❙tr❡❛♠ ❝✐♣❤❡r ❝r②♣t❛♥❛❧②s✐s ❆♥♥❡ ❈❛♥t❡❛✉t ❆♥♥❡✳❈❛♥t❡❛✉t❅✐♥r✐❛✳❢r ❤tt♣✿✴✴✇✇✇✲r♦❝q✳✐♥r✐❛✳❢r✴s❡❝r❡t✴❆♥♥❡✳❈❛♥t❡❛✉t✴ ■❝❡ ❇r❡❛❦ ✷✵✶✸
❑❡②str❡❛♠ ❣❡♥❡r❛t♦r s❡❝r❡t ❦❡② ♣✉❜❧✐❝ k ❜✐ts ✐♥✐t✐❛❧ ✈❛❧✉❡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ❫ ✡ ✢ ✐♥✐t✐❛❧✐③❛t✐♦♥ ✬✩ ✬✩ ❄ ✐♥t❡r♥❛❧ st❛t❡ x 0 x 1 ✲ ✲ ✲ ✲ Φ Φ n ❜✐ts ✫✪ ✫✪ · · ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❅ f � ❅ f � ✜❧t❡r ❅ � ❅ � tr❛♥s✐t✐♦♥ ❄ ❄ s 0 s 1 ❦❡②str❡❛♠ ✶
❆✈♦✐❞✐♥❣ ❣❡♥❡r✐❝ ❛tt❛❝❦s • ❚❤❡ ✐♥t❡r♥❛❧ st❛t❡ ♠✉st ❜❡ ❛t ❧❡❛st t✇✐❝❡ ❧❛r❣❡r t❤❛♥ t❤❡ ❦❡② ✭❚✐♠❡✲♠❡♠♦r②✲❞❛t❛ tr❛❞❡✲♦✛ ❬●♦❧✐❝ ✾✺❪❬❇❛❜❜❛❣❡ ✾✺❪✮ • ❚❤❡ ❣❡♥❡r❛t♦r ♠✉st ♣❛ss t❤❡ st❛t✐st✐❝❛❧ t❡sts✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ✜❧t❡r✐♥❣ ❢✉♥❝t✐♦♥ f ♠✉st ❜❡ ❜❛❧❛♥❝❡❞✳ • ❆t ❧❡❛st ♦♥❡ ❢✉♥❝t✐♦♥ ❛♠♦♥❣ Φ ❛♥❞ f ♠✉st ❜❡ ♥♦♥❧✐♥❡❛r✳ • Φ ❤❛s ♥♦ s❤♦rt ❝②❝❧❡s✳ ✷
❈❤♦♦s✐♥❣ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ Φ ❚✇♦ str❛t❡❣✐❡s✿ • ❈❤♦♦s❡ ❛ r❛♥❞♦♠✲❧♦♦❦✐♥❣ ♠❛♣♣✐♥❣✴♣❡r♠✉t❛t✐♦♥ ♦♣❡r❛t✐♥❣ ♦♥ ❛ ❧❛r❣❡ ✐♥t❡r♥❛❧ st❛t❡✿ t❤❡ ♣❡r✐♦❞ ♦❢ ( x t ) t ≥ 0 ✐s ❡①♣❡❝t❡❞ t♦ ❜❡ ❝❧♦s❡ n t♦ 2 2 ✳ ❙❤♦rt ❝②❝❧❡s ❡①✐st ❜✉t ❛r❡ ✉♥❧✐❦❡❧② t♦ ♦❝❝✉r✳ ❊❣✿ ❘❈✹✳ • ❈❤♦♦s❡ ❛ ♣❡r♠✉t❛t✐♦♥ ✇✐t❤ s♦♠❡ ❦♥♦✇♥ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦♣❡rt✐❡s ♦♣❡r❛t✐♥❣ ♦♥ ❛ s♠❛❧❧ ✐♥t❡r♥❛❧ st❛t❡✿ t❤❡ ♣❡r✐♦❞ ♦❢ ( x t ) t ≥ 0 ❝❛♥ ❜❡ ♣r♦✈❡❞ t♦ ❜❡ ❝❧♦s❡ t♦ 2 n ✳ ❙❤♦rt ❝②❝❧❡s ❛r❡ ❛✈♦✐❞❡❞✳ ❊❣✿ ▲❋❙❘ ✭❚♦r✬s ❧❡❝t✉r❡✮✳ ✸
❖✉t❧✐♥❡ • ❙t❛t✐st✐❝❛❧ ❛tt❛❝❦s ❛❣❛✐♥st ✜❧t❡r❡❞ ▲❋❙❘s • ❆❧❣❡❜r❛✐❝ ❛tt❛❝❦s ❛❣❛✐♥st ✜❧t❡r❡❞ ▲❋❙❘s • ❈♦rr❡❧❛t✐♦♥ ❛tt❛❝❦s ❛❣❛✐♥st ❝♦♠❜✐♥❛t✐♦♥ ❣❡♥❡r❛t♦rs ✹
❙t❛t✐st✐❝❛❧ ❛tt❛❝❦s ❛❣❛✐♥st ✜❧t❡r❡❞ ▲❋❙❘s ✺
Pr✐♥❝✐♣❧❡ s t ✻ ✟ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ f ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✻ ✻ ✻ ✻ ✻ ✻ u t ✲ ✲ u t + γ 1 u t + γ 2 u t + γ 3 . . . u t + γ n ■❢ f ✐s ❜❛❧❛♥❝❡❞✱ t❤❡♥ t❤❡ ♦✉t♣✉t s❡q✉❡♥❝❡ ❝♦♥t❛✐♥s r♦✉❣❤❧② t❤❡ s❛♠❡ ♥✉♠❜❡rs ♦❢ 0 ❛♥❞ 1 ✳ ◗✉❡st✐♦♥✿ ❲❤❛t ✐❢ ✇❡ ❝♦♥s✐❞❡r ♣❛✐rs ♦❢ ❜✐ts❄ ❉♦ ✇❡ ❣❡t ❛ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥❄ ✻
❊①❛♠♣❧❡ ♦✉t♣✉t ✻ ✎☞ ✍✌ ✲ ✰ ✐ P P ✏ ✶ ✏✏ P P ✎☞ ✎☞ ✍✌ ✍✌ × × ❨ ❍ ❍ ❨ ✯ ✟ ✟ ✯ ✟✟ ❍ ✟✟ ❍ ❍ ❍ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✎☞ ❄ ✍✌ ✛ ✰ ❣❡♥❡r❛t❡s t❤❡ 127 ✲❜✐t s❡q✉❡♥❝❡ ✵✶✵✵✵✵✶✶✵✵✵✵✵✶✵✵✵✵✶✶✶✶✶✵✶✶✶✵✶✵✵✵✶✵✵✶✶✶✵✶✶✶✵✶✶✵✵✶✶✵✵✵✶✶✵✵✶✶✶✶ ✵✶✶✵✶✶✶✵✵✶✶✵✵✶✶✵✵✶✵✵✶✵✶✵✵✶✶✵✵✶✵✶✶✶✵✵✶✶✶✵✶✵✶✵✵✶✶✵✶✶✵✵✶✵✵✵✶✶✶✵ ✵✶✶✵✶✵✵ → ■t ❤❛s 64 ♦♥❡s✳ ✼
❊①❛♠♣❧❡ ♦✉t♣✉t ✻ ✎☞ ✍✌ ✲ ✰ ✐ P P ✶ ✏ ✏✏ P P ✎☞ ✎☞ ✍✌ ✍✌ × × ❨ ❍ ❍ ❨ ✯ ✟ ✯ ✟ ✟✟ ❍ ✟✟ ❍ ❍ ❍ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✎☞ ❄ ✍✌ ✛ ✰ ❣❡♥❡r❛t❡s t❤❡ 127 ✲❜✐t s❡q✉❡♥❝❡ ✵✶✵✵✵✵✶✶✵✵✵✵✵✶✵✵✵✵✶✶✶✶✶✵✶✶✶✵✶✵✵✵✶✵✵✶✶✶✵✶✶✶✵✶✶✵✵✶✶✵✵✵✶✶✵✵✶✶✶✶ ✵✶✶✵✶✶✶✵✵✶✶✵✵✶✶✵✵✶✵✵✶✵✶✵✵✶✶✵✵✶✵✶✶✶✵✵✶✶✶✵✶✵✶✵✵✶✶✵✶✶✵✵✶✵✵✵✶✶✶✵ ✵✶✶✵✶✵✵ (0 , 0) 30 t✐♠❡s (0 , 1) 32 t✐♠❡s ( s t , s t +1 ) = (1 , 0) 32 t✐♠❡s (1 , 1) 32 t✐♠❡s ✽
❊①❛♠♣❧❡ ♦✉t♣✉t ✻ ✎☞ ✍✌ ✲ ✰ ✐ P P ✶ ✏ ✏✏ P P ✎☞ ✎☞ ✍✌ ✍✌ × × ❨ ❍ ❍ ❨ ✯ ✟ ✯ ✟ ✟✟ ❍ ✟✟ ❍ ❍ ❍ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✎☞ ❄ ✍✌ ✛ ✰ ❣❡♥❡r❛t❡s t❤❡ 127 ✲❜✐t s❡q✉❡♥❝❡ ✵✶✵✵✵✵✶✶✵✵✵✵✵✶✵✵✵✵✶✶✶✶✶✵✶✶✶✵✶✵✵✵✶✵✵✶✶✶✵✶✶✶✵✶✶✵✵✶✶✵✵✵✶✶✵✵✶✶✶✶ ✵✶✶✵✶✶✶✵✵✶✶✵✵✶✶✵✵✶✵✵✶✵✶✵✵✶✶✵✵✶✵✶✶✶✵✵✶✶✶✵✶✵✶✵✵✶✶✵✶✶✵✵✶✵✵✵✶✶✶✵ ✵✶✶✵✶✵✵ (0 , 0) 22 t✐♠❡s (0 , 1) 40 t✐♠❡s ( s t , s t +2 ) = (1 , 0) 39 t✐♠❡s (1 , 1) 24 t✐♠❡s ✾
❊①❛♠♣❧❡ ❆t t✐♠❡ t ✿ ❆t t✐♠❡ t + 2 ✿ ♦✉t♣✉t ✻ ✎☞ ✍✌ ✲ ✰ ✐ P P ✶ ✏ ✏✏ P P ✎☞ ✎☞ ✍✌ ✍✌ × × ❨ ❍ ❍ ❨ ✟ ✯ ✟ ✯ ✟✟ ❍ ✟✟ ❍ ❍ ❍ ✲ c ✲ b ✲ ✲ ✲ ✲ ✲ a ✲ ✎☞ ❄ ✍✌ ✛ ✰ s t = f ( c, b, a ) ✶✵
❊①❛♠♣❧❡ ❆t t✐♠❡ t ✿ ❆t t✐♠❡ t + 2 ✿ ♦✉t♣✉t ♦✉t♣✉t ✻ ✻ ✎☞ ✎☞ ✍✌ ✍✌ ✲ ✰ ✲ ✰ ✐ P ✐ P P P ✶ ✏ ✶ ✏ ✏✏ P ✏✏ P P P ✎☞ ✎☞ ✎☞ ✎☞ ✍✌ ✍✌ ✍✌ ✍✌ × × × × ❨ ❍ ❨ ❍ ❍ ❨ ❨ ❍ ✯ ✟ ✟ ✯ ✟ ✯ ✯ ✟ ✟✟ ❍ ✟✟ ❍ ✟✟ ❍ ✟✟ ❍ ❍ ❍ ❍ ❍ ✲ c ✲ b ✲ c ✲ b ✲ ✲ ✲ ✲ ✲ a ✲ ✲ ✲ ✲ d ✲ ✲ ✲ ✎☞ ✎☞ ❄ ❄ ✍✌ ✍✌ ✛ ✛ ✰ ✰ s t = f ( c, b, a ) s t +2 = f ( d, c, b ) ✶✶
❊①❛♠♣❧❡ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ( s t , s t +2 ) ✐s ❣✐✈❡♥ ❜② t❤❡ tr✉t❤ t❛❜❧❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ( a, b, c, d ) �→ ( f ( c, b, a ) , f ( d, c, b )) d c b a y 1 y 2 d c b a y 1 y 2 ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✶ ✶ ✶ ✵ ✶ ✵ ✶ ✶ ✶ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✵ ✶ ✵ ✶ ✶ ✵ ✶ ✶ ✵ ✶ ✶ ✶ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✶ ✶ ✵ ✵ ✶ ✵ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ✶✷
❊①❛♠♣❧❡ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ( s t , s t +2 ) ✐s ❣✐✈❡♥ ❜② t❤❡ tr✉t❤ t❛❜❧❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ( a, b, c, d ) �→ ( f ( c, b, a ) , f ( d, c, b )) d c b a y 1 y 2 d c b a y 1 y 2 ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✶ ✶ ✶ ✵ ✶ ✵ ✶ ✶ ✶ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✶ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✶ ✵ ✶ ✶ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✶ ✶ ✵ ✵ ✶ ✵ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ → Pr[ s t � = s t +2 ] = 5 8 > 1 2 ✶✸
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