●❡♥❡ ❘❛♥❦✐♥❣ ❛♥❞ ❉✐✛❡r❡♥t✐❛❧ ❊①♣r❡ss✐♦♥ ▼✐✐❦❛ ❆♥ ❡✣❝✐❡♥t ❛♣♣r♦❛❝❤ ❢♦r ❧❛r❣❡ s❝❛❧❡ ❆❤❞❡s♠ä❦✐ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❱❡r❡♥❛ ♣r❡❞✐❝t✐♦♥ ❛♥❞ ❢❡❛t✉r❡ s❡❧❡❝t✐♦♥ ❩✉❜❡r ❛♥❞ ❑♦r❜✐♥✐❛♥ ❙tr✐♠♠❡r ❇❛❝❦❣r♦✉♥❞ ▼✐✐❦❛ ❆❤❞❡s♠ä❦✐ ♦♥ ▲❉❆ ❛♥❞ ❉❊ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❱❡r❡♥❛ ❩✉❜❡r ❛♥❞ ❑♦r❜✐♥✐❛♥ ❙tr✐♠♠❡r ❈♦rr❡❧❛t✐♦♥ ❆❞❥✉st❡❞ t✲❙❝♦r❡ ■♥st✐t✉t ❢ür ▼❡❞✐③✐♥✐s❝❤❡ ■♥❢♦r♠❛t✐❦✱ ❙t❛t✐st✐❦ ✉♥❞ ❊♣✐❞❡♠✐♦❧♦❣✐❡ ✭■▼■❙❊✮✱ ❉❡✜♥✐t✐♦♥ ❯♥✐✈❡rs✐t② ♦❢ ▲❡✐♣③✐❣ ●❡♥❡ ❙❡ts ❊st✐♠❛t✐♦♥ ❊✣❝✐❡♥t ❏✉❧② ✾✱ ✷✵✵✾ ❍✐❣❤✲ ❉✐♠❡♥s✐♦♥❛❧ ✉s❡❘ ❈❧❛ss✐✜❝❛t✐♦♥ P♦♦❧❡❞ ❚❤❡ ❘ ❯s❡r ❈♦♥❢❡r❡♥❝❡ ✷✵✵✾ ❈❡♥tr♦✐❞ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ▲❉❆ ▼✉❧t✐✲❈❧❛ss ❈❆❚ ❙❝♦r❡ ❋◆❉❘✲❜❛s❡❞ ❋❡❛t✉r❡ ❙❡❧❡❝t✐♦♥ ❘❡s✉❧ts ♦♥ ❞✐✛❡r❡♥t ✲♦♠✐❝s ❞❛t❛ s❡ts ❈♦♥❝❧✉s✐♦♥s
●❡♥❡ ❘❛♥❦✐♥❣ ❛♥❞ ▼♦t✐✈❛t✐♦♥ ❉✐✛❡r❡♥t✐❛❧ ❊①♣r❡ss✐♦♥ ▼✐✐❦❛ ❆❤❞❡s♠ä❦✐ ▼✉❝❤ ♦❢ ❝✉rr❡♥t ❛♥❛❧②s✐s ♦❢ ✯♦♠✐❝s ❞❛t❛ ❢♦❝✉s❡s ♦♥ ❜✐♦♠❛r❦❡r ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❱❡r❡♥❛ ❞✐s❝♦✈❡r②✳ ❊①❛♠♣❧❡s✿ ❩✉❜❡r ❛♥❞ ❑♦r❜✐♥✐❛♥ ❙tr✐♠♠❡r • ❲❤✐❝❤ ❣❡♥❡s ❛r❡ ❞✐✛❡r❡♥t✐❛❧❧② ❡①♣r❡ss❡❞❄ ❇❛❝❦❣r♦✉♥❞ • ❲❤✐❝❤ ❢❡❛t✉r❡s ♦❢ ❛ ♣r♦t❡♦♠✐❝ s♣❡❝tr✉♠ ❝❛♥ ❜❡ ✉s❡❞ t♦ ♦♥ ▲❉❆ ❛♥❞ ❉❊ ❞✐st✐♥❣✉✐s❤ ❜❡t✇❡❡♥ ❝❛♥❝❡r ❛♥❞ ❤❡❛❧t❤② t✐ss✉❡❄ ❈♦rr❡❧❛t✐♦♥ ❆❞❥✉st❡❞ t✲❙❝♦r❡ ❚❤❡r❡ ❤❛✈❡ ❜❡❡♥ ✈❡r② ♠❛♥② s✉❣❡st✐♦♥s ❤♦✇ t♦ ❝♦♥❞✉❝t ❉❡✜♥✐t✐♦♥ ●❡♥❡ ❙❡ts st❛t✐st✐❝❛❧ ❛♥❛❧②s❡s ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡①♣r❡ss✐♦♥ ❛♥❞ ❝❧❛ss✐✜❝❛t✐♦♥✳ ❊st✐♠❛t✐♦♥ ❊✣❝✐❡♥t ❍✐❣❤✲ ❉✐♠❡♥s✐♦♥❛❧ ❙♦♠❡ ✈❡r② ❣♦♦❞ ✭❛♥❞ ✇❡❧❧✲❦♥♦✇♥✮ ❝❤♦✐❝❡s✿ ❈❧❛ss✐✜❝❛t✐♦♥ P♦♦❧❡❞ • ❙❆▼ ♦r ✏♠♦❞❡r❛t❡❞ t ✑ ❢♦r ❣❡♥❡ r❛♥❦✐♥❣✱ ❈❡♥tr♦✐❞ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ▲❉❆ • P❆▼ ❛❧❣♦r✐t❤♠ ❢♦r ❝❧❛ss✐✜❝❛t✐♦♥ ❛♥❞ ♣r❡❞✐❝t✐♦♥ ▼✉❧t✐✲❈❧❛ss ❈❆❚ ❙❝♦r❡ ❋◆❉❘✲❜❛s❡❞ ❋❡❛t✉r❡ ❙❡❧❡❝t✐♦♥ ❘❡s✉❧ts ♦♥ ❞✐✛❡r❡♥t ✲♦♠✐❝s ❞❛t❛ s❡ts ❈♦♥❝❧✉s✐♦♥s
●❡♥❡ ❘❛♥❦✐♥❣ ❛♥❞ ▼♦t✐✈❛t✐♦♥ ■■ ❉✐✛❡r❡♥t✐❛❧ ❊①♣r❡ss✐♦♥ ❖✉r st❛rt✐♥❣ ♣♦✐♥t✿ ❛♥❛❧②s✐s ♦❢ ❛ ♣r♦t❡♦♠✐❝s ❞❛t❛ s❡t ✭st✉❞② ♦❢ ▼✐✐❦❛ ❆❤❞❡s♠ä❦✐ ♣❛♥❝r❡❛s ❝❛♥❝❡rs✮ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❱❡r❡♥❛ ❩✉❜❡r ❛♥❞ Pr♦♣❡rt✐❡s✿ ❑♦r❜✐♥✐❛♥ ❙tr✐♠♠❡r • ✈❡r② str♦♥❣ ✴ ♣❡r✈❛s✐✈❡ ❝♦rr❡❧❛t✐♦♥ ♣❛tt❡r♥ ❛♠♦♥❣ ❢❡❛t✉r❡s ❇❛❝❦❣r♦✉♥❞ • ❞✐♠❡♥s✐♦♥ ❧❡ss ❡①tr❡♠❡ t❤❛♥ ✐♥ ❣❡♥❡ ❡①♣r❡ss✐♦♥ ❞❛t❛ ♦♥ ▲❉❆ ❛♥❞ ❉❊ ❈♦rr❡❧❛t✐♦♥ ◗✉❡st✐♦♥ ✶✿ ✐s ✉♥✐✈❛r✐❛t❡ ❢❡❛t✉r❡ s❡❧❡❝t✐♦♥ ❛♣♣r♦♣r✐❛t❡ ✐❢ ❢❡❛t✉r❡s ❆❞❥✉st❡❞ t✲❙❝♦r❡ ❛r❡ ❝♦rr❡❧❛t❡❞ ❄ ❉❡✜♥✐t✐♦♥ ●❡♥❡ ❙❡ts ◗✉❡st✐♦♥ ✷✿ ✇❤❛t r♦❧❡ ❞♦ ❣❡♥❡ ✴ ❢❡❛t✉r❡ s❡ts ♣❧❛② ✐♥ t❤❡ ❊st✐♠❛t✐♦♥ ❛♥❛❧②s✐s❄ ❊✣❝✐❡♥t ❍✐❣❤✲ ◗✉❡st✐♦♥ ✸✿ ✐s t❤❡ ❋❉❘ ❢r❛♠❡✇♦r❦ s✉✐t❛❜❧❡ ❢♦r ❛ss✐❣♥✐♥❣ ❉✐♠❡♥s✐♦♥❛❧ ❈❧❛ss✐✜❝❛t✐♦♥ s✐❣♥✜❝❛♥❝❡ t♦ ❢❡❛t✉r❡s ❄ P♦♦❧❡❞ ❈❡♥tr♦✐❞ ◗✉❡st✐♦♥ ✹✿ ❛r❡ t❤❡r❡ ❝♦♠♣✉t❛t✐♦♥❛❧❧② ❡✣❝✐❡♥t ♣r♦❝❡❞✉r❡s❄ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ▲❉❆ ▼✉❧t✐✲❈❧❛ss ❈❆❚ ❙❝♦r❡ ❋◆❉❘✲❜❛s❡❞ ▼❛✐♥ t❤❡♠❡s✿ r❛♥❦✐♥❣ ❛♥❞ ❢❡❛t✉r❡ s❡❧❡❝t✐♦♥ ✉♥❞❡r ❞❡♣❡♥❞❡♥❝❡✱ ❋❡❛t✉r❡ ❙❡❧❡❝t✐♦♥ ❛♣♣❧✐❝❛t✐♦♥ t♦ ❝❧❛ss✐✜❝❛t✐♦♥ ❘❡s✉❧ts ♦♥ ❞✐✛❡r❡♥t ✲♦♠✐❝s ❞❛t❛ s❡ts ❈♦♥❝❧✉s✐♦♥s
●❡♥❡ ❘❛♥❦✐♥❣ ❛♥❞ ❉✐✛❡r❡♥t✐❛❧ ❊①♣r❡ss✐♦♥ ▼✐✐❦❛ ❆❤❞❡s♠ä❦✐ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❱❡r❡♥❛ ❩✉❜❡r ❛♥❞ ❑♦r❜✐♥✐❛♥ ❙tr✐♠♠❡r ■✳ ❉✐✛❡r❡♥t✐❛❧ ❊①♣r❡ss✐♦♥ ❛♥❞ ❈❧❛ss✐✜❝❛t✐♦♥ ❇❛❝❦❣r♦✉♥❞ ♦♥ ▲❉❆ ❛♥❞ ❉❊ ❈♦rr❡❧❛t✐♦♥ ❆❞❥✉st❡❞ t✲❙❝♦r❡ ❉❡✜♥✐t✐♦♥ ●❡♥❡ ❙❡ts ❊st✐♠❛t✐♦♥ ❊✣❝✐❡♥t ❍✐❣❤✲ ❉✐♠❡♥s✐♦♥❛❧ ❈❧❛ss✐✜❝❛t✐♦♥ P♦♦❧❡❞ ❈❡♥tr♦✐❞ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ▲❉❆ ▼✉❧t✐✲❈❧❛ss ❈❆❚ ❙❝♦r❡ ❋◆❉❘✲❜❛s❡❞ ❋❡❛t✉r❡ ❙❡❧❡❝t✐♦♥ ❘❡s✉❧ts ♦♥ ❞✐✛❡r❡♥t ✲♦♠✐❝s ❞❛t❛ s❡ts ❈♦♥❝❧✉s✐♦♥s
●❡♥❡ ❘❛♥❦✐♥❣ ❛♥❞ ❉✐✛❡r❡♥t✐❛❧ ❊①♣r❡ss✐♦♥ ✲ ❙❡t✉♣ ❉✐✛❡r❡♥t✐❛❧ ❊①♣r❡ss✐♦♥ ▼✐✐❦❛ ❆❤❞❡s♠ä❦✐ • ❙❡t✲❯♣✿ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❱❡r❡♥❛ • ❑ = ✷ ❣r♦✉♣s ✭❡✳❣✳ ♣❛t✐❡♥t ❛♥❞ ❝♦♥tr♦❧✮ ❩✉❜❡r ❛♥❞ ❑♦r❜✐♥✐❛♥ • ▲❛r❣❡ s❡t ♦❢ ✐ ∈ ✶ , ..., ♣ ❣❡♥❡s ❙tr✐♠♠❡r • ❆ s♠❛❧❧ s❡t ♦❢ ♥ ✶ + ♥ ✷ = ♥ ♠❡❛s✉r❡♠❡♥ts ❇❛❝❦❣r♦✉♥❞ • ◗✉❡st✐♦♥✿ ❲❤✐❝❤ ❣❡♥❡s ❛r❡ ❞✐✛❡r❡♥t✐❛❧❧② ❡①♣r❡ss❡❞ ♦♥ ▲❉❆ ❛♥❞ ❉❊ ✭s❤♦✇ ❛ ❞✐✛❡r❡♥t ❡①♣r❡ss✐♦♥ ♣r♦✜❧❡✮❄ ❈♦rr❡❧❛t✐♦♥ • ●♦❛❧✿ ❘❛♥❦✐♥❣ t❤❡ ♣ ❣❡♥❡s ❛❝❝♦r❞✐♥❣ t♦ ❆❞❥✉st❡❞ t✲❙❝♦r❡ t❤❡✐r ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❣r♦✉♣s ❉❡✜♥✐t✐♦♥ ●❡♥❡ ❙❡ts ❊st✐♠❛t✐♦♥ • ❚♦♦❧s✿ ❚❤❡r❡ ❡①✐sts ❛♥ ❛❜✉♥❞❛♥❝❡ ♦❢ r❛♥❦✐♥❣ st❛t✐st✐❝s ❊✣❝✐❡♥t ♠♦st❧② ♠♦❞✐✜❝❛t✐♦♥s ♦❢ t❤❡ ♦r❞✐♥❛r② ❙t✉❞❡♥t t ✲st❛t✐st✐❝✿ ❍✐❣❤✲ ❉✐♠❡♥s✐♦♥❛❧ ❈❧❛ss✐✜❝❛t✐♦♥ µ ✶ ( ✐ ) − µ ✷ ( ✐ ) P♦♦❧❡❞ ❈❡♥tr♦✐❞ t st✉❞ ( ✐ ) = ❋♦r♠✉❧❛t✐♦♥ � ♦❢ ▲❉❆ ( ✶ ♥ ✶ + ✶ σ ( ✐ ) ♥ ✷ ) ▼✉❧t✐✲❈❧❛ss ❈❆❚ ❙❝♦r❡ ❋◆❉❘✲❜❛s❡❞ ❋❡❛t✉r❡ ❙❡❧❡❝t✐♦♥ ❘❡s✉❧ts ♦♥ ❞✐✛❡r❡♥t ✲♦♠✐❝s ❞❛t❛ s❡ts ❈♦♥❝❧✉s✐♦♥s
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