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▲✐♣s❝❤✐t③✲❑✐❧❧✐♥❣ ❝✉r✈❛t✉r❡s ♦❢ ❡①❝✉rs✐♦♥ s❡ts ❢♦r ✷❉ r❛♥❞♦♠ ✜❡❧❞s

❍❡r♠✐♥❡ ❇✐❡r♠é✱ ▲▼❆✱ ❯♥✐✈❡rs✐t② ♦❢ P♦✐t✐❡rs

▼❛r❝❤✱ ✶✷t❤ ✷✵✶✾✱ ■❍P✱ P❛r✐s

❞❡❢❛✉❧t

❈♦❧❧❛❜♦r❛t✐♦♥

❆❣♥ès ❉❡s♦❧♥❡✉① ✭❈◆❘❙✱ ❈▼▲❆✱ ❊◆❙ P❛r✐s✲❙❛❝❧❛②✮ ❊❧❡♥❛ ❉✐ ❇❡r♥❛r❞✐♥♦ ✭❈◆❆▼✱ P❛r✐s✮ ❈é❧✐♥❡ ❉✉✈❛❧ ✭▼❆P✺✱ ❯♥✐✈❡rs✐té P❛r✐s ❉❡s❝❛rt❡s✮ ❆♥♥❡ ❊str❛❞❡ ✭▼❆P✺✱ ❯♥✐✈❡rs✐té P❛r✐s ❉❡s❝❛rt❡s✮

❞❡❢❛✉❧t

❖✉t❧✐♥❡s

✶ ▼♦t✐✈❛t✐♦♥ ✿ ❙❤♦t ♥♦✐s❡ r❛♥❞♦♠ ✜❡❧❞s ❙❤♦t ♥♦✐s❡ r❛♥❞♦♠ ✜❡❧❞s ❊①❝✉rs✐♦♥ s❡ts ✿ ❡①❛♠♣❧❡s ✷ ▲✐♣s❝❤✐t③ ❑✐❧❧✐♥❣ ❈✉r✈❛t✉r❡s ❢♦r ❡①❝✉rs✐♦♥ s❡ts ▲✐♣s❝❤✐t③✲❑✐❧❧✐♥❣ ❝✉r✈❛t✉r❡s ❛♥❞ ❞❡♥s✐t✐❡s ❈❛s❡ ♦❢ s♠♦♦t❤ ❞❡t❡r♠✐♥✐st✐❝ ❢✉♥❝t✐♦♥s ❲❡❛❦ ❢♦r♠✉❧❛ ❢♦r st❛t✐♦♥❛r② r❛♥❞♦♠ ✜❡❧❞ ✸ ❊①❛♠♣❧❡s ♦❢ st❛t✐♦♥❛r② ✐s♦tr♦♣✐❝ r❛♥❞♦♠ ✜❡❧❞s

• ❛✉ss✐❛♥ r❛♥❞♦♠ ✜❡❧❞s

❈❤✐✷ r❛♥❞♦♠ ✜❡❧❞s ❙t✉❞❡♥t r❛♥❞♦♠ ✜❡❧❞s ❙❤♦t ♥♦✐s❡ r❛♥❞♦♠ ✜❡❧❞s ✹ ❊①❛♠♣❧❡ ♦❢ r❡❛❧ ❞❛t❛

❞❡❢❛✉❧t

▼♦t✐✈❛t✐♦♥ ✿ ❙❤♦t ♥♦✐s❡ r❛♥❞♦♠ ✜❡❧❞s

❆ ✭P♦✐ss♦♥✮ s❤♦t ♥♦✐s❡ r❛♥❞♦♠ ✜❡❧❞ ✐s ❛ r❛♥❞♦♠ ❢✉♥❝t✐♦♥ X : Rd → R ❣✐✈❡♥ ❜② ∀x ∈ Rd, X(x) =

• i∈I

gmi(x − xi), ✇❤❡r❡ {xi}i∈I ✐s ❛ P♦✐ss♦♥ ♣♦✐♥t ♣r♦❝❡ss ♦❢ ✐♥t❡♥s✐t② λ > ✵ ✐♥ Rd✱ {mi}i∈I ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ✓ ♠❛r❦s ✔ ✇✐t❤ ❞✐str✐❜✉t✐♦♥ F(dm) ♦♥ Rk✱ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ {xi}i∈I✳ ❚❤❡ ❢✉♥❝t✐♦♥s gm ❛r❡ r❡❛❧✲✈❛❧✉❡❞ ❞❡t❡r♠✐♥✐st✐❝ ❢✉♥❝t✐♦♥s✱ ❝❛❧❧❡❞ s♣♦t ❢✉♥❝t✐♦♥s✱ s✉❝❤ t❤❛t

• Rk
• Rd |gm(y)| dy F(dm) < +∞.

❍❡r❡ ✇❡ ❝♦♥s✐❞❡r d = ✷ ❛♥❞✱ ❢♦r s❛❦❡ ♦❢ s✐♠♣❧✐❝✐t②✱ k ≤ ✷ ✇✐t❤ ❛ s✐♥❣❧❡ L✶(R✷) ❢✉♥❝t✐♦♥ g r❛♥❞♦♠❧② ✇❡✐❣❤t❡❞ ❛♥❞ ❞✐❧❛t❡❞ ✿ (W , R) ∼ F ✐s ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ R × (✵, +∞) ❛♥❞ ❢♦r m = (w, r) gm(x) = wg(x/r).

❞❡❢❛✉❧t

❊①❛♠♣❧❡ ✶ ✿ ❞✐s❦ ✇✐t❤ r❛♥❞♦♠ r❛❞✐✉s

▲❡t d = ✷✱ T ❛ ❜♦✉♥❞❡❞ ❝❧♦s❡❞ r❡❝t❛♥❣❧❡ ♦❢ R✷ ❛♥❞ g = ✶D✳ ❈♦♥s✐❞❡r r❛♥❞♦♠ ❞✐s❦ ♦❢ r❛❞✐✉s r = r✶ ♦r r = r✷ ✇✐t❤ ✵ < r✶ < r✷ ✭❡❛❝❤ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶/✷✮✱ s❛♠❡ ✇❡✐❣❤ts W = ✶ ❛✳s✳ ❛♥❞ ✐♥t❡♥s✐t② λ > ✵ ❚❤❡ ♥✉♠❜❡r n ♦❢ ❝❡♥t❡rs ✐♥ T ✐s ❛ P♦✐ss♦♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦❢ ♣❛r❛♠❡t❡r λ|T| ❚❤❡ ❝❡♥t❡rs x✶, . . . , xn ❛r❡ t❤r♦✇♥ ✉♥✐❢♦r♠❧②✱ ✐♥❞❡♣❡♥❞❡♥t❧② ♦♥ T ❚❤❡ r❛❞✐✉s R✶, . . . , Rn ❛r❡ ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ❝❡♥t❡r ❜② ✢✐♣♣✐♥❣ ❛ ❝♦✐♥ t♦ ❝❤♦♦s❡ ❜❡t✇❡❡♥ r✶ ♦r r✷✳

❞❡❢❛✉❧t

❊①❝✉rs✐♦♥ s❡t

❲❡ ❝♦♥s✐❞❡r t❤❡ ❡①❝✉rs✐♦♥ s❡t ♦r t❤❡ ❧❡✈❡❧ s❡t ♦❢ ❧❡✈❡❧ u ∈ R ♦❢ X ✐♥ T ❞❡✜♥❡❞ ❜② EX(u) ∩ T := {x ∈ T; X(x) ≥ u} ✇✐t❤ EX(u) = {X ≥ u}. ✈✐❡✇ ✸❉ ✈✐❡✇ ✷❉ s♦♠❡ ❧❡✈❡❧ ❧✐♥❡s u = ✵.✺ u = ✶.✺ u = ✷.✺

❞❡❢❛✉❧t

❊①❛♠♣❧❡ ✷ ✿ ●❛✉ss✐❛♥ ❦❡r♥❡❧

▲❡t ✉s ❝❤♦♦s❡ g(x) = e− x✷

✷

✐♥st❡❛❞ ♦❢ ✶D✳ ✈✐❡✇ ✸❉ ✈✐❡✇ ✷❉ s♦♠❡ ❧❡✈❡❧ ❧✐♥❡s u = ✵.✺ u = ✶ u = ✶.✺

❞❡❢❛✉❧t

▼❛✐♥ q✉❡st✐♦♥s

❲❤❛t ❝❛♥ ❜❡ s❛✐❞ ❛❜♦✉t ✧♠❡❛♥✧ ❣❡♦♠❡tr② ♦❢ ❡①❝✉rs✐♦♥ s❡ts ❄ ❆r❡❛ ❄ P❡r✐♠❡t❡r ❄ ❊✉❧❡r ❈❤❛r❛❝t❡r✐st✐❝❂# ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ✕ # ❤♦❧❡s ❄ ❑♥♦✇♥ r❡s✉❧ts ❢♦r ❇♦♦❧❡❛♥ ♠♦❞❡❧ ✿ ▼❡❝❦❡ ✭✷✵✵✶✮✱ ▼❡❝❦❡✱ ❲❛❣♥❡r ✭✶✾✾✶✮ ❙♠♦♦t❤ ●❛✉ss✐❛♥ ❛♥❞ r❡❧❛t❡❞ r❛♥❞♦♠ ✜❡❧❞s ✿ ❆❞❧❡r ✭✷✵✵✵✮✱ ❆❞❧❡r✱ ❚❛②❧♦r ✭✷✵✵✼✮✱ ❆③❛ïs✱ ❲s❝❤❡❜♦r ✭✷✵✵✾✮✱ ✳✳✳ ❍✐❣❤ ❧❡✈❡❧s ❢♦r s♦♠❡ ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ✜❡❧❞s ✿ ❆❞❧❡r✱ ❙❛♠♦r♦❞♥✐ts❦②✱ ❚❛②❧♦r ✭✷✵✶✵✱✷✵✶✸✮✱✳✳✳ ❚✇♦ ❞✐✛❡r❡♥t ❢r❛♠❡✇♦r❦s

✶ P✐❡❝❡✇✐③❡ ❝♦♥st❛♥t ✜❡❧❞s ✭❡❧❡♠❡♥t❛r②✮ ✷ ❙♠♦♦t❤ ✜❡❧❞s ✿ ❛t ❧❡❛st C ✷

❞❡❢❛✉❧t

❈✉r✈❛t✉r❡ ♠❡❛s✉r❡s

▲❡t E ⊂ R✷ ❜❡ ❛ ✧♥✐❝❡ s❡t✧✳ ■ts ❝✉r✈❛t✉r❡ ♠❡❛s✉r❡s Φj(E, ·)✱ ❢♦r j = ✵, ✶, ✷✱ ❛r❡ ❞❡✜♥❡❞ ❢♦r ❛♥② ❇♦r❡❧ s❡t U ⊂ R✷ ❜② Φ✷(E, U) = |E ∩ U|✱ Φ✶(E, U) = ✶

✷ H✶(∂E ∩ U) = ✶ ✷P❡r(E, U)

Φ✵(E, U) =

✶ ✷π❚❈(∂E, U)✱

✇❤❡r❡ H✶(∂E ∩ U) ✐s t❤❡ ❧❡♥❣❤t ❛♥❞ ❚❈(∂E, U) t❤❡ t♦t❛❧ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ♣♦s✐t✐✈❡❧② ♦r✐❡♥t❡❞ ❝✉r✈❡ ∂E ✐♥ U✳

❘❡❢ ✿ ❙❝❤♥❡✐❞❡r✱ ❲❡✐❧✱ ❙t♦❝❤❛st✐❝ ❛♥❞ ■♥t❡❣r❛❧ ●❡♦♠❡tr②

❞❡❢❛✉❧t

P✐❡❝❡✇✐s❡ r❡❣✉❧❛r ❝✉r✈❡

❆ ❏♦r❞❛♥ ❝✉r✈❡ Γ ⊂ R✷ ✐s ♣✐❡❝❡✇✐s❡ r❡❣✉❧❛r ✐❢ Γ = RΓ ∪ CΓ ✇✐t❤ #CΓ < +∞ ❢♦r x ∈ RΓ ♦♥❡ ❤❛s x = γ(s) ❢♦r s♦♠❡ s ∈ (✵, ε) ✇✐t❤ γ : (✵, ε) → Γ C ✷✱ ❛r❝ ❧❡♥❣t❤ ♣❛r❛♠❡tr✐③❡❞✳ ❚❤❡♥✱ |γ(✵, ε)|✶ = ε✳ ❚❤❡ s✐❣♥❡❞ ❝✉r✈❛t✉r❡ κΓ(x) ♦❢ Γ ❛t x ✐s κΓ(x) = γ′′(s), γ′(s)⊥. ❢♦r x ∈ CΓ ♦♥❡ ❤❛s x = γ(✵) ✇✐t❤ γ : (−ε, ε) → Γ ❝♦♥t✐♥✉♦✉s ❛♥❞ C ✷ ♦♥ (−ε, ε) {✵} s✳t✳ γ′ ❛❞♠✐ts ❧✐♠✐ts γ′(✵−) ∈ S✶ ❛♥❞ γ′(✵+) ∈ S✶ ❛t ✵✳ ❚❤❡♥✱ |γ(−ε, ε)|✶ = ✷ε✳ ❚❤❡ t✉r♥✐♥❣ ❛♥❣❧❡ ❛t ❛ ❝♦r♥❡r ♣♦✐♥t x = γ(✵) ∈ CΓ ✐s t❤❡ ❛♥❣❧❡ αΓ(x) ∈ (−π, π) ❜❡t✇❡❡♥ t❤❡ t❛♥❣❡♥t ✏❜❡❢♦r❡✑ ❛♥❞ t❤❡ ♦♥❡ ✏❛❢t❡r✑ x αΓ(x) = ❆r❣ γ′(✵+) − ❆r❣ γ′(✵−) ∈ (−π, π).

❞❡❢❛✉❧t

❚♦t❛❧ ❝✉r✈❛t✉r❡ ❛♥❞ ❊✉❧❡r ❝❤❛r❛❝t❡r✐st✐❝

❚❤❡ t♦t❛❧ ❝✉r✈❛t✉r❡ ♦❢ Γ ✐♥ U ✐s ❞❡✜♥❡❞ ❛s ❚❈(Γ, U) :=

• RΓ∩U

κΓ(x)H✶(dx) +

• x∈CΓ∩U

αΓ(x).

• ❛✉ss✲❇♦♥♥❡t ❚❤❡♦r❡♠ ✿ ▲❡t E ⊂ U ❜❡ ❛ r❡❣✉❧❛r r❡❣✐♦♥ ✐❡ E =
• E s✉❝❤

t❤❛t ∂E ✐s ❢♦r♠❡❞ ❜② n ♣✐❡❝❡✇✐s❡ r❡❣✉❧❛r ♣♦s✐t✐✈❡❧② ♦r✐❡♥t❡❞ ❞✐s❥♦✐♥t ❏♦r❞❛♥ ❝✉r✈❡s Γ✶, . . . , Γn t❤❡♥ ❚❈(∂E, U) :=

n

• i=✶

❚❈(Γi, U) = ✷πχ(E), ✇❤❡r❡ χ(E) ✐s t❤❡ ❊✉❧❡r ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ E✱ χ(E) = #❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts − # ❤♦❧❡s. ■t ❢♦❧❧♦✇s t❤❛t Φ✵(E, U) = χ(E)✳

❞❡❢❛✉❧t

• ❡♦♠❡tr② ♦❢ ❡①❝✉rs✐♦♥ s❡ts

▲❡t X = (X(x))x∈R✷ ❜❡ ❛ st❛t✐♦♥❛r② ✧♥✐❝❡✧ r❛♥❞♦♠ ✜❡❧❞ ❛♥❞ T ❛ ❜♦✉♥❞❡❞ ❝❧♦s❡❞ r❡❝t❛♥❣❧❡ ✇✐t❤

• T = ∅✳ ❋♦r u ∈ R✱ ✇❡ ❝♦♥s✐❞❡r t❤❡

❡①❝✉rs✐♦♥ s❡t ♦❢ ❧❡✈❡❧ u ✐♥ T EX(u) ∩ T := {x ∈ T; X(x) ≥ u}. t❤❡ ▲❑ ❝✉r✈❛t✉r❡s ♦❢ t❤❡ ❡①❝✉rs✐♦♥ s❡t EX(u) ✇✐t❤✐♥ T ❛r❡ Cj(X, u, T) := Φj(EX(u) ∩ T, T), ❢♦r j = ✵, ✶, ✷. ❛♥❞✱ ❛ss✉♠✐♥❣ t❤❡ ❧✐♠✐ts ❡①✐st✱ t❤❡ ❛ss♦❝✐❛t❡❞ ▲❑ ❞❡♥s✐t✐❡s ❛r❡ C ∗

j (X, u) := lim TրR✷

E[Cj(X, u, T)] |T| , ❢♦r j = ✵, ✶, ✷, ✇❤❡r❡ lim

TրR✷ st❛♥❞s ❢♦r t❤❡ ❧✐♠✐t ❛❧♦♥❣ ❛♥② s❡q✉❡♥❝❡ ♦❢ ❜♦✉♥❞❡❞

r❡❝t❛♥❣❧❡s t❤❛t ❣r♦✇s t♦ R✷✳ ◆♦t❡ t❤❛t C ∗

j (σX + m, u) = C ∗ j (X, (u − m)/σ).

❞❡❢❛✉❧t

▲❑ ❞❡♥s✐t✐❡s

◆♦t❡ t❤❛t Cj(X, T, u) = Φj(EX(u) ∩ T, T) = Φj(EX(u),

• T) + Φj(T ∩ EX(u), ∂T).

❆❝t✉❛❧❧②✱ C ∗

j (X, u) = E[Φj(EX (u), ˚ T)] |T|

. ▼♦r❡♦✈❡r✱ ❜② st❛t✐♦♥❛r✐t② C ∗

✷ (X, u) = P(X(✵) ≥ u).

❞❡❢❛✉❧t

❲❤❛t ❢♦r s♠♦♦t❤ ❞❡t❡r♠✐♥✐st✐❝ ❢✉♥❝t✐♦♥s ❄

❆ss✉♠❡ t❤❛t f : W → R ✐s C ✷ ✇✐t❤ W ♦♣❡♥ s✳t✳ T ⊂ W ❛♥❞ ♥♦t❡ t❤❛t ∂Ef (u) ∩ ˚ T = {x ∈ ˚ T; f (x) = u}✳

✶ ❇② ▼♦rs❡✲❙❛r❞ t❤❡♦r❡♠✱ t❤❡ ✐♠❛❣❡ ❜② f ♦❢ t❤❡ s❡t ♦❢ ❝r✐t✐❝❛❧ ✈❛❧✉❡s

♦❢ f ❤❛s ♠❡❛s✉r❡ ✵ ✐♥ R✳

✷ ▲❡t u ❜❡ s✉❝❤ ❛ ♥♦♥✲❝r✐t✐❝❛❧ ✈❛❧✉❡✳ ❋♦r ❛ ❝✉r✈❡ γ ❣✐✈❡♥ ❜② ❛♥ ✐♠♣❧✐❝✐t

❢♦r♠ f (γ(s)) = u✱ ✇❡ ❤❛✈❡ γ′(s)⊥ = ∇f (γ(s))/||∇f (γ(s))|| ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ ❝✉r✈❛t✉r❡ ❛t x = γ(s) ✐s ❣✐✈❡♥ ❜② κf (x) = −D✷f (x).(∇f ⊥(x), ∇f ⊥(x)) ||∇f (x)||✸ .

✸ ❚❤❡ ❝♦❛r❡❛ ❢♦r♠✉❧❛ ❢♦r ▲✐♣s❝❤✐t③ ♠❛♣♣✐♥❣s st❛t❡s t❤❛t✱ ❢♦r ❛♥②

L✲✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ g✱

• R
• ∂Ef (u)∩ ˚

T

g(x)H✶(dx) du =

• ˚

T

g(x)∇f (x) dx.

❞❡❢❛✉❧t

❲❡❛❦ ❢♦r♠✉❧❛ ❢♦r Φ✶ ❛♥❞ Φ✵

▲❡t ✉s ❝❤♦♦s❡ h : R → R ❛ ❜♦✉♥❞❡❞ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ✭t❡st ❢✉♥❝t✐♦♥✮✳ ❈♦❛r❡❛ ❢♦r♠✉❧❛ ✇✐t❤ g(x) = h(f (x)) ✿

• R

h(u)Φ✶(Ef (u), ˚ T)du = ✶ ✷

• ˚

T

h(f (x))∇f (x) dx. ❈♦❛r❡❛ ❢♦r♠✉❧❛ ✇✐t❤ g(x) = h(f (x))κf (x) ❢♦r κf (x) = −D✷f (x).(∇f (x)⊥, ∇f (x)⊥) ∇f (x)✸ ✶∇f (x)>✵,

• R

h(u)Φ✵(Ef (u), ˚ T)du = − ✶ ✷π

• ˚

T

h(f (x))D✷f (x).(∇f (x)⊥, ∇f (x)⊥) ∇f (x)✷ ✶∇f (x)>✵ dx.

❞❡❢❛✉❧t

❊①♣❡❝t❛t✐♦♥ ✉♥❞❡r st❛t✐♦♥❛r✐t②

❲❤❡♥ X ✐s ❛ st❛t✐♦♥❛r② ✜❡❧❞ ❛✳s✳ C ✷ ✇✐t❤ X(✵)✱ ∇X(✵) ❛♥❞ D✷X(✵) L✶

• R

h(u)C ∗

✶ (X, u)du = ✶

✷E (h(X(✵))∇X(✵))

• R

h(u)C ∗

✵ (X, u)du = −✶

✷π E

• h(X(✵))D✷X(✵).(∇X(✵)⊥, ∇X(✵)⊥)

∇X(✵)✷ ✶∇X(✵)>✵

• ➥ ❆❧❧♦✇s ✐♥❢♦r♠❛t✐♦♥ ❢♦r ❛✳❡ u ∈ R

❲❡ ❛❧s♦ ♦❜t❛✐♥ ❢♦r h = ✶ TV ∗(X) = ✷

• R C ∗

✶ (X, u)du ❀

LTC ∗(X) = ✷π

• R C ∗

✵ (X, u)du.

❞❡❢❛✉❧t

❊①♣❡❝t❛t✐♦♥ ✉♥❞❡r st❛t✐♦♥❛r✐t② ❛♥❞ ✐s♦tr♦♣②

❲❤❡♥ X ✐s ❛❧s♦ ✐s♦tr♦♣✐❝✱ ✇❡ ❣❡t E (h(X(✵))∇X(✵)) = π ✷ E (h(X(✵))|X✶(✵)|) . ▼♦r❡♦✈❡r✱ E

• h(X(✵))D✷X(✵).(∇X(✵)⊥, ∇X(✵)⊥)

∇X(✵)✷ ✶∇X(✵)>✵

• = α✵(h) + ✷α✷(h),

✇✐t❤ α✵(h) = E (h(X(✵))X✶✶(✵))✱ ❛♥❞ α✷(h) = −✷E

• h(X(✵))X✶✷(✵)X✶(✵)X✷(✵)

∇X(✵)✷ ✶∇X>✵

• .

❚❤❡r❡❢♦r❡ TV ∗(X) = π ✷ E (|X✶(✵)|) ❛♥❞ LTC ∗(X) = ✹E

• X✶✷(✵)X✶(✵)X✷(✵)

∇X(✵)✷ ✶∇X>✵

• .

❞❡❢❛✉❧t

• ❛✉ss✐❛♥ ❝❛s❡

❋♦r X ❛ st❛t✐♦♥❛r② st❛♥❞❛r❞ ✐s♦tr♦♣✐❝ C ✷ ●❛✉ss✐❛♥ r❛♥❞♦♠ ✜❡❧❞ ✇❡ ♥♦t❡ ρ(x) = ❈♦✈(X(x), X(✵))✱ ❛♥❞ t❤❡ s❡❝♦♥❞ s♣❡❝tr❛❧ ♠♦♠❡♥t λ✷ = −∂✷

kρ(✵) = −❈♦✈(X(✵), Xkk(✵)) = ❱❛r(Xk(✵)).

❇② st❛t✐♦♥❛r✐t② ❈♦✈(X(✵), Xk(✵)) = ❈♦✈(Xk(✵), X✶✷(✵)) = ✵ ❛♥❞ E (h(X(✵))|X✶(✵)|) = E (h(X(✵))) E (|X✶(✵)|) =

• ✷λ✷

π E (h(X(✵))) ; α✵(h) = E (h(X(✵))X✶✶(✵)) = E (h(X(✵))E (X✶✶(✵)|X(✵))) = −λ✷E (h(X(✵))X(✵)) ; α✷(h) = −✷E

• h(X(✵))X✶✷(✵)X✶(✵)X✷(✵)

∇X(✵)✷ ✶∇X>✵

• =

−✷E (h(X(✵))X✶✷(✵)) E X✶(✵)X✷(✵) ∇X(✵)✷ ✶∇X>✵

• = ✵.

❍❡♥❝❡ TV ∗(X) =

• πλ✷

✷

❛♥❞ LTC ∗(X) = ✵✳

❞❡❢❛✉❧t

• ❛✉ss✐❛♥ ❝❛s❡

❋♦r X(✵) ∼ N(✵, ✶)✱ t❤✐s ②✐❡❧❞s t♦ ❢♦r ❛✳❡✳ u ∈ R C ∗

✵ (X, u) =

✶ (✷π)✸/✷ λ✷ u e− u✷

✷ ❛♥❞ C ∗

✶ (X, u) = ✶

✹ λ✶/✷

✷

e− u✷

✷ .

ρ(x) = e−κ✷x✷✱ ❢♦r κ = ✶✵✵/✷✶✵ ✐♥ ❛ ❞♦♠❛✐♥ ♦❢ s✐③❡ ✷✶✵ × ✷✶✵ ♣✐①❡❧s✳

❞❡❢❛✉❧t

❈♦♠♠❡♥ts

■❢ ♦♥❡ ❦♥♦✇s t❤❛t u → C ∗

✶ (X, u) ♦r u → C ∗ ✵ (X, u) ❛r❡ ❝♦♥t✐♥✉♦✉s

t❤❡♥ ❛✳❡✳ ✐s ❡♥♦✉❣❤ ✦ ■♥ ❇❡r③✐♥✱ ▲❛t♦✉r✱ ▲❡♦♥ ✭✷✵✶✼✮ ❣❡♥❡r❛❧ ❛ss✉♠♣t✐♦♥s t♦ ❡♥s✉r❡ t❤❛t u → C ∗

✶ (X, u) ✐s ❝♦♥t✐♥✉♦✉s ❀

❋♦r ✐s♦tr♦♣✐❝ st❛t✐♦♥❛r② C ✸ ●❛✉ss✐❛♥ ✜❡❧❞ t❤❡ ❢♦r♠✉❧❛s ❤♦❧❞ ❢♦r ❛❧❧ ❧❡✈❡❧ ✭✇❡❛❦❡st ❛ss✉♠♣t✐♦♥s ❝❢ ❆❞❧❡r✱ ❚❛②❧♦r ✭✷✵✵✼✮✮ ❀ ▼♦r❡♦✈❡r ❜② ❦✐♥❡♠❛t✐❝ ❢♦r♠✉❧❛ E[C✵(X, T, u)] = C ∗

✵ (X, u)|T| + ✶

π C ∗

✶ (X, u)H✶(∂T) + C ∗ ✷ (X, u),

E[C✶(X, T, u)] = C ∗

✶ (X, u)|T| + ✶

✷C ∗

✷ (X, u)H✶(∂T).

❉✉❡ t♦ ❆❞❧❡r ❛♥❞ ❚❛②❧♦r✱ ✉s✐♥❣ ●❛✉ss✐❛♥ ❦✐♥❡♠❛t✐❝ ❛♥❞ ❚✉❜❡ ❢♦r♠✉❧❛s✱ ❝♦♠♣✉t❛t✐♦♥s ❢♦r ✜❡❧❞s ♦❢ ●❛✉ss✐❛♥ t②♣❡ ✿ X = F(●) ✇❤❡r❡ F : Rk → R C ✷ ❛♥❞ ● = (G✶, . . . , Gk) ✇✐t❤ G✶, . . . , Gk ✐✐❞ C ✸ ❤♦♠♦❣❡♥❡♦✉s ●❛✉ss✐❛♥ r❢✳

❞❡❢❛✉❧t

❈❤✐✷ ❝❛s❡

❋♦r k ≥ ✶✱ Zk = G ✷

✶ + . . . + G ✷ k ❛♥❞ ♥♦r♠❛❧✐③❡❞ ✜❡❧❞

• Zk(t) :=

✶ √ ✷k (Zk(t) − k), t ∈ R✷. ❚❤❡♥✱ ❢♦r ❛❧❧ u ∈ R✱ C ∗

✵ (

Zk, u) = λ✷ π✷k/✷Γ(k/✷)

• k + u

√ ✷k (k−✷)/✷ u √ ✷k + ✶

• exp
• −k + u

√ ✷k ✷

• C ∗

✶ (

Zk, u) = √πλ✷ ✷(k+✶)/✷Γ(k/✷)

• k + u

√ ✷k (k−✶)/✷ exp

• −k + u

√ ✷k ✷

• ,

C ∗

✷ (

Zk, u) = P

• χ✷

k ≥ k + u

√ ✷k

• .

❍❡♥❝❡ TV ∗( Zk) =

• ✷πλ✷

k Γ((k+✶)/✷) Γ(k/✷)

❛♥❞ LTC ∗( Zk) =

• ✷

k λ✷✳

❞❡❢❛✉❧t

❈❤✐✷ ❝❛s❡

• Zk ❢♦r k = ✷ ❛♥❞ ❢♦r ✐✐❞ st❛♥❞❛r❞ G✶, . . . , Gk ✇✐t❤ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥

ρ(x) = e−κ✷x✷✱ ❢♦r κ = ✶✵✵/✷✶✵ ✐♥ ❛ ❞♦♠❛✐♥ ♦❢ s✐③❡ ✷✶✵ × ✷✶✵ ♣✐①❡❧s✳

❞❡❢❛✉❧t

❙t✉❞❡♥t ❝❛s❡

❋♦r k ≥ ✸✱ Tk = Gk+✶/

• Zk/k ❛♥❞ ♥♦r♠❛❧✐③❡❞ ✜❡❧❞
• Tk(t) :=
• (k − ✷)/kTk(t), t ∈ R✷.

❚❤❡♥✱ C ∗

✵ (

Tk, u) = λ✷(k − ✶) ✹π

✸ ✷

u √ k − ✷ Γ k−✶

✷

• Γ

k

✷

• ✶ +

u✷ k − ✷ ✶−k

✷

, C ∗

✶ (

Tk, u) = √λ✷ ✹

• ✶ +

u✷ k − ✷ ✶−k

✷

, C ∗

✷ (

Tk, u) = P(Student(k) ≥ u

• k/(k − ✷)).

❍❡♥❝❡ TV ∗( Tk) =

• (k − ✷)πλ✷

Γ((k−✷)/✷) ✷Γ((k−✶)/✷) ❛♥❞ LTC ∗(

Tk) = ✵✳

❞❡❢❛✉❧t

❙t✉❞❡♥t ❝❛s❡

• Tk ❢♦r k = ✹ ❛♥❞ ❛♥❞ ❢♦r ✐✐❞ st❛♥❞❛r❞ G✶, . . . , Gk+✶ ✇✐t❤ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥

ρ(x) = e−κ✷x✷✱ ❢♦r κ = ✶✵✵/✷✶✵ ✐♥ ❛ ❞♦♠❛✐♥ ♦❢ s✐③❡ ✷✶✵ × ✷✶✵ ♣✐①❡❧s✳

❞❡❢❛✉❧t

❙❤♦t ♥♦✐s❡ ✜❡❧❞s

❚❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ✐s ❣✐✈❡♥ ❜② E

• eitX(✵)

= exp

• λ
• Rk×R✷[ei[tgm(x)] − ✶]F(dm)dx
• .

❲❤❡♥ gm ✐s s♠♦♦t❤✱ ✇❡ ❤❛✈❡ ❛❧s♦ ❛❝❝❡ss t♦ ❥♦✐♥t ❧❛✇ ♦❢ (X(✵), ∇X(✵), D✷X(✵)) ✈✐❛ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ❛♥❞ s✐♠✐❧❛r ✐♥t❡❣r❛❧ ❡①♣r❡ss✐♦♥✳ ■♥ ♣❛rt✐❝✉❧❛r t❤❡ ❥♦✐♥t ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ♦❢ X ❛♥❞ ∂✶X ✐s ϕ(t, s) = E

• eitX(✵)+is∂✶X(✵)

= exp

• λ
• [eitgm(x)+is∂✶gm(x) − ✶] F(dm) dx

❞❡❢❛✉❧t

■s♦tr♦♣✐❝ s♠♦♦t❤ ❙❤♦t ♥♦✐s❡ ✜❡❧❞s

❚❤❡ ♠❛✐♥ ✐❞❡❛ ✐s t❤❡r❡❢♦r❡ t♦ t❛❦❡ ht(u) = eitu t♦ ❝♦♠♣✉t❡

• C ∗

j (t) =

• R eituC ∗

j (X, u)du✳ ❲❡ ♦❜t❛✐♥ ✐♥t❡❣r❛❧ ❢♦r♠✉❧❛s ✿

• C ∗

✶ (t) = ✶

✷ +∞

✵

ϕ(t, s) s S✵(t, s)ds.

• C ∗

✵ (t) = S✶(t)ϕ(t, ✵) +

+∞

✵

ϕ(t, s) s S✷(t, s)ds, ✇✐t❤ S✵(t) = −iλ

• R
• R✷ ∂✶gm(x)ei[tgm(x)+s∂✶gm(x)] dx F(dm)

S✶(t) = − λ ✷π

• R
• R✷ ∂✷

✶gm(x)eitgm(x) dx F(dm)

S✷(t, s) = λ ✷π

• R
• R✷[∂✷

✷gm(x) − ∂✷ ✶gm(x)]ei[tgm(x)+s∂✶gm(x)] dx F(dm)

❞❡❢❛✉❧t

❙❤♦t ♥♦✐s❡ ●❛✉ss✐❛♥ ❡①❛♠♣❧❡s

• 4
• 2

• 4
• 2

• 400
• 200

• 8
• 6
• 4
• 2

• 8
• 6
• 4
• 2

• 400
• 200

❲❡ ❝❤♦s❡ gm(x) = we− x/r✷

✷

✇✐t❤ R = ✶/a > ✵ ❛✳s✳ ❚♦♣ ✿ W ∼ E(µ)✱ ✇❡ ✜♥❞ ϕ(t) =

• µ

µ−it

✷πλ/a ❛♥❞ X(✵) ∼ γ(µ, ✷πλ/a) ❀ ❇♦tt♦♠ ✿ W ∼ L(µ)✱ ϕ(t) =

• µ✷

µ✷+t✷

πλ/a ❛♥❞ X(✵) ∼ GSL(µ, πλ/a)✳

❞❡❢❛✉❧t

❙❤♦t ♥♦✐s❡ ❞✐s❦ ❡①❛♠♣❧❡s

❈♦♥s✐❞❡r✐♥❣ gm(x) = ✶rD(x) ✇❡ ♥♦t❡ ¯ a = πE(R✷) ❛♥❞ ¯ p = ✷πE(R) ❛♥❞ ❣❡t X(✵) ∼ P(λ¯ a). ▼♦r❡♦✈❡r✱ ❢♦r u ∈ R+ \ Z+✱ ✐t ❤♦❧❞s t❤❛t C ∗

✵ (u)

= e−λ¯

a (λ¯

a)⌊u⌋ ⌊u⌋! λ

• ✶ − λ ¯

p✷ ✹π + ⌊u⌋ ¯ p✷ ✹π¯ a

• ,

C ∗

✶ (u)

= ✶ ✷e−λ¯

a (λ¯

a)⌊u⌋ ⌊u⌋! λ¯ p C ∗

✷ (u)

= e−λ¯

a k>u

(λ¯ a)k k! .

❞❡❢❛✉❧t

❙❤♦t ♥♦✐s❡ ❞✐s❦ ❝❛s❡

R = ✺✵ ♦r R = ✶✵✵ ❡❛❝❤ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶/✷ ❛♥❞ λ = ✺ × ✶✵−✹ ✐♥ ❛ ❞♦♠❛✐♥ ♦❢ s✐③❡ ✷✶✵ × ✷✶✵ ♣✐①❡❧s✳

❞❡❢❛✉❧t

❇✐♦❧♦❣✐❝❛❧ ♠♦✈✐❡s

❈❛❧❝✐✉♠ ✐♠❛❣✐♥❣ r❡❝♦r❞✐♥❣s ❢♦r ♠♦✉s❡ ♥❛s❛❧ ❡①♣❧❛♥ts ❈♦♥t❡①t ✿ st✉❞② ♦❢ ♣✉❧s❛t✐✈✐t② ❢♦r ●♦♥❛❞♦tr♦♣✐♥✲❘❡❧❡❛s✐♥❣ ❍♦r♠♦♥❡✲✶ ♥❡✉r♦♥s ❛♥❞ ❝❛❧❝✐✉♠ ❡✈❡♥t s②♥❝❤r♦♥✐③❛t✐♦♥ ❈♦❧❧❛❜♦r❛t✐♦♥ ✿ ❆✳ ❉✉✐tt♦③ ✭■◆❘❆ ◆♦✉③✐❧❧②✮ ❛♥❞ ❈✳ ●❡♦r❣❡❧✐♥ ✭■❉P ❚♦✉rs✮

❞❡❢❛✉❧t

❚❤❡ ❞❛t❛

D = ✼✵✵ ✐♠❛❣❡s ♦❢ s✐③❡ ✷✺✼ × ✸✷✺ ♣✐①❡❧s ✐♥ ❘●❇ ❝♦❞❡❞ ✐♥ ✽ ❜✐t ✿ ❘

• ❇

❞❡❢❛✉❧t

❙t❛t✐st✐❝s ♦❢ t❤❡ ❞❛t❛

❊✈♦❧✉t✐♦♥ ❛❝❝♦r❞✐♥❣ t♦ t✐♠❡ ♠❡❛♥ TV ∗ × |T| LTC ∗ × |T| C ∗

✵ /

• |C ∗

✵ |

C ∗

✶ /TV ∗

C ∗

✷

▼❡❛♥ ♦✈❡r t✐♠❡ ✴ ❛❝❝♦r❞✐♥❣ t♦ ✈❛❧✉❡s

❞❡❢❛✉❧t

❚❤❡ t✐♠❡ ❞✐✛❡r❡♥❝❡s

❈♦♥s✐❞❡r t❤❡ ❞✐✛❡r❡♥❝❡ ✈s t✐♠❡ ∆Xt = Xt+✶ − Xt ❘

• ❇

❞❡❢❛✉❧t

❙t❛t✐st✐❝s ♦❢ t✐♠❡ ❞✐✛❡r❡♥❝❡s

❊✈♦❧✉t✐♦♥ ❛❝❝♦r❞✐♥❣ t✐♠❡ ♠❡❛♥ TV ∗ × |T| LTC ∗ × |T| C ∗

✵ /

• |C ∗

✵ |

C ∗

✶ /TV ∗

C ∗

✷

▼❡❛♥ ♦✈❡r t✐♠❡ ✴ ❛❝❝♦r❞✐♥❣ t♦ ✈❛❧✉❡s

❞❡❢❛✉❧t

P❡rs♣❡❝t✐✈❡s

❊✛❡❝t ♦❢ ❞✐s❝r❡t✐③❛t✐♦♥ ♦❢ r❛♥❞♦♠ ✜❡❧❞s ❯s❡ ❢♦r st❛t✐st✐❝❛❧ ✐♥❢❡r❡♥❝❡ ❉❡t❡❝t ♠❛✐♥ t♦♣♦❧♦❣✐❝❛❧ ❝❤❛♥❣❡s ✐♥ ❛ s❡q✉❡♥❝❡ ♦❢ ✐♠❛❣❡s ❇❡②♦♥❞ ❡①♣❡❝t❛t✐♦♥ ✿ ❤✐❣❤❡r ♦r❞❡r ♠♦♠❡♥ts ✭✈❛r✐❛♥❝❡ ❛t ❧❡❛st✱ ❛♥❞ t❤❡♥ ❈▲❚✮ ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥

❞❡❢❛✉❧t

❘❡❢❡r❡♥❝❡s

❘✳ ❆❞❧❡r✱ ❏✳ ❚❛②❧♦r ✿ ❘❛♥❞♦♠ ✜❡❧❞s ❛♥❞ ●❡♦♠❡tr②✳ ❙♣r✐♥❣❡r✱ ◆❨ ✭✷✵✵✼✮✳ ❏✳▼✳ ❆③❛✐s✱ ▼✳ ❲s❝❤❡❜♦r ✿ ▲❡✈❡❧ s❡ts ❛♥❞ ❡①tr❡♠❛ ♦❢ r❛♥❞♦♠ ♣r♦❝❡ss❡s ❛♥❞ ✜❡❧❞s✳ ❲✐❧❡② ✭✷✵✵✾✮✳ ❈✳ ❇❡r③✐♥✱ ❆✳ ▲❛t♦✉r✱ ❏✳ ▲❡♦♥ ✿ ❑❛❝✲❘✐❝❡ ❢♦r♠✉❧❛s ❢♦r r❛♥❞♦♠ ✜❡❧❞s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ✿ r❛♥❞♦♠ ❣❡♦♠❡tr②✱ r♦♦ts ♦❢ r❛♥❞♦♠ ♣♦❧②♥♦♠✐❛❧s ❛♥❞ s♦♠❡ ❡♥❣✐♥❡❡r✐♥❣ ♣r♦❜❧❡♠s✳ ■♥st✐t✉t♦ ❱❡♥❡③♦❧❛♥♦ ❞❡ ■♥✈❡st✐❣❛❝✐♦♥❡s ❈✐❡♥tí✜❝❛s✱ ✭✷✵✶✼✮✳ ❍✳ ❇✐❡r♠é✱ ❆✳ ❉❡s♦❧♥❡✉① ✿ ❖♥ t❤❡ ♣❡r✐♠❡t❡r ♦❢ ❡①❝✉rs✐♦♥ s❡ts ♦❢ s❤♦t ♥♦✐s❡ r❛♥❞♦♠ ✜❡❧❞s✳ ❆♥♥❛❧s ♦❢ Pr♦❜❛❜✐❧✐t②✱ ✹✹✭✶✮✱ ✺✷✶✲✺✹✸✱ ✭✷✵✶✻✮✳ ❍✳ ❇✐❡r♠é✱ ❆✳ ❉❡s♦❧♥❡✉① ✿ ▼❡❛♥ ●❡♦♠❡tr② ♦❢ ✷❉ r❛♥❞♦♠ ✜❡❧❞s ✿ ❧❡✈❡❧ ♣❡r✐♠❡t❡r ❛♥❞ ❧❡✈❡❧ t♦t❛❧ ❝✉r✈❛t✉r❡ ✐♥t❡❣r❛❧s✳ r❡✈✐s❡❞ ✈❡rs✐♦♥✱ ✷✵✶✽✳ ❍✳ ❇✐❡r♠é✱ ❊✳ ❉✐ ❇❡r♥❛r❞✐♥♦✱ ❈✳ ❉✉✈❛❧✱ ❆✳ ❊str❛❞❡ ✿ ▲✐♣s❝❤✐t③ ❑✐❧❧✐♥❣ ❝✉r✈❛t✉r❡s ♦❢ ❡①❝✉rs✐♦♥ s❡ts ❢♦r ✷❉ r❛♥❞♦♠ ✜❡❧❞s t♦ ❛♣♣❡❛r ✐♥ ❊❏❙