Xấp xỉ chuẩn bằng phương pháp stein cho cặp hoán đổi được và một số ứng dụng

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖

❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍

◆●❯❨➍◆ ❈❍➓ ❉Ơ◆●

❳❻P ❳➓ ❈❍❯❽◆ ❇➀◆● P❍×❒◆● P❍⑩P ❙❚❊■◆
❈❍❖ ❈➄P ❍❖⑩◆ ✣✃■ ✣×Đ❈ ❱⑨ ▼❐❚ ❙➮ Ù◆● ❉Ư◆●

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

◆❣❤➺ ❆♥ ✲ ✷✵✶✾


❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖
❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍

◆●❯❨➍◆ ❈❍➓ ❉Ơ◆●

❳❻P ❳➓ ❈❍❯❽◆ ❇➀◆● P❍×❒◆● P❍⑩P ❙❚❊■◆
❈❍❖ ❈➄P ❍❖⑩◆ ✣✃■ ìẹ ệ


ị ❙❯❻❚ ❱⑨ ❚❍➮◆● ❑➊ ❚❖⑩◆ ❍➴❈
▼➣ sè✿ ✽✹✻✵✶✵✻
▲❯❾◆ ❱❿◆ ❚❍❸❈
ở ữợ ồ
P ❚❤➔♥❤

◆❣❤➺ ❆♥ ✲ ✷✵✶✾

ị
ì ề

R

t ủ sè t❤ü❝

R+

t➟♣ ❤đ♣ ❝→❝ sè t❤ü❝ ❦❤ỉ♥❣ ➙♠

B(X )

σ ✲ ✤↕✐ sè ❇♦r❡❧ ❝õ❛ X

❧♦❣x

❧♦❣❛r✐t ❝ì sè tü ♥❤✐➯♥ ❝õ❛ số tỹ ữỡ x

exp(x)

số ụ ợ ỡ số e sè ♠ô ❧➔ x

EX

❦➻ ✈å♥❣ ❝õ❛ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ X
❝❤✉➞♥ ❝õ❛ ❤➔♠ sè

.
V ar(X)

♣❤÷ì♥❣ s❛✐ ❝õ❛ X

I(A)

❤➔♠ ❝❤➾ t✐➯✉ ừ t ủ A



t tú ự

C

ỵ ởt ❤➡♥❣ sè ❞÷ì♥❣ ✈➔ ❝â t❤➸ ❦❤ỉ♥❣ ❣✐è♥❣
♥❤❛✉ ð ♠é✐ t
d

X=Y

ỵ X, Y ❝ò♥❣ ♣❤➙♥ ♣❤è✐




ử ử
ữỡ Pữỡ t ố ợ ✤ê✐ ✤÷đ❝

✺

✶✳✶

✣➦❝ tr÷♥❣ ❙t❡✐♥ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❙t❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

✶✳✷

P❤÷ì♥❣ ♣❤→♣ ❙t❡✐♥ ❝❤♦ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

ữỡ số tr ố ợ ổ N tr ợ
trữớ tr



ổ N tr ợ trữớ tr ởt số t q q✉❛♥ ✳ ✷✵

✷✳✷

❙❛✐ sè tr♦♥❣ ①➜♣ ①➾ ❝❤✉➞♥ ✤è✐ ✈ỵ✐ ổ N tr ợ trữớ tr
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸




ớ õ
ỵ tt st ỷ t ✷✵ ✤➣ ❝â ♥❤ú♥❣ t❤➔♥❤ tü✉ ✈÷đt ❜➟❝ tr♦♥❣
✈✐➺❝ ❝❤ù♥❣ ỵ ợ ờ ữ t số ợ t rt
ỵ ợ tr t ❝❤♦ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✳✳✳ P❤÷ì♥❣ ♣❤→♣ ❝ê
ự ỵ ợ tr t ỹ ✈➔♦ ❤➔♠ ✤➦❝ tr÷♥❣✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣

❦❤ỉ♥❣ ✤ë❝ ❧➟♣ t❤➻ ♣❤÷ì♥❣ ♣❤→♣ ❤➔♠ ✤➦❝ tr÷♥❣ r➜t ❦❤â →♣ ❞ư♥❣ ✈➔ t❤÷í♥❣ ❦❤ỉ♥❣
t➻♠ ✤÷đ❝ tè❝ ✤ë ❤ë✐ tư✳
◆➠♠ ✶✾✼✷✱ ❈❤❛r❧❡s ❙t❡✐♥ ợ t ởt ữỡ ợ ồ
ữỡ t ự ỵ ợ ❤↕♥ tr✉♥❣ t➙♠ ❝❤♦ ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉
♥❤✐➯♥ m✲♣❤ö t❤✉ë❝✳ ▼ư❝ ✤➼❝❤ ❜❛♥ ✤➛✉ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❙t❡✐♥ ❧➔ ①➜♣ ①➾ ♠ët t❤è♥❣
❦➯ ♥➔♦ ✤â ♠➔ t❛ ✤❛♥❣ q✉❛♥ t➙♠ ợ ố s số Pữỡ
ữợ ữủ tữớ ừ s số ①➜♣ ①➾ ❦❤✐ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
t❤ä❛ ♠➣♥ ♥❤✐➲✉ ❝➜✉ tró❝ ♣❤ư t❤✉ë❝ ❦❤→❝ ♥❤❛✉✳ ◆❤í ♥❤ú♥❣ ÷✉ ✤✐➸♠ ♥➔② ♠➔ ♥â ♥❣➔②
❝➔♥❣ ✤÷đ❝ ù♥❣ ❞ư♥❣ ♥❤✐➲✉ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉✳ ❚ø ✈✐➺❝ t➻♠ ❤✐➸✉ ❝❤õ ✤➲
♥➔② ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➔✐ ❧✐➺✉ ❧✐➯♥ q✉❛♥✱ ❝❤ó♥❣ tỉ✐ ❧ü❛ ❝❤å♥ ✤➲ t➔✐ ❝❤♦ ❧✉➟♥ ✈➠♥ ❝❛♦
❤å❝ ❝õ❛ ♠➻♥❤ ❧➔✿

✏❳➜♣ ①➾ ❝❤✉➞♥ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❙t❡✐♥ ❝❤♦ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝ ✈➔

♠ët sè ù♥❣ ❞ư♥❣✳✑

▲✉➟♥ ✈➠♥ ✤÷đ❝ t❤ü❝ ❤✐➺♥ t↕✐ trữớ ồ ữợ sỹ ữợ ừ t
▲➯ ❱➠♥ ❚❤➔♥❤✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ❝õ❛ ♠➻♥❤ ✤➳♥ t❤➛②✱




ữớ trỹ t t t ữợ ú ✤ï t→❝ ❣✐↔ r➜t ♥❤✐➲✉ tr♦♥❣ q✉→ tr➻♥❤
❤å❝ t➟♣ ✈➔ ự ỗ tớ t ụ ỷ ớ ❝↔♠ ì♥ ✤➳♥ ❝→❝ t❤➛② ❝ỉ
❣✐→♦ ð ❱✐➺♥ s÷ ♣❤↕♠ tü ♥❤✐➯♥ ✈➔ ♣❤á♥❣ s❛✉ ✤↕✐ ❤å❝ tr÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ ❣✐↔♥❣
❞↕② ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳
❚→❝ ❣✐↔ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧✱ ✤➦❝ ❜✐➺t tr ợ ồ ỵ
tt s✉➜t ✈➔ t❤è♥❣ ❦➯ t♦→♥ ❤å❝ ✤➣ ❣✐ó♣ ✤ï✱ ✤ë♥❣ ỗ ũ t
tr sốt q tr ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳
❉♦ tr➻♥❤ ✤ë ✈➔ t❤í✐ ❣✐❛♥ ❤↕♥ ❝❤➳✱ ♠➦❝ ❞ị ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣✱ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥

❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚→❝ ❣✐↔ rt ữủ ỳ ỵ
õ õ ừ t❤➛②✱ ❝ỉ ❣✐→♦ ✈➔ ❜↕♥ ❜➧ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❚→❝ ❣✐↔
①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦




ữỡ

Pữỡ t ố ợ ờ
ữủ
ỵ ợ tr t ởt tr ỳ ỵ q trồ ừ ỵ tt
st ởt {Xn }n1 ỵ ợ tr t➙♠ ❦❤➥♥❣
✤à♥❤ r➡♥❣ ✈ỵ✐ ♠ët sè ✤✐➲✉ ❦✐➺♥ ♥➔♦ ✤â✱ t❛ ❝â

lim P

n→∞

tr♦♥❣ ✤â Sn =

n

Sn − ESn
√
≤ x = Φ(x) ✈ỵ✐ ♠å✐ x ∈ R,
V arSn

Xk , n ≥ 1

k=1

tố ở ở tử ừ ỵ ợ tr t ữớ t ũ ởt
số ữ r ssrst ợ
X Y trữợ õ

dW = sup |Eh(X) − Eh(Y )|
h ≤1

✈➔

dK = sup|P (X ≤ x) − P (Y ≤ x)|
x∈R

❧➛♥ ❧÷đt ✤÷đ❝ ❣å✐ ❧➔ ❦❤♦↔♥❣ ❝→❝❤ ❲❛ss❡rst❡✐♥ ✈➔ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❧♠♦❣♦r♦✈ ❣✐ú❛ ❤❛✐


✻

❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X, Y ✳

✶✳✶ ✣➦❝ tr÷♥❣ ❙t❡✐♥ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❙t❡✐♥
❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ s➩ tr➻♥❤ ❜➔② ✈➲ ✤➦❝ tr÷♥❣ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❙t❡✐♥ ❝ơ♥❣
♥❤÷ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❙t❡✐♥✳ ❇ê ✤➲ s❛✉ ✤➙② ♥➯✉ ❧➯♥ ✤➦❝
tr÷♥❣ ❝ì ❜↔♥ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝â ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ t➢❝✳

✶✳✶✳✶ ❇ê ✤➲ ✭✣➦❝ tr÷♥❣ ❙t❡✐♥✮✳ ◆➳✉ Z ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝â ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ t➢❝
N (0, 1)

t❤➻

✭✶✳✶✮

Ef (Z) = EZf (Z),
✈ỵ✐ ♠å✐ ❤➔♠ ❧✐➯♥ tử tt ố

ữủ
ợ

E|f (Z)| <

ự


t

f :RR

tọ ♠➣♥

E|f (Z)| < ∞.

✤ó♥❣ ✈ỵ✐ ♠å✐ ❤➔♠ ❧✐➯♥ tư❝✱ ❜à ❝❤➦♥✱ ❦❤↔ ✈✐ tr➯♥ tø♥❣ ❦❤♦↔♥❣

Z ∼ N (0, 1).

❈❤✐➲✉ t❤✉➟♥✿ ●✐↔ sû Z ∼ N (0, 1).

❱ỵ✐ ❤➔♠ f : R −→ R ❧✐➯♥ tö❝ t✉②➺t ✤è✐ t❤ä❛ ♠➣♥ E|f (Z)| < ∞✱ t❛ ❝â
∞

Ef (Z) =
−∞

1 − x2
f (x). √ e 2 dx
2π
 0

∞



x2
x2
1
= √  f (x).e− 2 dx + f (x).e− 2 dx
2π
−∞
0
 0

x
∞
∞
u2
u2
1
= √  f (x)
−ue− 2 dudx + f (x) ue− 2 dudx

2π

−∞

−∞

0

x


✼

0



0



∞





u





−u2
−u2
1
= √   −f (x)ue 2 dx du +  f (x)ue 2 dx du
2π
−∞
u
0
0

 0
∞
u2
u2
1
= √  (f (u) − f (0))ue− 2 du + (f (u) − f (0))ue− 2 du
2π

−∞
∞

1
=√
2π
1
=√
2π

0

u2

u2

f (u)ue− 2 − f (0)ue− 2

du

−∞
∞
u2

f (u)ue− 2 du = EZf (Z).
−∞

❈❤✐➲✉ ♥❣❤à❝❤✳ ●✐↔ sû Ef (Z) = EZf (Z) ✈ỵ✐ ♠å✐ ❤➔♠ ❧✐➯♥ tư❝✱ ❜à ❝❤➦♥✱ ❦❤↔ ✈✐ tr➯♥
tø♥❣ ❦❤♦↔♥❣ t❤ä❛ ♠➣♥ E|f (Z)| < .
ợ x R t ý t ữỡ tr ♣❤➙♥
✭✶✳✷✮

f (ω) − ωf (ω) = I(ω ≤ x) − Φ(x),
tr♦♥❣ ✤â Φ(x) ❧➔ ❤➔♠ ♣❤➙♥ ♣❤è✐ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝â ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ t➢❝✳
ω2

◆❤➙♥ ❝↔ ❤❛✐ ✈➳ ❝õ❛ ữỡ tr ợ e 2 t ữủ
2

2

e 2 (f (ω) − ωf (ω)) = e− 2 (I(ω ≤ x) − Φ(x)),
❤❛②
ω2

e− 2 f (ω)

ω2

= e− 2 I(ω ≤ x) − Φ(x) ,

❚ø ✤➙② s✉② r❛

2

e

− ω2

f (ω) =

 ∞
2

− u2

−
e
(I(u ≤ x) − Φ(x))du


ω





♥➳✉ ω > x,

(3)

♥➳✉ ω ≤ x.

(4)

ω

−∞

u2

e− 2 (I(u ≤ x) − Φ(x))du




tữỡ ữỡ ợ

2

u2

e
(x)du

2

2

e

f () =





u2


e 2 (1 − Φ(x))du
−∞
√
 2π.Φ(x)(1 − Φ(ω))
= √
 2π.Φ(ω)(1 − Φ(x))

♥➳✉

ω > x,

♥➳✉

ω ≤ x.

♥➳✉ ω > x,
♥➳✉ ω ≤ x.

❉♦ ✤â

√
2
 2π.e ω2 Φ(x)(1 − Φ(ω))
f (ω) = √
 2π.e ω22 Φ(ω)(1 − Φ(x))

♥➳✉ ω > x,
♥➳✉ ω ≤ x.

✭✶✳✸✮

❍➔♠ f ♥❤÷ tr➯♥ ❧➔ ❧✐➯♥ tư❝✱ ❜à ❝❤➦♥ ✈➔ ❝â ✤↕♦ ❤➔♠ t↕✐ ♠å✐ ✤✐➸♠ ❝❤➾ trø t↕✐ ✤✐➸♠

ω = x✳
◆❤÷ ✈➟② ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ ❝â ♥❣❤✐➺♠ ❧✐➯♥ tư❝✱ ❜à ❝❤➦♥ ❞✉② ♥❤➜t✳
Ð ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ t❛ t❤❛② ω ❜ð✐ Z ✱ ❧➜② ❦ý ✈å♥❣ ❤❛✐ ✈➳ ✈➔ ❝❤ó þ ❣✐↔ t❤✐➳t

Ef (Z) = EZf (Z) t❛ ✤÷đ❝
0 = Ef (Z) − EZf (Z) = E(I(Z ≤ x)) − Φ(x),
❤❛②

P (Z ≤ x) = Φ(x) ✈ỵ✐ ♠å✐ x ∈ R.
◆❤÷ ✈➟② Z ∼ N (0, 1).
❚✐➳♣ t❤❡♦ ❝❤ó♥❣ t❛ s➩ tr➻♥❤ ❜➔② ❤❛✐ ❜ê ✤➲ ✈➲ t➼♥❤ ❝❤➜t ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
❙t❡✐♥✳ ❈→❝ ❜ê ✤➲ ♥➔② ✤÷đ❝ sû ❞ư♥❣ ✤➸ ❝❤➦♥ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❧♠♦❣♦r♦✈ ✈➔ ❦❤♦↔♥❣ ❝→❝❤
❲❛ss❡rst❡✐♥ ❝❤♦ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝✳

✶✳✶✳✷ ❇ê ✤➲✳ ●✐↔ sû x ∈ R ✈➔ f ❧➔ ♥❣❤✐➺♠ ❜à ❝❤➦♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
f (ω) − ωf (ω) = I(ω ≤ x) − Φ(x)✳

❑❤✐ ✤â

ωf (ω)

❧➔ ❤➔♠ t➠♥❣ ❝õ❛

ω✳

❍ì♥ ♥ú❛✱ ✈ỵ✐


✾

♠å✐ sè t❤ü❝

u, v

|wf (w)| ≤ 1,

✈➔

w✱

t❛ ❝â

|wf (w) − uf (u)| ≤ 1✱

|f (w)| ≤ 1, √|f (w) − f (u)| ≤ 1✱
2π
0 < f (w) ≤
✱
4

√

|(w + u)f (w + u) − (w + v)f (w + v)| ≤

❈❤ù♥❣ ♠✐♥❤✳

|w| +

2π
4

(|u| + |v|).

❚ø ✭✶✳✷✮ ✈➔ ✭✶✳✸✮ t❛ s✉② r❛

f (ω) = ωf (ω) + I(ω ≤ x) − Φ(x)


ωf (ω) + 1 − Φ(x) ✈ỵ✐ ω < x,
=
ωf (ω) − Φ(x)
✈ỵ✐ ω > x,
√
( 2πωeω2 /2 Φ(ω) + 1)(1 − Φ(x)) ✈ỵ✐ ω < x,
=
(√2πωeω2 /2 (1 − Φ(ω)) − 1)Φ(x) ✈ỵ✐ ω > x,

✭✶✳✹✮

✈➔


√
ω

 2π(1 − Φ(x)) (1 + ω 2 )eω2 /2 Φ(ω) +
2π
(ωf (ω)) = √
ω
2

2
ω
/2
 2πΦ(x) (1 + ω )e
(1 − Φ(ω)) −
2π

✈ỵ✐

ω < x,

ợ

> x,



ú ỵ r ợ > 0 t




ex

2

/2

dx ≤

ω

✈➔

2

e−ω /2

x −x2 /2
e
dx =
,
ω
ω

ω
∞

2

e

−x2 /2

ωe−ω /2
dx ≥
,
1 + ω2

ω

❞♦ ✤â

2

2

ωe−ω /2

e−ω /2
√ ≤ 1 − Φ(ω) ≤ √ .
(1 + ω 2 ) 2π
ω 2π

✭✶✳✻✮




ỷ ử ữợ tr t tự tr tr÷í♥❣ ❤đ♣ ω > x ❝❤♦ ❜✐➸✉ t❤ù❝

(ωf (ω)) t❛ s✉② r❛ (ωf (ω)) > 0✱ ❞♦ ✤â ωf (ω) ❧➔ ❤➔♠ t➠♥❣ ❝õ❛ ω ✳
❚ø ✭✶✳✹✮ t❛ ❝â

lim f (ω) = 0, lim f (ω) = 0.

ω→−∞

ω→∞

❚ø ❦➳t ❧✉➟♥ tr➯♥ ✈➔ sû ❞ư♥❣ ✭✶✳✹✮ ❧➛♥ ♥ú❛ t❛ ✤÷đ❝

lim ωf (ω) = Φ(x) − 1, lim ωf (ω) = Φ(x).

ω→−∞

ω→∞

✭✶✳✼✮

❑➳t ❤đ♣ ✈ỵ✐ ωf (ω) ❧➔ ❤➔♠ t➠♥❣ ❝õ❛ ω t❛ s✉② r❛

|wf (w)| ≤ 1, |wf (w) − uf (u)| ≤ 1.
❇➙② ❣✐í sû ❞ư♥❣ ωf (ω) ❧➔ ❤➔♠ t➠♥❣ ❝õ❛ ω ✱ ✭✶✳✹✮✱ ✭✶✳✻✮ ✈➔ ✭✶✳✼✮ t❛ ✤÷đ❝

0 < f (ω) ≤ xf (x) + 1 − Φ(x) < 1 ✈ỵ✐ ω < x

✭✶✳✽✮

✈➔

− 1 < xf (x) − (x) f () < 0

ợ

> x,



ữ |f (w)| ≤ 1✳
❱ỵ✐ u, w ❜➜t ❦ý t❛ ❝â

|f (w) − f (u)| ≤ xf (x) + 1 − Φ(x) − (xf (x) − Φ(x)) = 1.

✭✶✳✶✵✮

❚ø ✭✶✳✽✮ ✈➔ ✭✶✳✾✮ t❛ s✉② r❛ f (ω) ✤↕t ❣✐→ trà ❧ỵ♥ ♥❤➜t t↕✐ x✳ ❉♦ ✤â

0 < f (ω) ≤ f (x) =

✣➦t

√

2πex

2

/2

Φ(x)(1 − Φ(x)).

2

e−x /2
1
x 2Φ(x)
, g1 (x) = √ + − √ .
g(x) = Φ(x)(1 − Φ(x)) −
4
2π 4
2π

❑❤✐ ✤â

g (x) = e−x

2

/2

g1 (x), g1 (0) = 0, g1 (0) < 0, g1 (x) =

x −x2 /2
e
, lim g1 (x) = ∞.
x→∞
π




õ g1 (x) ỗ tr [0, ) tỗ t↕✐ x1 > 0 s❛♦ ❝❤♦ g1 (x) < 0 ✈ỵ✐ x < x1 ✈➔

g1 (x) > 0 ✈ỵ✐ x > x1 ✳ ✣➦❝ ❜✐➺t✱ tr➯♥ [0, ∞) ❤➔♠ sè g(x) t➠♥❣ ❦❤✐ x > x1 ✈➔ ❣✐↔♠
❦❤✐ x < x1 ✱ ✈➻ ✈➟② ♥â ♣❤↔✐ ✤↕t ❣✐→ trà ❧ỵ♥ ♥❤➜t t↕✐ ❤♦➦❝ x = 0 ❤♦➦❝ x = ∞✱ tù❝ ❧➔

g(x) ≤ max (g(0), g(∞)) = 0 ✈ỵ✐ ♠å✐ x [0.).

2
t ủ ợ t ữủ
tữỡ ữỡ ợ f (x)
4

2
0 < f (w)
.
4

ố ❝ò♥❣ t❛ ❝â

(w+u)f (w+u)−(w+v)f (w+v) = w(f (w+u)−f (w+v))+uf (w+u)−vf (w+v),
ử ỵ tr tr ✈➔ ✭✶✳✶✶✮✱ ✭✶✳✶✵✮ t❛ ✤÷đ❝

√
|(w + u)f (w + u) − (w + v)f (w + v)| ≤

|w| +

2π
4

(|u| + |v|).

❚✐➳♣ t❤❡♦ ❝❤ó♥❣ tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ♠ët ❜ê ✤➲ ❦❤→❝ ✈➲ t➼♥❤ ❝❤➜t ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ ❙t❡✐♥ ❞↕♥❣ tê♥❣ q✉→t✳ ❈❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝❤♦ ❜ê ✤➲ ♥➔② ❝â t❤➸ ①❡♠ tr♦♥❣ ❬✸✱
tr❛♥❣ ✸✽❪✳

✶✳✶✳✸ ❇ê ✤➲✳ ●✐↔ sû h : R → R ❧➔ ❤➔♠ ✤♦ ✤÷đ❝✱ ❦❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤
f (ω) − ωf (ω) = h(ω) − Eh(Z)

•

◆➳✉

h

f ≤

•

◆➳✉

h

❝â ♥❣❤✐➺♠

f

❜à ❝❤➦♥ ❞✉② ♥❤➜t✳

❜à ❝❤➦♥ t❤➻

π
h − Eh(Z)
2

✈➔

f

≤ 2 h − Eh(Z) .

❧✐➯♥ tö❝ t✉②➺t ✤è✐ t❤➻

f ≤2 h , f

≤

2
h , f
π

≤2 h .


✶✷

✶✳✷ P❤÷ì♥❣ ♣❤→♣ ❙t❡✐♥ ❝❤♦ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝
❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ t❛ s➩ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❙t❡✐♥ ❝❤♦ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝✱
♠ët ♥ë✐ ❞✉♥❣ tr✉♥❣ t➙♠ ❝õ❛ ♣❤÷ì♥❣ t rữợ t ú t s ợ t
✈➲ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝✳

✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❍❛✐ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ W ✈➔ W
♠❡tr✐❝

X

✤÷đ❝ ❣å✐ ❧➔ ♠ët ❝➦♣ ❤♦→♥ ờ ữủ

ố tự ợ ồ

A, B B(X )✱

♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤æ♥❣ ❣✐❛♥

(W, W )

✈➔

(W , W )

❝ò♥❣ ♣❤➙♥

t❛ ❝â

P (W ∈ A, W ∈ B) = P (W ∈ B, W ∈ A).

✶✳✷✳✷ ❇ê ✤➲✳ ◆➳✉ (X, Y ) ❧➔ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝ t❤➻ (f (X), f (Y )) ụ
ờ ữủ ợ ♠å✐ ❤➔♠

❈❤ù♥❣ ♠✐♥❤✳

f

✤♦ ✤÷đ❝ ❜➜t ❦ý✳

d

❱➻ (X, Y ) ❧➔ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝ ♥➯♥ (X, Y ) = (Y, X) t ủ ợ
d

f ữủ t s r❛ (f (X), f (Y )) = (f (Y ), f (X))✳ ✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔
(f (X), f (Y )) ❝ơ♥❣ ❧➔ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝✳

✶✳✷✳✸ ❇ê ✤➲✳ ●✐↔ sû (X, Y ) ❧➔ ♠ët ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝✱ õ
Eg(X, Y ) = 0,
ợ ồ ữủ ố ự

ự

g(x, y)

s ý ồ tỗ t

g(x, y) ❧➔ ❤➔♠ ♣❤↔♥ ✤è✐ ①ù♥❣ ♥➯♥

g(X, Y ) + g(Y, X) = 0.
❑➳t ❤đ♣ ✈ỵ✐ (X, Y ) ❧➔ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝ t❛ s✉② r❛

2Eg(X, Y ) = Eg(X, Y ) + Eg(X, Y ) = Eg(X, Y ) + Eg(Y, X) = 0.


✶✸

✶✳✷✳✹ ❱➼ ❞ö✳ ❈❤♦ X, Y

❧➔ ❤❛✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ❝ò♥❣ ♣❤➙♥ ♣❤è✐✳ ❑❤✐ ✤â ❞➵

t❤➜② (X, Y ) ❧➔ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝✳

✶✳✷✳✺ ❱➼ ❞ư✳ ❈❤♦ {Xi, 1 ≤ i ≤ n} ❧➔ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✳ ●å✐

Xi , 1 ≤ i ≤ n

❧➔ ❜↔♥ s❛♦ ✤ë❝ ❧➟♣ ❝õ❛ {Xi , 1 ≤ i ≤ n} . ●✐↔ sû I ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ♣❤➙♥ ♣❤è✐ ✤➲✉
tr➯♥ t➟♣ {1, 2, ..., n} ✈➔ ✤ë❝ ❧➟♣ ✈ỵ✐ Xi , Xi , 1 ≤ i ≤ n . ❑❤✐ ✤â (Sn , SnI ) ❧➔ ❝➦♣ ❤♦→♥
✤ê✐ ✤÷đ❝✱ tr♦♥❣ ✤â

n

Xi , SnI = Sn − XI + XI .

Sn =
i=1

❚ê♥❣ q✉→t ❤ì♥ ♥➳✉ ✤➦t

X = (X1 , X2 , ..., Xn ),
Y = (X1I , X2I , ..., XnI ),
tr♦♥❣ ✤â


Xk ♥➳✉ I = k
I
Xk =
X ♥➳✉ I = k.
k
❑❤✐ ✤â (X, Y ) ❧➔ ❝➦♣ ❤♦→♥ ờ ữủ õ ợ ồ x1 , x2 , ..., xn , y1 , y2 , ..., yn ∈

R t❛ ❝â

P (X1 ≤ x1 , X2 ≤ x2 , ..., Xn ≤ xn , X1I ≤ y1 , X2I ≤ y2 , ..., XnI ≤ yn )
= P (X1 ≤ y1 , X2 ≤ y2 , ..., Xn ≤ yn , X1I ≤ x1 , X2I ≤ x2 , ..., XnI ≤ xn ).

✭✶✳✶✷✮

❚❤➟t ✈➟② t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ✭✶✳✶✷✮✳ ❱➳ tr→✐ ❝õ❛ ✭✶✳✶✷✮ ✤÷đ❝ ❜✐➳♥ ✤ê✐ ♥❤÷ s❛✉✿
n

P (X1 ≤ x1 , ..., Xn ≤ xn , X1I ≤ y1 , ..., XnI ≤ yn , I = k)
k=1
n

=
k=1
n

=
k=1

1
P (X1 ≤ x1 , ..., Xn ≤ xn , X1 ≤ y1 , ..., XkI ≤ yk , ..., Xn ≤ yn )
n
1
P (Xi ≤ min {xi , yi }).P (Xk ≤ xk ).P (XkI ≤ yk ).
n i=k

❇➡♥❣ ❝→❝❤ t÷ì♥❣ tü t❛ ❝ơ♥❣ ❜✐➳♥ ✤ê✐ ✤÷đ❝ ✈➳ ♣❤↔✐ ❝õ❛ ✭✶✳✶✷✮ ✈➲ ❞↕♥❣ ♥❤÷ tr➯♥✳ ❱➟②

(X, Y ) ❧➔ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝✳


✶✹

❳➨t ❤➔♠ ✤♦ ✤÷đ❝

f : Rn −→ R
①→❝ ✤à♥❤ ❜ð✐

f (x1 , x2 , ..., xn ) = x1 + x2 + ... + xn .
❑❤✐ ✤â Sn = f (X), SnI = f (Y ). ❱➻ (X, Y ) ❧➔ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝ ♥➯♥ (Sn , SnI ) ❝ơ♥❣
❧➔ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝✳
❚✐➳♣ t❤❡♦ ❝❤ó♥❣ t❛ s➩ t❤✐➳t ❧➟♣ ❝❤➦♥ tr➯♥ ✤è✐ ✈ỵ✐ ❦❤♦↔♥❣ ❝→❝❤ ❲❛ss❡rst❡✐♥ ✈➔
❦❤♦↔♥❣ ❝→❝❤ ❑♦❧♠♦❣♦r♦✈ ❝❤♦ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝✳

✶✳✷✳✻ ✣à♥❤ ❧➼✳ ●✐↔ sû (W, W ) ❧➔ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝ t❤ä❛ ♠➣♥
E(W − W |W ) = λW

✈ỵ✐

0 < λ < 1.

❑❤✐ ✤â

1
sup |P (W ≤ x) − Φ(x)| ≤ 2 E 1 − E(∆2 |W )
2λ
x∈R
x
tr♦♥❣ ✤â

∆ = W − W , Φ(x) =
−∞

❈❤ù♥❣ ♠✐♥❤✳

2

+

E|∆|3
,
λ

1
t2
√ e− 2 dt.
2π

❱ỵ✐ x ∈ R, > 0 ❜➜t ❦ý✱ ①➨t ❤➔♠ sè



1
♥➳✉ ω ≤ x,



ω−x
h(ω) = 1 −
♥➳✉ x < ω ≤ x + ,




0
♥➳✉ ω ≥ x + .

❑❤✐ ✤â t❛ ❝â I(ω ≤ x) ≤ h(ω) ≤ I(ω ≤ x + ).
❚❤❛② ω ❜ð✐ W ✈➔ ❧➜② ❦ý ✈å♥❣ t❛ ✤÷đ❝
✭✶✳✶✸✮

P (W ≤ x) ≤ Eh(W ) ≤ P (W ≤ x + ).
◆❣♦➔✐ r❛ t❛ ❝á♥ ❝â h ❜à ❝❤➦♥✱ ❧✐➯♥ tö❝ t✉②➺t ✤è✐ ✈➔ h ≤ 1, h

1
≤ .


✶✺

❳➨t ♣❤÷ì♥❣ tr➻♥❤ ❙t❡✐♥ f (ω) − ωf (ω) = h(ω) − Eh(Z),
tr♦♥❣ ✤â Z ∼ N (0, 1) ✈➔ ✈➻ h ❜à ❝❤➦♥✱ ❧✐➯♥ tö❝ t✉②➺t ✤è✐ ♥➯♥ t❤❡♦ ❇ê ✤➲ ✶✳✶✳✸ t❛ ❝â

f ≤

π
h − Eh(Z) , f
2

≤2 h .

❱➻ (W, W ) ❧➔ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝ ♥➯♥

0 = E(W − W )(f (W ) + f (W ))
= E(W − W )(f (W ) − f (W )) + 2E(W − W )f (W )
= E(W − W )(f (W ) − f (W )) + 2E(E(W − W )f (W )|W )
= E(W − W )(f (W ) − f (W )) + 2E(f (W )E(W − W )|W )

= E(W − W )(f (W ) − f (W )) + 2E(f (W )λW )
= E(W − W )(f (W ) − f (W )) + 2λE(W f (W )).
❉♦ ✤â t❛ ❝â E(W f (W )) =

1
E∆(f (W ) − f (W )).
2λ

❚r♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❙t❡✐♥ f (ω) − ωf (ω) = h(ω) − Eh(Z), t❤❛② ω ❜ð✐ W ✈➔
❧➜② ❦ý ✈å♥❣ ❤❛✐ ✈➳ t❛ ✤÷đ❝

Eh(W ) − Eh(Z) = Ef (W ) − EW f (W )
1
= Ef (W ) + E∆(f (W ) − f (W ))
2λ
1
1
= Ef (W ) 1 − E(∆2 |W ) + E f (W ) E(∆2 |W )
2λ
2λ
1
+ E∆(f (W ) − f (W ))
2λ
1
1
= Ef (W ) 1 − E(∆2 |W ) + E(E(f (W )∆2 |W ))
2λ
2λ
1
+ E∆(f (W ) − f (W ))

2λ
1
1
1
= Ef (W ) 1 − E(∆2 |W ) + Ef (W )∆2 + E∆(f (W ) − f (W ))
2λ
2λ
2λ
= R1 + R2 ,


✶✻

tr♦♥❣ ✤â

R1 = Ef (W ) 1 −

R2 =

1
E(∆2 |W ) ,
2λ

1
E(∆(f (W ) − f (W )) + ∆2 f (W )).
2λ

❚❛ ❝â

1

E(∆2 |W )
2λ
1
≤ E f (W ) 1 − E(∆2 |W )
2λ
1
≤ 2 h − Eh(Z) E 1 − E(∆2 |W )
2λ
1
≤ 2E 1 − E(∆2 |W )
2λ

|R1 | = Ef (W ) 1 −

1
≤ 2 E 1 − E(∆2 |W )
2λ

2

.

❚❤❡♦ ❝æ♥❣ t❤ù❝ ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r t❛ ❝â

f (x ) − f (x) =
✈ỵ✐ x
˜ ♥➡♠ ❣✐ú❛ ① ✈➔ ①✬.

f (x)
f (˜

x)
(x − x) +
(x − x)2
1!
2!
∼

❚❤❛② x , x, x
˜ ❧➛♥ ❧÷đt ❜ð✐ W , W, W t❛ ❝â
∼

f (W )
f (W )
f (W ) − f (W ) =
(W − W ) +
(W − W )2
1!
2!
∼

f (W )
= −∆f (W ) + ∆2
.
2!
❉♦ ✤â

1
E(∆(f (W ) − f (W )) + ∆2 f (W ))
2λ 


∼
1  3 f (W ) 
E ∆
.
=
2λ
2!

R2 =


✶✼

❚ø ✤â t❛ s✉② r❛

1
1
E|∆3 |. f ≤
E|∆3 |.2 h
4λ
4λ
1 2
1
=
. E|∆3 | =
E|∆3 |.
4λ
2λ

|R2 | ≤

◆❤÷ ✈➟② t❛ ❝â

1
|Eh(W ) − Eh(Z)| ≤ 2 E 1 − E(∆2 |W )
2λ

2

+

1
E|∆3 |.
2λ

❚ø ✭✶✳✶✸✮ t❛ ❝â

P (W ≤ x) − Φ(x) ≤ Eh(W ) − Φ(x)
= Eh(W ) − Eh(Z) + Eh(Z) − Φ(x)
≤ |R1 | + |R2 | + Φ(x + ) − Φ(x)
≤ |R1 | + |R2 | + √

2π

.

❚÷ì♥❣ tü t❛ ❝â

P (W ≤ x) − Φ(x) ≥ −|R1 | − |R2 | − √

2π

.

❚ø ✤â t❛ s✉② r❛

|P (W ≤ x) − Φ(x)| ≤ |R1 | + |R2 | + √

2π

1
≤ 2 E 1 − E(∆2 |W )
2λ
❈❤å♥

=

4

π
.
2

2

+

1
E|∆3 | + √ .
2λ

2π

E|∆|3
t❛ ✤÷đ❝
2

1
|P (W ≤ x) − Φ(x)| ≤ 2 E 1 − E(∆2 |W )
2λ

2

1
≤ 2 E 1 − E(∆2 |W )
2λ

2

+

2E|∆|3
√
2πλ

+

E|∆|3
.
λ

✶✽

✶✳✷✳✼ ▼➺♥❤ ✤➲✳ ●✐↔ sû (W, W ) ❧➔ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝ t❤ä❛ ♠➣♥ E(W − W |W ) =
λ(W + R)

✈ỵ✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥

sup |Eh(W ) − Eh(Z)| ≤
h ≤1
tr♦♥❣ ✤â

h:R→R

❈❤ù♥❣ ♠✐♥❤✳

R = R(W )

✈➔

0 < λ < 1✱

❦❤✐ ✤â

2
1
1
E 1 − E(∆2 |W ) + E|∆|3 + 2E|R|,
π
2λ

2λ

❧➔ ❤➔♠ ✤♦ ✤÷đ❝ t❤ä❛ ♠➣♥

h

≤1

✈➔

E|h(Z)| < ∞.

●✐↔ sû h : R → R ❧➔ ❤➔♠ ✤♦ ✤÷đ❝ t❤ä❛ ♠➣♥ h

≤ 1 ✈➔ E|h(Z)| < ∞

✈➔ ❣✐↔ sû f := fh ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❙t❡✐♥

f (ω) − ωf (ω) = h(ω) − Eh(Z).
❱➻ (W, W ) ❧➔ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝ ✈➔ E(W − W |W ) = λ(W + R) ♥➯♥

0 = E(W − W )(f (W ) + f (W ))
= E(W − W )(f (W ) − f (W )) + 2Ef (W )(W − W )
= E(W − W )(f (W ) − f (W )) + 2E(E(f (W )(W − W )|W ))
= E∆(f (W ) − f (W )) + 2λEW f (W ) + 2λEf (W )R.
❚ø ✤â s✉② r❛

|Eh(W ) − Eh(Z)| = |E(f (W ) − W f (W ))|
1
= Ef (W ) + E∆(f (W ) − f (W )) + Ef (W )R

2λ
1
1
= E f (W ) 1 − E(∆2 |W ) + ∆(f (W ) − f (W ) + ∆f (W )) + f (W )R
2λ
2λ
1
1
f E|∆|3 + f E|R|.
≤ f E 1 − E(∆2 |W ) +
2λ
4λ
✭✶✳✶✹✮
❚❤❡♦ ❇ê ✤➲ ✷✳✹ tr♦♥❣ ❬✸❪ t❛ ❝â

f ≤ 2, f

≤

2
, f
π

≤ 2.

❑➳t ❧✉➟♥ ❝õ❛ ♠➺♥❤ ✤➲ ✤÷đ❝ s✉② r❛ tø ✭✶✳✶✹✮ ✈➔ ✭✶✳✶✺✮✳

✭✶✳✶✺✮

r tữớ sỷ ử ỡ tr ỵ t❤✉②➳t ①→❝ s✉➜t
✈➔ t❤è♥❣ ❦➯✱ t✉② ♥❤✐➯♥ ♥â ❧↕✐ t❤÷í♥❣ õ ỷ ỵ ỡ ssrst
❩❤❛♥❣ ❬✶✵❪ ❝❤ù♥❣ ♠✐♥❤ ♠ët ❜à ❝❤➦♥ tê♥❣ q✉→t ❤ì♥✳ t q ừ ồ
ữ s

ữợ t❤✐➳t ❝õ❛ ▼➺♥❤ ✤➲ ✶✳✷✳✼✱ t❛ ❝â
sup |P (W ≤ z) − Φ(z)| ≤ E 1 −
z∈R
tr♦♥❣ ✤â

∆∗ := ∆∗ (W, W )

∆∗ (W , W )

✈➔

1
1
E(∆2 |W ) + E|E(∆∆∗ |W )| + E|R|,
2λ
λ

❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❜➜t ❦ý t❤ä❛ ♠➣♥

∆∗ (W, W ) =

∆∗ ≥ |∆|.

❚r♦♥❣ tr÷í♥❣ ❤đ♣ || a t q ữợ q trü❝ t✐➳♣ ❝õ❛ ▼➺♥❤
✤➲ ✶✳✷✳✽✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② ❜à ❝❤➦♥ ✤ì♥ ❣✐↔♥ ❤ì♥ ♥❤✐➲✉ s♦ ✈ỵ✐ ▼➺♥❤ ✤➲ ✶✳✷✳✽✳

✶✳✷✳✾ ❍➺ q✉↔✳ ◆➳✉ |∆| ≤ a✱ ❦❤✐ ✤â
sup |P (W ≤ z) − Φ(z)| ≤ E 1 −
z∈R

❈❤ù♥❣ ♠✐♥❤✳

1
E(∆2 |W ) + (E|W | + 1)a + E|R|.
2λ

✭✶✳✶✻✮

❚r♦♥❣ ▼➺♥❤ ✤➲ ✶✳✷✳✽ ❝❤å♥ ∆∗ = a✱ ❦❤✐ ✤â

E|E(∆∆∗ |W )| = aE|E(∆|W )| ≤ aλ(E|W | + E|R|).

✭✶✳✶✼✮

◆➳✉ E|R| ≥ 1✱ ✭✶✳✶✻✮ ❤✐➸♥ ♥❤✐➯♥ ✤ó♥❣✳ ◆➳✉ E|R| < 1✱ ❦❤✐ ✤â tø ✭✶✳✶✼✮ ✈➔ ▼➺♥❤
✤➲ ✶✳✷✳✽ t❛ ❝â

1
1
E(∆2 |W ) + E|E(∆∆∗ |W )| + E|R|
2λ
λ
1

1
≤ E 1 − E(∆2 |W ) + aλ(E|W | + E|R|) + E|R|
2λ
λ
1
≤ E 1 − E(∆2 |W ) + (E|W | + 1)a + E|R|.
2λ

sup |P (W ≤ z) − Φ(z)| ≤ E 1 −
z∈R




ữỡ

số tr ố ợ ổ
N tr ợ trữớ tr
ổ N tr ợ trữớ tr ởt số t
q q
sû N ≥ 2 ✈➔ SN −1 ❧➔ ♠➦t ❝➛✉ ✤ì♥ ✈à tr♦♥❣ RN ✳ ❚↕✐ ♠é✐ ✤➾♥❤ i ❝õ❛ ởt ỗ
t ừ n t ởt s σi ✳ ✣➦t Ωn = (SN −1 )n ✱ tr➯♥ n t ở t

Pn = à ì ... ì µ✱ tr♦♥❣ ✤â µ ❧➔ ✤ë ✤♦ ①→❝ s✉➜t ✤➲✉ tr➯♥ SN −1 ✳
❱ỵ✐ σ = (σ1 , σ2 , · · · , σn ) ∈ Ωn ✱ ①➨t ❤➔♠ ♥➠♥❣ ❧÷đ♥❣ ❍❛♠✐❧t♦♥

1
Hn (σ) = −
2n

n

n

σi , σj ,
i=1 j=1

tr õ Ã, Ã t ổ ữợ tr Rn .
❱ỵ✐ β ≥ 0✱ ①➨t ✤ë ✤♦ ●✐❜❜s

dPn,β =

1
exp(−βHn (σ))dPn ,
Zn,β


✷✶

tr♦♥❣ ✤â Zn,β ❧➔ ❤➡♥❣ sè ❝❤✉➞♥ ❤â❛

Zn,β =

exp(−βHn (σ))dPn .
Ωn

▼ỉ ❤➻♥❤ ♥➔② ❣å✐ ❧➔ ♠ỉ ❤➻♥❤ N ✲✈❡❝t♦r ✈ỵ✐ tr÷í♥❣ tr✉♥❣ ❜➻♥❤✱ ❤❛② ❝á♥ ✤÷đ❝ ❣å✐
❧➔ ♠ỉ ❤➻♥❤ O(N ) ợ trữớ tr õ ữủ ồ ổ ❤➻♥❤ XY ✱ ♠æ ❤➻♥❤
❍❡✐s❡♥❜❡r❣✱ ♠æ ❤➻♥❤ ❚♦② ❦❤✐ N ữủt 2, 3, 4.
rữợ t ú tổ ợ t ởt số ỵ ữ s r t ở ữỡ

Z ỵ ởt õ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ t➢❝✱ Sn =

n

σi ✱ t❛ ✈✐➳t

i=1

f = sup|f (x)|.
x

❚r♦♥❣ ♠æ ❤➻♥❤ ❍❡✐s❡♥❜❡r❣✱ ❑✐r❦♣❛tr✐❝❦ ✈➔ ▼❡❝❦❡s ❬✺❪ ✤➣ t❤✐➳t ❧➟♣ ❝→❝ ❦➳t q✉↔
✈➲ ✤ë ❧➺❝❤ ❧ỵ♥ ✤è✐ ✈ỵ✐ s♣✐♥ t♦➔♥ ♣❤➛♥ Sn =

n

σi tr♦♥❣ tr÷í♥❣ ❤đ♣ β = 3. ❈→❝ ❦➳t

i=1

q✉↔ tr♦♥❣ ❬✺❪ ✤÷đ❝ tê♥❣ q✉→t ❜ð✐ ❑✐r❦♣❛tr✐❝❦ ✈➔ ◆❛✇❛③ ❬✻❪ ✤è✐ ✈ỵ✐ ♠ỉ ❤➻♥❤ N ✲✈❡❝t♦r
✈ỵ✐ n ≥ 2✳ ◆❣♦➔✐ r❛✱ ❚ù ✈➔ ❝→❝ ❝ë♥❣ sü ❝ô♥❣ ♠ð rë♥❣ ❝→❝ ❦➳t q✉↔ ✈➲ ✤ë ❧➺❝❤ ❧ỵ♥ ❝õ❛
❑✐r❦♣❛tr✐❝❦ ✈➔ ▼❡❝❦❡s ❬✺❪ ❜➡♥❣ ❝→❝❤ t❤✐➳t ❧➟♣ ✤ë ❧➺❝❤ ❧ỵ♥ tr ổ sr
ợ tứ trữớ
sỷ I ❤➔♠ ❇❡ss❡❧ ❝↔✐ ❜✐➳♥ ❧♦↕✐ ♠ët ✭♠♦❞✐❢✐❡❧ ❇❡ss❡❧ ❢✉♥❝t✐♦♥✮ ✈➔ ✤÷đ❝ ①→❝
✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝

z
Iν (z) =
2

ν

∞

k=0

(z 2 /4)k
,
k!Γ(ν + k + 1)
1
2

tr♦♥❣ ✤â Γ(z) ❧➔ ❤➔♠ ❣❛♠♠❛✳ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ν > − ✱ ❤➔♠ ❇❡ss❡❧ ❝↔✐ ❜✐➳♥ ❧♦↕✐
♠ët Iν ❝á♥ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝
π

1
z
Iν (z) = √
πΓ(ν + 1/2) 2

ν

exp(z cos θ) sin2ν θdθ.
0


✷✷

✣➦t

f (x) =

I N (x)
2

I N −1 (x)

,

x > 0.

2

❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✷✱ t❛ ❝â

f (x)
x

< 0 ✈ỵ✐ ♠å✐ x > 0.

✭✷✳✶✮

❈❤ó♥❣ t❛ ❝ơ♥❣ ❝â

1
f (x)
f (x)
=

✈➔ lim
= 0.
x→∞ x
x→0
x
N
❚r♦♥❣ ♠ö❝ ♥➔② t❛ ①➨t β > N ✱ ❦❤✐ ✤â tø ✭✷✳✶✮ ✈➔ ✭✷✳✷✮ s✉② r❛ ♣❤÷ì♥❣ tr➻♥❤
lim+

x − βf (x) = 0.

✭✷✳✷✮

✭✷✳✸✮

❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❜➡♥❣ b✳
✣➦t

Wn :=

√

n

β2
|Sn |2 − 1 ,
2
2
nb

4β 2
(N − 1)f (b)
B =
− (f (b))2 .
1
−
2
(1 − βf (b))b
b
2

✭✷✳✹✮
✭✷✳✺✮

●✐↔ sû r➡♥❣ σ = {σ1 , σ2 , ..., σn } ❧➔ ❜↔♥ s❛♦ ✤ë❝ ❧➟♣ ❝õ❛ σ = {σ1 , σ2 , ..., σn } ✈➔ I ❧➔
❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ♣❤➙♥ ♣❤è✐ ✤➲✉ tr➯♥ {1, 2, ..., n} . ✣➦t

Wn :=
tr♦♥❣ ✤â Sn =

n
i=1

√

β2
n
|S |2 − 1 .
n2 b2 n

✭✷✳✻✮

σi − σI + σI ✳ ❑❤✐ ✤â (Wn , Wn ) ❧➔ ♠ët ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝✳

❈❤➦♥ tr➯♥ ❦❤♦↔♥❣ ❝→❝❤ ❲❛ss❡rst❡✐♥ ✈➔ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❧♠♦❣♦r♦✈ ❣✐ú❛ Wn /B
✈➔ Z ❧➔ C(ln n/n)1/4 ♥❤➟♥ ✤÷đ❝ ❜ð✐ ❑✐r❦♣❛tr✐❝❦ ✈➔ ◆❛✇❛③ tr♦♥❣ ❬✻❪✱ tr♦♥❣ ❧✉➟♥ ✈➠♥


✷✸

❝❤ó♥❣ tỉ✐ t❤✐➳t ❧➟♣ ❝❤➦♥ tr➯♥ ❦❤♦↔♥❣ ❝→❝❤ ❲❛ss❡rst❡✐♥ ✈➔ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❧♠♦❣♦r♦✈
❣✐ú❛ Wn /B ✈➔ Z ✈ỵ✐ tè❝ ✤ë tè✐ ÷✉ ❤ì♥ ❜➡♥❣ Cn−1/2 ✳

✷✳✷ ❙❛✐ sè tr♦♥❣ ①➜♣ ố ợ ổ N tr ợ
trữớ tr
rữợ tr ở ừ ử ❝❤ó♥❣ tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ♠ët sè
❜ê ✤➲ sû ❞ư♥❣ tr♦♥❣ ự ừ ỵ s

ờ ợ β > N ✱ Sn =

n

σi

✈➔

b

❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

✭✷✳✸✮

t❤➻

i=1

2
β|Sn |
C
−b ≤ .
E
n
n

Sn
❬✻✱ ▼➺♥❤ ✤➲ ✷❪ ✈➔ ❧➟♣ ❧✉➟♥ tr♦♥❣ tr
n
t õ t ự r tỗ t > 0 s❛♦ ❝❤♦

❈❤ù♥❣ ♠✐♥❤✳

❚❤❡♦ ✤ë ❧➺❝❤ ❧ỵ♥ ❝❤♦

P
✈ỵ✐ ♠å✐ 0 ≤ x ≤ ✳ ❚ø
t❛ s✉② r❛

β|Sn |
2
− b ≥ x ≤ e−Cnx

n
β|Sn |
− b ≤ C,
n