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r r strs t ts Ptr r t r t r
P❛tr✐❝ ❇♦♥♥✐❡r ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❈❤♦♥❣ ▲✐✉ ❛♥❞ ❍❛r❛❧❞ ❖❜❡r❤❛✉s❡r ▼❛t❤❡♠❛t✐❝❛❧ ■♥st✐t✉t❡✱ ❯♥✐✈❡rs✐t② ♦❢ ❖①❢♦r❞ ❆❧❣❡❜r❛✱ ❈♦♠❜✐♥❛t♦r✐❝s ❛♥❞ P❡rs♣❡❝t✐✈❡s ✐♥ ▼❛t❤❡✲ ♠❛t✐❝❛❧ s❝✐❡♥❝❡s✱ ▼❛② ✷✵✷✵
❖✈❡r✈✐❡✇ ❆❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s ♦♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❚❤❡ s✐❣♥❛t✉r❡ ♦❢ ❛ r❛♥❦ r ♣r♦❝❡ss ▼❡tr✐③✐♥❣ t❤❡ r❛♥❦ r ❛❞❛♣t❡❞ t♦♣♦❧♦❣② ❖✉t❧♦♦❦✫❙✉♠♠❛r②
❆ s❡q✉❡♥❝❡ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s X n ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ V ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ X ✐❢
❲❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s✿ ❙t♦❝❤❛st✐❝ ♣r♦❝❡ss (Xt)t∈{✵,...,T} ♦♥ V ⇐ ⇒ ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ V {✵,...,T} ■❣♥♦r❡s t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ✐♥❢♦r♠❛t✐♦♥ ❛ss♦❝✐❛t❡❞ t♦ ❛ ♣r♦❝❡ss✳
✶✴✷✺
❊①❛♠♣❧❡ ✭✶✮
p = ✵.✺ p = ✵.✺ ε p = ✶ p = ✶ p = ✶ p = ✵.✺ p = ✵.✺
❊①❛♠♣❧❡ ✭✷✮
❚❤❡ ♠❛♣ X → inf{E[Lτ] : τ ≤ T} ✐s ♥♦t ❝♦♥t✐♥✉♦✉s ✐♥ ✇❡❛❦ t♦♣♦❧♦❣② ❢♦r ❡✈❡r② ❛❞❛♣t❡❞ ❢✉♥❝t✐♦♥❛❧ (Lt)t∈{✵,...,T} t❤❛t ❞❡♣❡♥❞s ❝♦♥t✐♥✉♦✉s❧② ♦♥ t❤❡ tr❛❥❡❝t♦r② ♦❢ X ❬✶✱ ❙❡❝t✐♦♥ ✼❪✳
✷✴✷✺
❯s❡❢✉❧ t♦ ❤❛✈❡ ❛ str♦♥❣❡r t♦♣♦❧♦❣② ✐♥ ❛♣♣❧✐❝❛t✐♦♥s s✉❝❤ ❛s ◮ ♦♣t✐♠❛❧ st♦♣♣✐♥❣✱ ◮ q✉❡✉✐♥❣ t❤❡♦r②✱ ◮ st♦❝❤❛st✐❝ ♣r♦❣r❛♠♠✐♥❣ ✇❤❡r❡ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ✐♥❢♦r♠❛t✐♦♥ ✐s ✐♠♣♦rt❛♥t✳
✸✴✷✺
❊①t❡♥❞❡❞ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ = ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♣r❡❞✐❝t✐♦♥ ♣r♦❝❡ss ❬✷❪ ˆ Xt = P(X ∈ ·|Ft) ∈ M(I → M(I → V )) ❉❡s❝r✐❜❡s ❤♦✇ ✇❡❧❧ ♦♥❡ ❝❛♥ ♣r❡❞✐❝t X ❛t t✐♠❡ t✳
❊①❛♠♣❧❡
■❢ X = (X✶, X✷, X✸) ∈ V ✸ ˆ X✶ = P(X✶, X✷, X✸ ∈ ·|X✶), ˆ X✷ = P(X✶, X✷, X✸ ∈ ·|X✶, X✷), ˆ X✸ = P(X✶, X✷, X✸ ∈ ·|X✶, X✷, X✸) ❖t❤❡r r❡s❡❛r❝❤❡rs ❤❛✈❡ ♣r♦♣♦s❡❞ s✐♠✐❧❛r r❡♠❡❞✐❡s✳ ❚❤❡② ❞♦ ♥♦t ❝❤❛r❛❝t❡r✐③❡ t❤❡ ❢✉❧❧ str✉❝t✉r❡ ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss✳
✹✴✷✺
❊①❛♠♣❧❡
❚❤❡r❡ ❡①✐sts t✇♦ ♣r♦❝❡ss❡s X ε, Y ε ❞❡♣❡♥❞✐♥❣ ♦♥ ε s✉❝❤ t❤❛t ✇❤❡♥ ε → ✵✿ X ε − Y ε → ✵, ˆ X ε − ˆ Y ε → ✵, ❜✉t E[E[X ε
✹|F✸]✷|F✶] − E[E[Y ε ✹ |F✸]✷|F✶] → ✵.
❚❛❦❡ ❛✇❛②✿ ♠♦r❡ str✉❝t✉r❡ ❝❛♥ ❜❡ ❝❛♣t✉r❡❞ ❜② ✐t❡r❛t✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥✳
✺✴✷✺
t = ✵ t = ✶ t = ✷ t = ✸ t = ✹ ✵ ε −ε
✸ ✷ε
− ✸
✷ε ✶ ✷ε
− ✶
✷ε ✺ ✸ε
− ✺
✸ε ✹ ✸ε
− ✹
✸ε
ε −ε
✷ ✸ε
− ✷
✸ε
✶ ✶ ✷ ✷ ✶ ✶ ✷ ✷ ✷ ✶ ✷ ✶ ✷ ✶ ✷ ✶
✭❛✮ X
t = ✵ t = ✶ t = ✷ t = ✸ t = ✹ ✵ ε −ε
✸ ✷ε
− ✸
✷ε ✶ ✷ε
− ✶
✷ε ✺ ✸ε
− ✺
✸ε ✹ ✸ε
− ✹
✸ε
ε −ε
✷ ✸ε
− ✷
✸ε
✶ ✶ ✷ ✷ ✶ ✷ ✶ ✷ ✷ ✶ ✷ ✶ ✷ ✷ ✶ ✶
✭❜✮ Y
✻✴✷✺
t = ✵ t = ✶ t = ✷ t = ✸ E[f (X✹)] E[f (X✹)] E[f (X✹)] E[f (X✹)] E[f (X✹)] E[f (X✹)] E[f (X✹)] f (✶) f (✷) f (✶) f (✷) E[f (X✹)] E[f (X✹)] E[f (X✹)] E[f (X✹)]
✭❛✮ E[f (X✹)|F]
t = ✵ t = ✶ t = ✷ t = ✸ E[f (Y✹)] E[f (Y✹)] E[f (Y✹)] E[f (Y✹)] E[f (Y✹)] E[f (Y✹)] E[f (Y✹)] f (✶) f (✷) E[f (Y✹)] E[f (Y✹)] f (✶) f (✷) E[f (Y✹)] E[f (Y✹)]
✭❜✮ E[f (Y✹)|F]
✼✴✷✺
❍♦♦✈❡r✲❑❡✐s❧❡r ❬✸❪ ✐♥tr♦❞✉❝❡❞ t❤❡ ❝❧❛ss ♦❢ ❛❞❛♣t❡❞ ❢✉♥❝t✐♦♥❛❧s AF
❉❡✜♥✐t✐♦♥
✶✳ f ∈ Cb(V n, R)✱ t❤❡♥ X → f (X(t✶), . . . , X(tn)) ∈ AF ✷✳ f✶, . . . , fn ∈ AF✱ f ∈ Cb(Rn, R)✱ t❤❡♥ X → f (f✶(X), . . . , fn(X)) ∈ AF ✸✳ f ∈ AF✱ ✵ ≤ t ≤ T✱ t❤❡♥ X → E[f (X)|Ft] ∈ AF.
✽✴✷✺
❚❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡s t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ✐s t❛❦❡♥ ✐s ❝❛❧❧❡❞ t❤❡ r❛♥❦ ♦❢ f ✳ ◮ f (X) =
✷ ❤❛s r❛♥❦ ✶✳ ◮ f (X) = E
t✶|Ft✷]
❉❡✜♥✐t✐♦♥
X ❛♥❞ Y ❤❛✈❡ t❤❡ s❛♠❡ ❛❞❛♣t❡❞ ❞✐str✐❜✉t✐♦♥ ✉♣ t♦ r❛♥❦ r ✐❢✿ Ef (Y ) = Ef (X) ❢♦r ❛♥② f ∈ AF ♦❢ r❛♥❦ ≤ r. ❚❤❡ ❛❞❛♣t❡❞ t♦♣♦❧♦❣② ♦❢ r❛♥❦ r ✐s ❞❡✜♥❡❞ ❜② t❤❡ r❡q✉✐r❡♠❡♥t X n →r X ✐❢ Ef (X n) → Ef (X) ❢♦r ❛♥② f ∈ AF ♦❢ r❛♥❦ ≤ r. ◮ r❛♥❦ ✵ ✐s ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ◮ r❛♥❦ ✶ ✐s ❡①t❡♥❞❡❞ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡✳
✾✴✷✺
❲✐❧❧ ✐♥tr♦❞✉❝❡✿ ◮ ❘❛♥❦ r t❡♥s♦r ❛❧❣❡❜r❛s✳ ◮ ❘❛♥❦ r ♣❛t❤s ❛♥❞ r❛♥❦ r s✐❣♥❛t✉r❡s✳ ◮ ❘❛♥❦ r st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❛♥❞ r❛♥❦ r ❡①♣❡❝t❡❞ s✐❣♥❛t✉r❡s✳
✶✵✴✷✺
❉❡✜♥✐t✐♦♥ ✭❘❛♥❦ r t❡♥s♦r ❛❧❣❡❜r❛✮
t✵(V ) = V , tr(V ) =
⊗m. ❋♦r ❡✈❡r② r ≥ ✶✱ (tr(V ), ⊗(r)) ✐s ❛ ♠✉❧t✐✲❣r❛❞❡❞ ❛❧❣❡❜r❛ ♦✈❡r V ✳ ◮ t✶(V ) =
n≥✵
V ⊗(✶)n ◮ t✷(V ) =
V ⊗(✶)n✶ ⊗(✷) · · · ⊗(✷) V ⊗(✶)nk ◮ t✸(V ) =
✶,...,n✶ k✶≥✵
✳ ✳ ✳
nk✷
✶ ,...,nk✷ k✶≥✵
✶ ⊗(✷)· · ·⊗(✷)V ⊗(✶)n✶ k✶
⊗(✸) · · · ⊗(✸)
✶ ⊗(✷) · · · ⊗(✷) V ⊗(✶)nk✷ k✶ ✶✶✴✷✺
◮ Seq(S) := ✜♥✐t❡ s❡q✉❡♥❝❡s ♦♥ S✳ ◮ ❘❡❝✉rs✐✈❡❧②✿ Seqr(S) = Seq(Seqr−✶(S))✱ tr(V )❦ : = tr−✶(V )❦✶ ⊗(r) · · · ⊗(r) tr−✶(V )❦l, tr(V ) =
tr(V )❦
✶✷✴✷✺
❊①❛♠♣❧❡
▼♦♠❡♥ts ♦❢ ❛ ❜♦✉♥❞❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ X✿ µX : t✶(V ) → R, µX(ei✶ · · · eik) = E(Xi✶ · · · Xik) ■♥❞✉❝❡s t❤❡ ❛❧❣❡❜r❛ ❤♦♠♦♠♦r♣❤✐s♠ µ⋆
X : t✷(V ) → R✳ ❚❤❡
❝✉♠✉❧❛♥ts ♦❢ X ❛r❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ♦❢ µ⋆
X✳
κ(X✶, X✷, X✸) = E(X✶X✷X✸) − E(X✶)E(X✷X✸) − E(X✷)E(X✶X✸) − E(X✸)E(X✶X✷) + ✷E(X✶)E(X✷)E(X✸) = µ⋆
X(e✶e✷e✸) − µ⋆ X(e✶ ⊗(✷) e✷e✸) − µ⋆ X(e✷ ⊗(✷) e✶e✸)
− µ⋆
X(e✸ ⊗(✷) e✶e✷) + ✷µ⋆ X(e✶ ⊗(✷) e✷ ⊗(✷) e✸)
✶✸✴✷✺
· ♥♦r♠ ♦♥ V ⇒ ♥♦r♠ ♦♥ t✶(V )✳ ❙❡t ❚✶(V ) := t✶(V )✳ ❘❡❝❛❧❧ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ exp : V → ❚✶(V ), v →
✶ m!v⊗m
❉❡✜♥✐t✐♦♥ ✭❙✐❣♥❛t✉r❡ ♦❢ ❛ ♣❛t❤✮
S :
x → exp
s∈{✶,...,t}
exp
❈♦♥tr❛❝ts ❛ ♣❛t❤ ✐♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✐♥t♦ ❛♥ ❡❧❡♠❡♥t ♦❢ ❛ ✭❞✐✛❡r❡♥t✮ ❇❛♥❛❝❤ s♣❛❝❡✳
❘❡♠❛r❦
❖✈❡r❧❛♣s ✇✐t❤ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❈❤❡♥✬s ✐t❡r❛t❡❞ ✐♥t❡❣r❛❧s ♦❢ ❛ ❧✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥ ♦❢ t❤❡ ♣♦✐♥ts (✵, x✵, . . . , xt)✳
✶✹✴✷✺
❉❡✜♥✐t✐♦♥ ✭❘❛♥❦ r ♣❛t❤✮
V✵ := V , Vr :=
Vr = I → (I → · · · (I → (I → V )) · · · )
❊①❛♠♣❧❡
x ∈ V✶ = (I → V )✱ x(t) ∈ V . x ∈ V✷ = (I → (I → V ))✱ x(t) : I → V , s → x(t)(s).
✶✺✴✷✺
◮ ■❢ x ∈ V✶✱ t❤❡♥ S(x) ∈ ❚✶(V )✳ ◮ ■❢ x ∈ V✷✱ t❤❡♥ x(t) : I → V ✱ s♦ S(x(t)) ∈ ❚✶(V )✱ ❤❡♥❝❡ S✶(x) : I → ❚✶(V ), t → S✶(s → x(t)(s)) s♦ S✷(x) = S (S✶(x)) ∈ ❚✷(V ). ❚❛❦❡ ❛✇❛②✿ ❆♣♣❧②✐♥❣ S r❡♠♦✈❡s ♦♥❡ r❛♥❦ ♦❢ t❤❡ ♣❛t❤ x ∈ Vr ❛♥❞ ♣✉s❤❡s ✐t ♦♥❡ r❛♥❦ ✉♣ ✐♥ t❤❡ t❛r❣❡t s♣❛❝❡ ❚r(V )✳
✶✻✴✷✺
Vr =
❚r(V ) Sr−✶ S Sr
❉❡✜♥✐t✐♦♥ ✭❘❛♥❦ r s✐❣♥❛t✉r❡✮
Sr : Vr → ❚r(V ) ✐s ❞❡✜♥❡❞ ✐♥❞✉❝t✐✈❡❧②✿ S✶ = S, Sr(x) = S(Sr−✶(x)) = S(t → Sr−✶(x(t)))
✶✼✴✷✺
◆♦t❛t✐♦♥✿ M(U) = s♣❛❝❡ ♦❢ ❇♦r❡❧ ♠❡❛s✉r❡s ♦♥ ❛ s♣❛❝❡ U✳
❉❡✜♥✐t✐♦♥ ✭r❛♥❦ r st♦❝❤❛st✐❝ ♣r♦❝❡ss✮
M✵ := V , Mr := M
❜❡ ❉✐r❛❝ ♠❡❛s✉r❡s✳ ❯♥r❛✈❡❧❧❡❞ Mr = M
◮ M✷ = st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ♦♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ♦♥ ❱
✶✽✴✷✺
◮ X ∈ M✶ = M(I → V )✱ t❤❡♥ E[S(X)] ∈ ❚✶(V ) ◮ X ∈ M✷ = M(I → M✶)✱ t❤❡♥ E S(X) ∈ M(I → ❚✶(V )), (t, ω) → EZ∼Xt(ω)[S(Z)] s♦ E S(E S(X)) = E[S((t, ω) → EZ∼Xt(ω)[S(Z)])] ∈ ❚✷(V ) ❚❛❦❡ ❛✇❛②✿ ❆♣♣❧②✐♥❣ E S r❡♠♦✈❡s ♦♥❡ r❛♥❦ ♦❢ t❤❡ ♣r♦❝❡ss X ∈ Mr ❛♥❞ ♣✉s❤❡s ✐t ♦♥❡ r❛♥❦ ✉♣ ✐♥ t❤❡ t❛r❣❡t s♣❛❝❡ ❚r(V )✳
❉❡✜♥✐t✐♦♥ ✭❘❛♥❦ r ❡①♣❡❝t❡❞ s✐❣♥❛t✉r❡✮
¯ Sr : Mr(V ) → ❚r(V ), X → E S(¯ Sr−✶(X))
✶✾✴✷✺
❘❡❝❛❧❧✿ ♣r❡❞✐❝t✐♦♥ ♣r♦❝❡ss ♦❢ X✿ ˆ Xt = P(X ∈ ·|Ft) ∈ M✷ ❉❡✜♥❡✿ ˆ X r
t = P( ˆ
X r−✶ ∈ ·|Ft) ∈ Mr+✶
❊①❛♠♣❧❡
ˆ X ✵
t (ω) = Xt(ω) ∈ M✵(V ) = V ,
ˆ X ✶
t (ω) ∈ M✶(V ) = M(I → V ),
ˆ X ✷
t (ω) ∈ M✷(V ) = M(I → M(I → V )).
ˆ X r st♦❝❤❛st✐❝ ♣r♦❝❡ss ✇✐t❤ st❛t❡ s♣❛❝❡ Mr ⇒ ¯ Sr( ˆ X r) st♦❝❤❛st✐❝ ♣r♦❝❡ss ✇✐t❤ st❛t❡ s♣❛❝❡ ❚r(V )✳
✷✵✴✷✺
Pr♦♣♦s✐t✐♦♥
¯ Sr( ˆ X r
t ) s❛t✐s✜❡s t❤❡ r❡❝✉rs✐♦♥
¯ Sr( ˆ X r
t ) = E[S(¯
Sr−✶( ˆ X r−✶))|Ft], ❛♥❞ E¯ Sr( ˆ X r
t ) = ¯
Sr(▲❛✇( ˆ X r))
❊①❛♠♣❧❡
¯ S✶( ˆ X ✶
t ) =
¯ S✷( ˆ X ✷
t ) = E[S(s → E[S(X)|Fs])|Ft]
✷✶✴✷✺
❚❤❡♦r❡♠
▲❡t X, Y ❜❡ t✇♦ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s✳ ❋♦r ❡✈❡r② r ≥ ✵ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ✶✳ Ef (X) = Ef (Y ) ❢♦r ❛♥② f ∈ AF ♦❢ r❛♥❦ ≤ r✱ ✷✳ ▲❛✇( ˆ X r) = ▲❛✇( ˆ Y r)✱ ✸✳ ▲❛✇( ˆ X ✵, . . . , ˆ X r) = ▲❛✇( ˆ Y ✵, . . . , ˆ Y r)✱ ▼♦r❡♦✈❡r✱ ✐❢ X, Y ❤❛✈❡ ❛ ❝♦♠♣❛❝t st❛t❡ s♣❛❝❡✱ t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛❧s♦ ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❛❜♦✈❡✿ ✹✳ E¯ Sr+✶( ˆ X r+✶
t
) = E¯ Sr+✶( ˆ Y r+✶
t
)✳
✷✷✴✷✺
Pr♦♣♦s✐t✐♦♥
▲❡t (X k)k≥✵, X ❜❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ✇✐t❤ ❝♦♠♣❛❝t st❛t❡ s♣❛❝❡✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t ✶✳ E¯ Sr( ˆ X k,r
t
) − E¯ Sr( ˆ X r
t ) → ✵
✷✳ X k ❝♦♥✈❡r❣❡s t♦ X ✐♥ t❤❡ ❛❞❛♣t❡❞ t♦♣♦❧♦❣② ♦❢ r❛♥❦ r
✷✸✴✷✺
❙✉♠♠❛r②✿ ◮ ■♥tr♦❞✉❝❡❞ ❤✐❣❤❡r ♦r❞❡r ❛❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s✴♣r❡❞✐❝t✐♦♥ ♣r♦❝❡ss❡s✳ ◮ ■♥tr♦❞✉❝❡❞ ❤✐❣❤❡r r❛♥❦ ♣❛t❤s ✴ s✐❣♥❛t✉r❡s ❛♥❞ ❤✐❣❤❡r ♦r❞❡r st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ✴ ❡①♣❡❝t❡❞ s✐❣♥❛t✉r❡s✳ ◮ ❯s❡ t❤❡ ❛❜♦✈❡ t♦ ♣r♦❞✉❝❡ ❢❡❛t✉r❡ ♠❛♣s X → ¯ Sr( ˆ X r) t❤❛t ♠❡tr✐③❡ t❤❡ ❛❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s ♦❢ ❛♥② r❛♥❦
✷✹✴✷✺
❖✉t❧♦♦❦✿ ✶✳ ❈♦♥t✐♥✉♦✉s t✐♠❡✱ s✐❣♥❛t✉r❡ ♦❢ t❤❡ ♣r❡❞✐❝t✐♦♥ ♣r♦❝❡ss❄ ✷✳ ◆♦♥✲❝♦♠♣❛❝t st❛t❡ s♣❛❝❡ ✸✳ ❊✣❝✐❡♥t❧② ❡st✐♠❛t✐♥❣ ¯ Sr( ˆ X r) ✹✳ ▼♦r❡ str✉❝t✉r❡✱ ✐t❡r❛t❡❞ ❝♦♥str✉❝t✐♦♥s✿ ❙✐❣♥❛t✉r❡ ♦❢ ♣❛t❤ ♦❢ ♣❛t❤ → ♣❛t❤ ♦❢ s✐❣♥❛t✉r❡✱ ❊①♣❡❝t❡❞ s✐❣♥❛t✉r❡ ♦❢ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✈❛❧✉❡❞ st♦❝❤❛st✐❝ ♣r♦❝❡ss → st♦❝❤❛st✐❝ ♣r♦❝❡ss ♦♥ ❡①♣❡❝t❡❞ s✐❣♥❛t✉r❡s✳
✷✺✴✷✺
❇✱ ❈❤♦♥❣ ▲✐✉✱ ❛♥❞ ❍❛r❛❧❞ ❖❜❡r❤❛✉s❡r✱ ✧❆❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s ❛♥❞ ❤✐❣❤❡r r❛♥❦ s✐❣♥❛t✉r❡s✧✱ ❝♦♠✐♥❣ s♦♦♥ t♦ ❛♥ ❛r❳✐✈ ♥❡❛r ②♦✉
❏✳ ❇❛❝❦❤♦✛✲❱❡r❛❣✉❛s✱ ❉✳ ❇❛rt❧✱ ▼✳ ❇❡✐❣❧❜ö❝❦✱ ❛♥❞ ▼✳ ❊❞❡r✱ ✏❆❧❧ ❛❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s ❛r❡ ❡q✉❛❧✱✑ ❛r❳✐✈ ♣r❡♣r✐♥t ❛r❳✐✈✿✶✾✵✺✳✵✵✸✻✽✱ ✷✵✶✾✳ ❉✳ ❏✳ ❆❧❞♦✉s✱ ✏❲❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ❛♥❞ ❣❡♥❡r❛❧ t❤❡♦r② ♦❢ ♣r♦❝❡ss❡s✱✑ ❯♥♣✉❜❧✐s❤❡❞ ❞r❛❢t ♦❢ ♠♦♥♦❣r❛♣❤✱ ✶✾✽✶✳ ❉✳ ❍♦♦✈❡r ❛♥❞ ❏✳ ❑❡✐s❧❡r✱ ✏❆❞❛♣t❡❞ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s✱✑ ❚r❛♥s❛❝t✐♦♥s ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✶✾✽✹✳