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❘❡♥♦r♠❛❧✐③❛t✐♦♥ ❛♥❞ ❞✐s♦r❞❡r ✿ ❛ s✐♠♣❧❡ t♦② ♠♦❞❡❧ ❇❡r♥❛r❞ ❉❊❘❘■❉❆ ❈♦❧❧è❣❡ ❞❡ ❋r❛♥❝❡ ❈❤❛✐r❡ ❞❡ P❤②s✐q✉❡ ❙t❛t✐st✐q✉❡ ❆♥♥❡❝② ✶✹ ❙❡♣t❡♠❜❡r ✷✵✶✽

❈♦❧❧❛❜♦r❛t♦rs ◮ ❍❛❦✐♠ ❛♥❞ ❱❛♥♥✐♠❡♥✉s ✶✾✾✷ ◮ ●✐❛❝♦♠✐♥✱ ▲❛❝♦✐♥✱ ❚♦♥✐♥❡❧❧✐ ✷✵✵✼ ◮ ❘❡t❛✉① ✷✵✶✹ ◮ ❈❤❡♥✱ ❍✉✱ ▲✐❢s❤✐ts✱ ❙❤✐ ✷✵✶✼ ◮ ❉❛❣❛r❞ ✷✵✶✽

❖❯❚▲■◆❊ ❚❤❡ ✐❞❡❛ ♦❢ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❚❤❡ t♦② ♠♦❞❡❧ ❚❤❡ ❞❡♥❛t✉r❛t✐♦♥ ♦❢ ❉◆❆ ♣r♦❜❧❡♠ ■ts ❤✐❡r❛r❝❤✐❝❛❧ ❧❛tt✐❝❡ ✈❡rs✐♦♥

♣ ♣ P❤❛s❡ tr❛♥s✐t✐♦♥ ❘❡♥♦r♠❛❧✐③❛t✐♦♥✿ ❛♥ ❡①❛♠♣❧❡✱ t❤❡ ♣❡r❝♦❧❛t✐♦♥ ♣r♦❜❧❡♠ → ♣ ′ ♣ ♣ ′ = R ( ♣ )

❘❡♥♦r♠❛❧✐③❛t✐♦♥✿ ❛♥ ❡①❛♠♣❧❡✱ t❤❡ ♣❡r❝♦❧❛t✐♦♥ ♣r♦❜❧❡♠ → ♣ ′ ♣ ♣ ′ = R ( ♣ ) ♣ ∗ = R ( ♣ ∗ ) P❤❛s❡ tr❛♥s✐t✐♦♥

▲♦♦❦ ❢♦r t❤❡ ✜①❡❞ ♣♦✐♥t ♣ ❜ ♣ ▲✐♥❡❛r✐③❡ ❜ ♥❡❛r t❤❡ ✜①❡❞ ♣♦✐♥t ♣ ❜ ♣ ❈r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ❛♥❞ ✉♥✐✈❡rs❛❧✐t② ❚❤❡ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❣r♦✉♣ p’, ... p b L L ❘❡♥♦r♠❛❧✐③❛t✐♦♥ tr❛♥s❢♦r♠❛t✐♦♥ ( ♣ ′ , · · · ) = R ❜ ( ♣ , · · · )

❚❤❡ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❣r♦✉♣ p’, ... p b L L ❘❡♥♦r♠❛❧✐③❛t✐♦♥ tr❛♥s❢♦r♠❛t✐♦♥ ( ♣ ′ , · · · ) = R ❜ ( ♣ , · · · ) ( ♣ ∗ , · · · ) = R ❜ ( ♣ ∗ · · · ) ▲♦♦❦ ❢♦r t❤❡ ✜①❡❞ ♣♦✐♥t ( ♣ ′ , · · · ) = L ❜ ( ♣ · · · ) ▲✐♥❡❛r✐③❡ R ❜ ♥❡❛r t❤❡ ✜①❡❞ ♣♦✐♥t ⇒ ❈r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ❛♥❞ ✉♥✐✈❡rs❛❧✐t②

❚❤❡ ✢♦✇ ♦❢ r❡♥♦r♠❛❧✐③❛t✐♦♥ ρ ( ① , ✵ ) > ✵ ❣✐✈❡♥ ❢♦r ① > ✵ � ① ∂ρ ( ① , τ ) = ∂ρ ( ① , τ ) + ✶ ρ ( ① ✶ , τ ) ρ ( ① − ① ✶ , τ ) ❞① ✶ ∂τ ∂ ① ✷ ✵

◗✉❡st✐♦♥s✿ ❲❤❛t ✐s t❤❡ ❧✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ P ♥ ❳ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s ❳ ♥ ❲❤❛t ✐s t❤❡ ❧✐♠✐t ♦❢ ❳ ♥ ✷ ♥ ❆ ❝❧❛ss ♦❢ ♠♦❞❡❧s ❚✇♦ ✐♥❣r❡❞✐❡♥ts✿ ◮ ❙t❛rt ✇✐t❤ ❛♥ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ✐✳✐✳❞✳ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❳ ( ✶ ) · · · ❳ ( ❥ ) · · · ✵ ✵ ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❛ ❞✐str✐❜✉t✐♦♥ P ✵ ( ❳ ) ✳ ◮ ❆ ♥♦♥✲❧✐♥❡❛r ❢✉♥❝t✐♦♥ ● t♦ ✐t❡r❛t❡ t❤❡s❡ ✈❛r✐❛❜❧❡s (1) (2) X X n n � � ❳ ( ❥ ) ❳ ( ✷ ❥ − ✶ ) + ❳ ( ✷ ❥ ) = ● ♥ ♥ ♥ X = max[X + X −1 , 0] X n+1

❆ ❝❧❛ss ♦❢ ♠♦❞❡❧s ❚✇♦ ✐♥❣r❡❞✐❡♥ts✿ ◮ ❙t❛rt ✇✐t❤ ❛♥ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ✐✳✐✳❞✳ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❳ ( ✶ ) · · · ❳ ( ❥ ) · · · ✵ ✵ ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❛ ❞✐str✐❜✉t✐♦♥ P ✵ ( ❳ ) ✳ ◮ ❆ ♥♦♥✲❧✐♥❡❛r ❢✉♥❝t✐♦♥ ● t♦ ✐t❡r❛t❡ t❤❡s❡ ✈❛r✐❛❜❧❡s (1) (2) X X n n � � ❳ ( ❥ ) ❳ ( ✷ ❥ − ✶ ) + ❳ ( ✷ ❥ ) = ● ♥ ♥ ♥ X = max[X + X −1 , 0] X n+1 ◗✉❡st✐♦♥s✿ ◮ ❲❤❛t ✐s t❤❡ ❧✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ P ♥ ( ❳ ) ♦❢ t❤❡ ✈❛r✐❛❜❧❡s ❳ ♥ ◮ ❲❤❛t ✐s t❤❡ ❧✐♠✐t ♦❢ � ❳ ♥ � ✷ ♥

❚❤❡ t♦② ♠♦❞❡❧ ❈♦❧❧❡t✱ ●❧❛s❡r✱ ❊❝❦♠❛♥♥✱ ▼❛rt✐♥ ✶✾✽✹ (1) (2) (3) (4) X n−1 X n−1 X n−1 X n−1 (1) (2) X X n n X n+1 ❳ ♥ + ✶ = ♠❛① [ ❳ ( ✶ ) + ❳ ( ✷ ) − ✶ , ✵ ] ♥ ♥

❚❤❡ t♦② ♠♦❞❡❧ ❈♦❧❧❡t✱ ●❧❛s❡r✱ ❊❝❦♠❛♥♥✱ ▼❛rt✐♥ ✶✾✽✹ (1) (2) (3) (4) X n−1 X n−1 X n−1 X n−1 (1) (2) X X n n X n+1 ❳ ♥ + ✶ = ♠❛① [ ❳ ( ✶ ) + ❳ ( ✷ ) − ✶ , ✵ ] ♥ ♥

❚❤❡ t♦② ♠♦❞❡❧ ❈♦❧❧❡t✱ ●❧❛s❡r✱ ❊❝❦♠❛♥♥✱ ▼❛rt✐♥ ✶✾✽✹ (1) (2) (3) (4) X n−1 X n−1 X n−1 X n−1 (1) (2) X X n n X n+1 ❳ ♥ + ✶ = ♠❛① [ ❳ ( ✶ ) + ❳ ( ✷ ) − ✶ , ✵ ] ♥ ♥

P ✵ ① ✶ ❳ ✵ ❳ ▼❛✐♥ q✉❡st✐♦♥ ❳ ♥ + ✶ = ♠❛① [ ❳ ( ✶ ) + ❳ ( ✷ ) − ✶ , ✵ ] ♥ ♥ ❲❤❛t ✐s t❤❡ ❧✐♠✐t ❨ ♦❢ ❳ ♥ ✷ ♥ P ✵ ( ① ) = δ ❳ ,µ Y 1 µ

▼❛✐♥ q✉❡st✐♦♥ ❳ ♥ + ✶ = ♠❛① [ ❳ ( ✶ ) + ❳ ( ✷ ) − ✶ , ✵ ] ♥ ♥ ❲❤❛t ✐s t❤❡ ❧✐♠✐t ❨ ♦❢ ❳ ♥ ✷ ♥ P ✵ ( ① ) = δ ❳ ,µ P ✵ ( ① ) = ( ✶ − λ ) δ ❳ , ✵ + λ δ ❳ ,µ Y Y 1 µ µ c µ

▼❛✐♥ q✉❡st✐♦♥ ❳ ♥ + ✶ = ♠❛① [ ❳ ( ✶ ) + ❳ ( ✷ ) − ✶ , ✵ ] ♥ ♥ ❲❤❛t ✐s t❤❡ ❧✐♠✐t ❨ ♦❢ ❳ ♥ ✷ ♥ P ✵ ( ① ) = ( ✶ − λ ) δ ❳ + λ δ ❳ − µ P ✵ ( ① ) = δ ❳ − µ Y Y µ c µ 1 µ

❘❡♥♦r♠❛❧✐③❛t✐♦♥ ❳ ♥ + ✶ = ♠❛① [ ❳ ( ✶ ) + ❳ ( ✷ ) − ✶ , ✵ ] ♥ ♥ ❊①❛❝t r❡♥♦r♠❛❧✐③❛t✐♦♥ ✭ s♣❡❝✐❛❧ ❝❛s❡ ❳ ♥ ❛r❡ ✐♥t❡❣❡rs ✮ P ✵ ( ① ) ✐s ❣✐✈❡♥ ; P ♥ → P ♥ + ✶ ❳ P ♥ ( ❳ ) ③ ❳ ❍ ♥ ( ③ ) = � ❉❡✜♥❡ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❍ ♥ + ✶ ( ③ ) = ❍ ♥ ( ③ ) ✷ − ❍ ♥ ( ✵ ) ✷ + ❍ ♥ ( ✵ ) ✷ ③

❆ ♦♥❡ ♣❛r❛♠❡t❡r ❢❛♠✐❧② ♦❢ ✜①❡❞ ♣♦✐♥ts ◆♦♥❡ ♦❢ t❤❡♠ ✐s ❛❝❝❡ss✐❜❧❡ ❆ ♣❤❛s❡ tr❛♥s✐t✐♦♥ ♦❢ t❤❡ ❇❡r❡③✐♥s❦✐ ❑♦st❡r❧✐t③ ❚❤♦✉❧❡ss t②♣❡ ❆ ❢❡✇ ❢❛❝ts ❳ ♥ + ✶ = ♠❛① [ ❳ ( ✶ ) + ❳ ( ✷ ) − ✶ , ✵ ] ♥ ♥ ❍ ♥ + ✶ ( ③ ) = ❍ ♥ ( ③ ) ✷ − ❍ ♥ ( ✵ ) ✷ + ❍ ♥ ( ✵ ) ✷ ③ ◮ ❆ ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❈♦❧❧❡t✱ ●❧❛s❡r✱ ❊❝❦♠❛♥♥✱ ▼❛rt✐♥ ✶✾✽✹ P (X) 0 ❍ ✵ ( ③ ) = ✶ − λ + λ ③ ✷ ❋♦r ❡①❛♠♣❧❡ 1−λ λ ❝ = ✶ ⇒ λ ✺ 0 2 X

◆♦♥❡ ♦❢ t❤❡♠ ✐s ❛❝❝❡ss✐❜❧❡ ❆ ♣❤❛s❡ tr❛♥s✐t✐♦♥ ♦❢ t❤❡ ❇❡r❡③✐♥s❦✐ ❑♦st❡r❧✐t③ ❚❤♦✉❧❡ss t②♣❡ ❆ ❢❡✇ ❢❛❝ts ❳ ♥ + ✶ = ♠❛① [ ❳ ( ✶ ) + ❳ ( ✷ ) − ✶ , ✵ ] ♥ ♥ ❍ ♥ + ✶ ( ③ ) = ❍ ♥ ( ③ ) ✷ − ❍ ♥ ( ✵ ) ✷ + ❍ ♥ ( ✵ ) ✷ ③ ◮ ❆ ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❈♦❧❧❡t✱ ●❧❛s❡r✱ ❊❝❦♠❛♥♥✱ ▼❛rt✐♥ ✶✾✽✹ P (X) 0 ❍ ✵ ( ③ ) = ✶ − λ + λ ③ ✷ ❋♦r ❡①❛♠♣❧❡ 1−λ λ ❝ = ✶ ⇒ λ ✺ 0 2 X ◮ ❆ ♦♥❡ ♣❛r❛♠❡t❡r ❢❛♠✐❧② ♦❢ ✜①❡❞ ♣♦✐♥ts

❆ ♣❤❛s❡ tr❛♥s✐t✐♦♥ ♦❢ t❤❡ ❇❡r❡③✐♥s❦✐ ❑♦st❡r❧✐t③ ❚❤♦✉❧❡ss t②♣❡ ❆ ❢❡✇ ❢❛❝ts ❳ ♥ + ✶ = ♠❛① [ ❳ ( ✶ ) + ❳ ( ✷ ) − ✶ , ✵ ] ♥ ♥ ❍ ♥ + ✶ ( ③ ) = ❍ ♥ ( ③ ) ✷ − ❍ ♥ ( ✵ ) ✷ + ❍ ♥ ( ✵ ) ✷ ③ ◮ ❆ ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❈♦❧❧❡t✱ ●❧❛s❡r✱ ❊❝❦♠❛♥♥✱ ▼❛rt✐♥ ✶✾✽✹ P (X) 0 ❍ ✵ ( ③ ) = ✶ − λ + λ ③ ✷ ❋♦r ❡①❛♠♣❧❡ 1−λ λ ❝ = ✶ ⇒ λ ✺ 0 2 X ◮ ❆ ♦♥❡ ♣❛r❛♠❡t❡r ❢❛♠✐❧② ♦❢ ✜①❡❞ ♣♦✐♥ts ◮ ◆♦♥❡ ♦❢ t❤❡♠ ✐s ❛❝❝❡ss✐❜❧❡

❆ ❢❡✇ ❢❛❝ts ❳ ♥ + ✶ = ♠❛① [ ❳ ( ✶ ) + ❳ ( ✷ ) − ✶ , ✵ ] ♥ ♥ ❍ ♥ + ✶ ( ③ ) = ❍ ♥ ( ③ ) ✷ − ❍ ♥ ( ✵ ) ✷ + ❍ ♥ ( ✵ ) ✷ ③ ◮ ❆ ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❈♦❧❧❡t✱ ●❧❛s❡r✱ ❊❝❦♠❛♥♥✱ ▼❛rt✐♥ ✶✾✽✹ P (X) 0 ❍ ✵ ( ③ ) = ✶ − λ + λ ③ ✷ ❋♦r ❡①❛♠♣❧❡ 1−λ λ ❝ = ✶ ⇒ λ ✺ 0 2 X ◮ ❆ ♦♥❡ ♣❛r❛♠❡t❡r ❢❛♠✐❧② ♦❢ ✜①❡❞ ♣♦✐♥ts ◮ ◆♦♥❡ ♦❢ t❤❡♠ ✐s ❛❝❝❡ss✐❜❧❡ ◮ ❆ ♣❤❛s❡ tr❛♥s✐t✐♦♥ ♦❢ t❤❡ ❇❡r❡③✐♥s❦✐ ❑♦st❡r❧✐t③ ❚❤♦✉❧❡ss t②♣❡