5719 ❋❡rr❛t② ❛♥❞ ❱✐❡✉ ⇒ 0.2823

2 4 6 8 10 12 20 22 24 26 28 months temperature

❋✐❣✉r❡ ✿ ❊❧ ◆✐♥♦ tr✉❡ ❛♥❞ ♣r❡❞✐❝t❡❞ t❡♠♣❡r❛t✉r❡ ❝✉r✈❡s ❢♦r t❤❡ ②❡❛r ✷✵✶✶✳ ❚❤❡ s♦❧✐❞ ❧✐♥❡ ✐s t❤❡ tr✉❡ ❝✉r✈❡✳ ❚❤❡ ❞❛s❤❡❞ ❧✐♥❡ ✐s t❤❡ ♣r❡❞✐❝t❡❞ ❝✉r✈❡ ✇✐t❤ t❤❡ r❡❝✉rs✐✈❡ ❡st✐♠❛t♦r✳ ❚❤❡ ❞♦tt❡❞ ❧✐♥❡ ✐s t❤❡ ♣r❡❞✐❝t❡❞ ❝✉r✈❡ ✇✐t❤ t❤❡ ❡st✐♠❛t♦r ❢r♦♠ ❋❡rr❛t② ❛♥❞ ❱✐❡✉✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

➨ ▼❡❛♥ sq✉❛r❡ ❡rr♦r ❛♥❞ ❛❧♠♦st s✉r❡ ❝♦♥✈❡r❣❡♥❝❡ r❡s✉❧ts ➨ ❆s②♠♣t♦t✐❝ ♥♦r♠❛❧✐t② r❡s✉❧t ➨ ❊①♣❡❝t❡❞ ❜❡❤❛✈✐♦✉r ♦♥ s✐♠✉❧❛t✐♦♥s ➨ ❊①t❡♥s✐♦♥s ✿ ❛❧♠♦st ❝♦♠♣❧❡t❡ ❝♦♥✈❡r❣❡♥❝❡✱ ❞❡♣❡♥❞❛♥t ❞❛t❛✱ ❛✉t♦♠❛t✐❝ ❝❤♦✐❝❡ ♦❢ ❜❛♥❞✇✐❞t❤✱ ✳ ✳ ✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❇✐❜❧✐♦❣r❛♣❤②

❬✶❪ ❆♠✐r✐✱ ❆✳ ✭✷✵✶✷✮✳ ❘❡❝✉rs✐✈❡ r❡❣r❡ss✐♦♥ ❡st✐♠❛t♦rs ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥ t♦ ♥♦♥♣❛r❛♠❡tr✐❝ ♣r❡❞✐❝t✐♦♥✳ ❏✳ ◆♦♥♣❛r❛♠✳ ❙t❛t✐st✳✱ ✷✹✱ ✶✻✾✲✶✽✻✳ ❬✷❪ ❆♠✐r✐✱ ❆✳✱ ❈r❛♠❜❡s✱ ❈✳ ❛♥❞ ❚❤✐❛♠✱ ❇✳ ✭✷✵✶✷✮✳ ❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡✳ ❙✉❜♠✐tt❡❞✳ ❬✸❪ ❋❡rr❛t②✱ ❋✳✱ ▼❛s✱ ❆✳ ❛♥❞ ❱✐❡✉✱ P✳ ✭✷✵✵✼✮✳ ◆♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ♦♥ ❢✉♥❝t✐♦♥❛❧ ❞❛t❛ ✿ ✐♥❢❡r❡♥❝❡ ❛♥❞ ♣r❛❝t✐❝❛❧ ❛s♣❡❝ts✳ ❆✉st✳ ◆✳ ❩✳ ❏✳ ❙t❛t✐st✳✱ ✹✾✱ ✷✻✼✲✷✽✻✳ ❬✹❪ ❋❡rr❛t②✱ ❋✳ ❛♥❞ ❱✐❡✉✱ P✳ ✭✷✵✵✻✮✳ ◆♦♥♣❛r❛♠❡tr✐❝ ❋✉♥❝t✐♦♥❛❧ ❉❛t❛ ❆♥❛❧②s✐s✳ ❚❤❡♦r② ❛♥❞ Pr❛❝t✐❝❡✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇✲❨♦r❦✳ ❬✺❪ ❘❛♠s❛②✱ ❏✳❖✳ ❛♥❞ ❙✐❧✈❡r♠❛♥✱ ❇✳❲✳ ✭✷✵✵✺✮✳ ❋✉♥❝t✐♦♥❛❧ ❉❛t❛ ❆♥❛❧②s✐s ✭2nd ❊❞✳✮✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇✲❨♦r❦✳

❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡