▲✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣
✭❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ Ù♥❣ ❉ö♥❣✮

❚➮❈ ✣❐ ❍❐■ ❚Ö ❈Õ❆ ❉❶❨ ❇■➌◆ ◆●❼❯
◆❍■➊◆ ❚❘❖◆● ❈⑩❈ ✣➚◆❍ ▲➑ ●■❰■ ❍❸◆
❚❘❯◆● ❚❹▼
●✐→♦ ✈✐➯♥ ❤÷î♥❣ ❞➝♥✿ ❚❤✳s ▲➙♠ ❍♦➔♥❣ ❈❤÷ì♥❣
❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥✿ ◆❣✉②➵♥ ❚❤à ❍✉ý♥❤ ◆❤÷
▼❙❙❱✿ ✶✵✼✻✻✹✻

◆❣➔② ✷ t❤→♥❣ ✻ ♥➠♠ ✷✵✶✶


✷

▲❮■ ❈❷▼ ❒◆
❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❣✐→♠ ❤✐➺✉ ❚r÷í♥❣ ✣↕✐ ❍å❝ ❈➛♥ ❚❤ì✱ ❇❛♥ ❝❤õ ♥❤✐➺♠
❑❤♦❛ ❑❤♦❛ ❍å❝ ❚ü ◆❤✐➯♥ ❝ò♥❣ q✉þ ❚❤➛② ❈æ t❤✉ë❝ ❇ë ♠æ♥ ❚♦→♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐
❝❤♦ ❡♠ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✳
❈↔♠ ì♥ ❚❤➛② ▲➙♠ ❍♦➔♥❣ ❈❤÷ì♥❣ ✤➣ t➟♥ t➻♥❤ ❝❤➾ ❜↔♦✱ ❣✐ó♣ ✤ï ❡♠ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐✳
❈↔♠ ì♥ sü ❣✐ó♣ ✤ï ❝õ❛ ❣✐❛ ✤➻♥❤ ✈➔ ❝→❝ t❤➔♥❤ ✈✐➯♥ tr♦♥❣ ❧î♣ ❚♦→♥ Ù♥❣ ❉ö♥❣ ❑✸✸✳
❚r♦♥❣ q✉→ tr➻♥❤ ❧➔♠ ❜➔✐ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ s❛✐ sât✱ ❡♠ r➜t ♠♦♥❣ q✉þ ❚❤➛② ❈æ t❤æ♥❣
❝↔♠ ✈➔ ❣✐ó♣ ❡♠ ❦❤➢❝ ♣❤ö❝ ✤➸ ❧✉➟♥ ✈➠♥ ❝õ❛ ❡♠ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳
❑➼♥❤ ❝❤ó❝ q✉þ ❚❤➛② ❈æ ✤÷ñ❝ ♥❤✐➲✉ sù❝ ❦❤ä❡✳
❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦
❈➛♥ ❚❤ì✱ ♥❣➔② ✸✵ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✶
◆❣✉②➵♥ ❚❤à ❍✉ý♥❤ ◆❤÷

P é
ỵ ồ t ử ự
st tố q trồ ừ t ồ t ồ õ trỏ tt
ố ợ sỹ t tr ừ ồ ữ õ ồ t ỵ ồ ố tữủ
ự ừ st tố tữủ q t
ú t tữớ tr tỹ t
ợ ởt số ổ t ồ trứ tữủ ỵ tt st tố ợ
t tỹ t tr ở số tr tỹ ở ữ rừ r tr
ổ õ sỷ ử ỵ tt ở t tr tt s t
st ọ õ
r ỵ tt st ợ tr t t õ trỏ q
trồ õ t q sỹ ở tử ừ ởt ợ t õ
t q tờ ừ s ở tử ởt õ
rữớ ủ ỡ t ừ ợ tr t t t sỹ ở tử ừ
ở õ ũ ồ ữỡ s
ụ tỗ t sỹ ở tử tr trữớ ủ ổ ũ
ố ữ ổ õ õ ố trở ỡ
ữ ố ừ ữủ
r
r tr ợ tr t tố ở ở tử ừ ở
tr ởt số ợ tr t ụ õ trỏ q trồ ởt
ữủ rt t ồ q t õ ụ ử t t ữủ ừ


ố tữủ ự
r t ở ú t sỷ ử t tỷ rttr ữ ởt ổ ử
tố ở ở tử tr ợ tr t
é t t t s tố ở ở tử ổ ử t tỷ rttr
ữủ tr rttr tờ ừ ữủ






Pữỡ ự
tỹ sữ t ồ t õ q
tứ trt s t ổ q sỹ ú ù ừ ữợ s
ự tt tr ỗ tớ ụ õ ởt t ữ
ỵ

trú ừ
ữỡ






ữỡ 1 tỷ rttr tr ự ợ tr t
r ữỡ t s ự ợ tr t ữủ
õ ũ ố ợ tr t ữủ
ổ ũ ố t tỷ rttr
ữỡ 2 ố ở ở tử tr ợ tr t
r ữỡ t ự tố ở ở tử ừ ữủ tr ởt
số ợ tr t t tỷ rttr
ữỡ 3 tố ở ở tử tr ởt số ợ tr t ữỡ
tr rttr
r t sỷ ử tr rttr ự tố ở ở tử ừ
ở ũ ố t tố ở ở tử ừ tờ
ở ũ ố

ý ữ


▼ö❝ ❧ö❝
❈❤÷ì♥❣ ✶✳ ❚♦→♥ tû ❚r♦tt❡r tr♦♥❣ ❝→❝ ✤à♥❤ ❧➼ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠

✼

✶✳✶✳ ❚♦→♥ tû ❚r♦tt❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼
✶✳✷✳ ✣à♥❤ ❧þ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠ ❝❤♦ ❝→❝ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ❝â ❝ò♥❣ ♣❤➙♥
♣❤è✐✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✶✳✸✳ ✣à♥❤ ❧þ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠ ❝❤♦ ❝→❝ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ❦❤æ♥❣ ❝ò♥❣ ♣❤➙♥ ♣❤è✐✿ ✶✽

❈❤÷ì♥❣ ✷✳ ❚è❝ ✤ë ❤ë✐ tö tr♦♥❣ ✤à♥❤ ❧➼ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠

✷✺

❈❤÷ì♥❣ ✸✳ ▼❡tr✐❝ ①→❝ s✉➜t ❞ü❛ tr➯♥ t♦→♥ tû ❚r♦tt❡r

✹✶

✷✳✶✳ ▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✷✳✷✳ ❈→❝ ✤à♥❤ ❧➼ ✈➲ tè❝ ✤ë ❤ë✐ tö ❝õ❛ ❞➣② ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✸✳✶✳ ▼❡tr✐❝ ①→❝ s✉➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✳✶✳✶✳ ❱➼ ❞ö ✈➲ ♠ët sè ♠❡tr✐❝ ①→❝ s✉➜t t❤æ♥❣ ❞ö♥❣
✸✳✶✳✷✳ ◗✉❛♥ ❤➺ ❣✐ú❛ ❝→❝ ♠❡tr✐❝ ①→❝ s✉➜t ✳ ✳ ✳ ✳ ✳ ✳
✸✳✷✳ ❚♦→♥ tû ❚r♦tt❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✳✷✳✶✳ ❚➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✳✸✳ ▼❡tr✐❝ ❚r♦tt❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✳✹✳ Ù♥❣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳


❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✳
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✳
✳
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✳
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✹✶
✹✷
✹✸
✹✸
✹✸
✹✹
✹✺

✺✸

✺


✻

▼ö❝ ❧ö❝


ữỡ
tỷ rttr tr ợ tr t

r ữỡ ú t ỹ t tỷ rttr sỷ ử õ ữ ởt ữỡ
tr ự sỹ tỗ t ừ ợ tr t ờ

tỷ rttr
sỷ CB (R) t ủ tỹ tử tr (, )ợ ộ f CB (R)
q ã ữ s

ã

: CB (R) CB (R)
f

f
= sup |f (x)|
x

õ

ã

ởt ừ f tr CB (R)

t
ợ ồ f CB (R), f = sup
|f (x)| 0;
x
ợ ồ f CB (R), ợ ồ R,

f = 0 f = 0,

f = sup |f (x)| = ||. sup |f (x)| = || ||f ||
x

x

ợ ồ f, g C


B (R)

||f + g|| = sup |(f + g)(x)| sup |f (x)| + sup |g(x)| = f + g
x

x

A tr ợ C
ồ ởt t tỷ t t tr CB (R)

x

B (R)

tr tr CB (R) ữủ

A(f + g) = Af + Ag




✽

❈❤÷ì♥❣ ✶✳ ❚♦→♥ tû ❚r♦tt❡r tr♦♥❣ ❝→❝ ✤à♥❤ ❧➼ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠
∀f, g ∈ CB (R); α, β ∈ R✳

❚r♦♥❣ ♣❤➛♥ t✐➳♣ t❤❡♦ t❛ s➩ t❤❛② ❦þ ❤✐➺✉ Af ❜➡♥❣ A(f )✮✳

◆➳✉ A, B ❧➔ ♥❤ú♥❣ t♦→♥ tû t✉②➳♥ t➼♥❤ tr➯♥ CB (R)✱
(A + B)f = Af + Bf, ∀f ∈ CB (R)❀ t÷ì♥❣ tü A − B ✤÷ñ❝

❚➼❝❤ AB t❤➻ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ AB.f = A(Bf )❀ ✷ t♦→♥
AB = BA✳
❚♦→♥ tû A ❝â t➼♥❤ ❝❤➜t

Af ≤ f

t❤➻ tê♥❣ A + B ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐
①→❝ ✤à♥❤ ❜ð✐ (A − B)f = Af − Bf ✳
tû A, B ✤÷ñ❝ ❣å✐ ❧➔ ❣✐❛♦ ❤♦→♥ ♥➳✉

✈î✐ ∀f ∈ CB (R) ✤÷ñ❝ ❣å✐ ❧➔ t♦→♥ tû ❝♦✳

◆❤➟♥ ①➨t ✶✳✶✳ ❚ê♥❣✱ ❤✐➺✉ ✈➔ t➼❝❤ ❝õ❛ ✷ t♦→♥ tû t✉②➳♥ t➼♥❤ ❧➔ t♦→♥ tû t✉②➳♥ t➼♥❤✳
❈❤ù♥❣ ♠✐♥❤✳ ✿ ●å✐ A,

B

❧➔ ✷ t♦→♥ tû t✉②➳♥ t➼♥❤ tr➯♥ CB (R)✱ ✈î✐ f,

g ∈ CB (R); α, β ∈ R

❚❛ ❝â✿
A(αf + βg) = αAf + βAg
B(αf + βg) = αBf + βBg

❑❤✐ ✤â✿
✶✳
(A + B)(αf + βg) = A(αf + βg) + B(αf + βg)
= αAf + βAg + αBf + βBg
= α(A + B)f + β(A + B)g


✷✳
(A − B)(αf + βg) = A(αf + βg) − B(αf + βg)
= αAf + βAg − αBf − βBg
= α(A − B)f + β(A − B)g

✸✳
(A.B)(αf + βg) = AB(αf ) + AB(βg)
= α(AB)f + β(AB)g

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t tò② þ (Ω, A, P )✱ ❦❤✐ ✤â ❤➔♠ ♣❤➙♥ ♣❤è✐ F ❝õ❛ ✤↕✐
❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ X ✱ ❦þ ❤✐➺✉ FX (x) ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐✿

P {ω ∈ Ω : X(ω) ≤ x} = FX (x), ∀x


✾

✶✳✶✳ ❚♦→♥ tû ❚r♦tt❡r

◆➳✉ f ❧➔ ❤➔♠ sè ❜➜t ❦ý tr♦♥❣ CB (R),

∃ E{f (X)}
✂

E{f (X)} =

✈➔

f (x)dFX (x)


❚❛ ①→❝ ✤à♥❤ ♠ët t♦→♥ tû t✉②➳♥ t➼♥❤ ❧✐➯♥ ❦➳t ✈î✐ X t❤❡♦ ❝→❝❤ s❛✉✿
❈❤♦ ❜➜t ❦ý f ∈ CB (R)✱ t❛ ①→❝ ✤à♥❤✿

✂

(VX f )(y) = E{f (X + y)} =

f (x + y)dFX (x),

✈î✐ ∀y

❇ê ✤➲ ✶✳✶✳ ◆➳✉ f ❧✐➯♥ tö❝ ✤➲✉ t❤➻ V f ❝ô♥❣ ❧✐➯♥ tö❝ ✤➲✉✱ ❜à ❝❤➦♥ ✭V

Xf

X

❧➔ ❧➔ t♦→♥ tû ❝♦ tr➯♥ CB (R)✳
❈❤ù♥❣ ♠✐♥❤✳ ✿
VX

∈ CB (R)✮✳

❍ì♥ ♥ú❛

✶✳ ❈❤ù♥❣ ♠✐♥❤ VX f ❧✐➯♥ tö❝ ✤➲✉ ✈➔ ❜à ❝❤➦♥✳ ❚ù❝ ❧➔ t❛ ❝❤ù♥❣ ♠✐♥❤✿
|(VX f )(x1 ) − (VX f )(x2 )| ≤ sup |f (x + x1 ) − f (x + x2 )|
x


✈î✐ ∀x1,

x2

❚❛ ❝â✿
✂

|(VX f )(x1 ) − (VX f )(x2 )| =

✂

f (x + x1 )dFX (x) −

f (x + x2 )dFX (x)

✂

≤
≤

✂
✂

≤

[f (x + x1 ) − f (x + x2 )] dFX (x)
|f (x + x1 ) − f (x + x2 )|dFX (x)
sup |f (x + x1 ) − f (x + x2 )| dFX (x)
x


≤ sup |f (x + x1 ) − f (x + x2 )|
x

✷✳ ❈❤ù♥❣ ♠✐♥❤ VX ❧➔ t♦→♥ tû ❝♦✳
❚❛ ❝â✿
✂

|(VX )f (x)| =

f (x + y)dFX (x)

✂

≤

✂

≤
=
=⇒ |(VX f )(x)| ≤ f , ∀x =⇒ |VX f | ≤ f
=⇒ VX ❧➔ t♦→♥ tû ❝♦.

|f (x + y)| dFX (x)
sup |f |dFX (x)
x
✂

f

dFX (x) = f



✶✵

❈❤÷ì♥❣ ✶✳ ❚♦→♥ tû ❚r♦tt❡r tr♦♥❣ ❝→❝ ✤à♥❤ ❧➼ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠

❈❤ó þ ✶✳✶✳ ✶✳ ◆➳✉ ❤❛✐ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ❝â ❝ò♥❣ ♣❤➙♥ ♣❤è✐ t❤➻ t♦→♥ tû ❧✐➯♥
❦➳t ❧➔ ❣✐è♥❣ ♥❤❛✉✳

❈❤ù♥❣ ♠✐♥❤✳ ✿ ●✐↔ sû X1, X2 ✤ë❝ ❧➟♣ ❝â ❝ò♥❣ ♣❤➙♥ ♣❤è✐ F ✳ ❑❤✐ ✤â✱ t❛ ❝â✿
✂

VX1 f (y) =

✂

VX2 f (y) =

✂

f (x1 + y)dFX (x1 ) =

✂

f (x2 + y)dFX (x2 ) =

f (x + y)dFX
f (x + y)dFX ; ∀y

=⇒ VX1 f (y) = VX2 f (y)


✷✳ ◆➳✉ X1, X2 ❧➔ ❤❛✐ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ✈î✐ ❤➔♠ ♣❤➙♥ ♣❤è✐ F1,
t❤➻✿

F2

t÷ì♥❣ ù♥❣✱

VX1 +X2 = VX2 VX1

❈❤ù♥❣ ♠✐♥❤✳ ✿
●✐↔ sû X1, X2 ❧➔ ❤❛✐ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ✈î✐ ❤➔♠ ♣❤➙♥ ♣❤è✐ F1, F2✳ ❚❤❡♦ ✤à♥❤
♥❣❤➽❛✱
(VX1 +X2 )(x) = E {f (X1 + X2 + x)} .

❱➻ X1,

X2

✤ë❝ ❧➟♣

☎

E {f (X1 + X2 + x)} =
=
=

✂
✂


f (x1 + x2 + x)dF1X (x1 )dF2X (x2 )
✂

f (x1 + x2 + x)dF1X (x1 ) dF2X (x2 )
(VX1 f )(x2 + x)dF2X (x2 ) = (VX2 VX1 f )(x)

❉♦ ✤â✿ VX +X = VX VX
❈❤ó þ ✶✳✷✳ ❇➡♥❣ ❝→❝❤ ✤↔♦ ♥❣÷ñ❝ t❤ù tü ❝õ❛ ♣❤➨♣ ❧➜② t➼❝❤ ♣❤➙♥ ð tr➯♥ t❛ ❝â ❝→❝❤ ✈✐➳t ❦❤→❝
❧➔ VX VX ✱ ✤✐➲✉ ♥➔② ❝❤♦ t❤➜② VX , VX ❣✐❛♦ ❤♦→♥✳
1

1

2

2

2

1

1

2

❍ì♥ ♥ú❛✱ ✈î✐ n ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ X1, X2, . . . , Xn t❤➻✿
VX1 +V2 +...+Xn = VX1 .VX2 . . . . VXn

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ▼ët ❞➣② ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ X ,


✈î✐ ❤➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t
t÷ì♥❣ ù♥❣ F1, F2, . . . ✤÷ñ❝ ❣å✐ ❧➔ ❤ë✐ tö t❤❡♦ ♣❤➙♥ ♣❤è✐ ✤➳♥ ♠ët ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ X ✈î✐
❤➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t F ♥➳✉ n→∞
lim Fn (y) = F (y), ∀y ♠➔ t↕✐ ✤â F (y) ❧✐➯♥ tö❝ ✭ tù❝ ❤➔♠ ♣❤➙♥
♣❤è✐ ①→❝ s✉➜t ❦❤æ♥❣ ❣✐↔♠ ✈➔ ❧✐➯♥ tö❝ ♣❤↔✐✱ ①→❝ ✤à♥❤ ♠ët ❣✐→ trà t↕✐ ♠å✐ ✤✐➸♠ ♠➔ ♥â ❧✐➯♥ tö❝✮✳
1

X2 , . . .


✶✶

✶✳✶✳ ❚♦→♥ tû ❚r♦tt❡r

❇ê ✤➲ ✶✳✷✳ ◆➳✉✿ 0 ≤ f (x) ≤ 1 ✈➔
f (x) =

t❤➻

1 , x∈A
0 , x∈B

P {X ∈ A} ≤ E{f (X)} ≤ 1 − P {X ∈ B}

❈❤ù♥❣ ♠✐♥❤✳ ✿ ❚❛ ❝â✿
✂∞

E(f (X)) =

f (x)dFX (x)

−∞
✂

=

✂

f (x)dFX (x) +
A
✂

=

✂

dFX (x) +

R/A

f (x)dFX (x)
✂

A
✂

R/A

dFX (x) +

=


f (x)dFX (x)

A

f (x)dFX (x)
R/A

≥ P (X ∈ A)

✈➻

✁

f (x)dFX (x) ≥ 0

R/A

▼➦t ❦❤→❝ ✿
✂∞

E(f (X)) =

f (x)dFX (x)
−∞
✂

=

✂


f (x)dFX (x) +
✂

B

≤

f (x)dFX (x)
R/B

|f (x)|dFX (x)
R/B

= P (X ∈ R/B)
= P (X ∈ R) − P (X ∈ B)
= 1 − P (X ∈ B)

❚ø ✤â t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳
❇ê ✤➲ ✶✳✸✳ ●å✐ F, F1, F2, . . . ❧➔ ❤➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ❝õ❛ X, X1, X2, . . . ●✐↔ sû ② ❧➔ ✤✐➸♠
♠➔ t↕✐ ✤â F ❧✐➯♥ tö❝✳ ∀ε > 0, ∃δ s❛♦ ❝❤♦ |F (y + δ) − F (y − δ)| < ε✳
❈❤ù♥❣ ♠✐♥❤✳ ✿ ❱➻ F ❧✐➯♥ tö❝ t↕✐ y ♥➯♥ ∀ε > 0, ∃δ > 0 s❛♦ ❝❤♦ ∀x ∈ [y − δ; y + δ] t❤➻
|F (x) − F (y)| < 2ε ✳


✶✷

❈❤÷ì♥❣ ✶✳ ❚♦→♥ tû ❚r♦tt❡r tr♦♥❣ ❝→❝ ✤à♥❤ ❧➼ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠

❑❤✐ ✤â t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ❧✐➯♥ tö❝ t↕✐ ♠ët ✤✐➸♠ t❛ ❝â✿

✈➔

|F (y + δ) − F (y)| <

ε
2

|F (y − δ) − F (y)| <

ε
2

❙✉② r❛✿

|F (y + δ) − F (y − δ)| = |F (y + δ) − F (y) + F (y) − F (y − δ)|
≤ |F (y + δ) − F (y)| + |F (y − δ) − F (y)|
ε ε
+ =ε
2 2

<

❇ê ✤➲ ✶✳✹✳ ∀f, g ∈ C

2
B (R)

s❛♦ ❝❤♦ 0 ≤ f (x) ≤ g(x) ≤ 1, ∀x ✈➔
1 , ♥➳✉ x ≤ y − δ
f (x) =

0 , ♥➳✉ x ≥ y
1 ,
0 ,

g(x) =

❚ø ✤â s✉② r❛

♥➳✉
♥➳✉

x≤y
x≥y+δ

E{f (Xn )} ≤ Fn (y) ≤ E{g(Xn )}, ∀n

❈❤ù♥❣ ♠✐♥❤✳ ✿
✶✳

✂

✂

y−δ

P (X ≤ y − δ) =

dFX (x) = F (y − δ)

f (x)dFX (x) =

−∞

❚❤❡♦ ❜ê ✤➲ ✶✳✷

y−δ
−∞

=⇒ F (y − δ) ≤ E(f (X))

✷✳

✂

y

P (X ≤ y) =

dFX (x) = F (y)
−∞

✈➔

✂

+∞

E(f (X)) =

f (x)dFX (x)
✂−∞

y−δ

=

✂

dFX (x) +
✂−∞
y−δ

≤
= F (y)

+∞

f (x)dFX (x) +
✂y−δ
y

dFX (x) +
−∞

✂

y

dFX (x)
y−δ

f (x)dFX (x)

y


✶✸

✶✳✶✳ ❚♦→♥ tû ❚r♦tt❡r

✸✳

✂

+∞

g(x)dFX (x)

E(g(X)) =

✂

✂−∞
y

g(x)dFX (x) +

=
−∞

✂

✂


y+δ

+∞

g(x)dFX (x) +
y

g(x)dFX (x)
y+δ

y+δ

g(x)dFX (x)

= F (y) +
y

≥ F (y)

✹✳

✂

+∞

E{g(X)} =

g(x)dFX (x)
✂−∞

y+δ

=

✂

+∞

g(x)dFX (x) +
✂−∞
y+δ

≤

g(x)dFX (x)
y+δ

g(x)dFX (x)
−∞

= F (y + δ)

❚â♠ ❧↕✐✱
❍ì♥ ♥ú❛✱

F (y − δ) ≤ E {f (X)} ≤ F (y) ≤ E {g(X)} ≤ F (y + δ)
E {f (Xn )} ≤ E {f (X)} ≤ Fn (y) ≤ E {g(Xn )} , ∀n

❇ê ✤➲ ✶✳✺✳ ✣✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ ♠ët ❞➣② ❝→❝ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ X , X , . . . ❤ë✐ tö t❤❡♦ ♣❤➙♥
1


♣❤è✐ ✤➳♥ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ X ❧➔✿

2

∗

lim VXn f − VX ∗ f = 0, ∀f ∈ CB2 (R)

n→∞

❈❤ù♥❣ ♠✐♥❤✳ ✿ ▲➜② ❝→❝ ❤➔♠ f, g ∈ CB2 (R) t❤ä❛ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❜ê ✤➲ ✶✳✺

❚❛ ❝â✿
lim VXn f − VX ∗ f

n→∞

= 0

⇐⇒ lim sup |VXn f − VX ∗ f | = 0
n→∞

x

=⇒ lim |(VXn f − VX ∗ f )(0)| = 0
n→∞

⇐⇒ lim (VXn f )(0) = (VX f )(0)
n→∞


=⇒ lim E{f (Xn )} = E{f (X)}
n→∞

❚÷ì♥❣ tü✿

E{g(X)} = lim E{g(Xn )}
n→∞


✶✹

❈❤÷ì♥❣ ✶✳ ❚♦→♥ tû ❚r♦tt❡r tr♦♥❣ ❝→❝ ✤à♥❤ ❧➼ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠

❑❤✐ ✤â✱

E{f (X)} ≤ lim inf Fn (y) ≤ lim sup Fn (y) ≤ E{g(X)}
n→∞

❚ø ✤â

n→∞

F (y) − ε ≤ lim inf Fn (y) ≤ lim sup Fn (y) ≤ F (y) + ε,
n→∞

n→∞

✈➔ ✤✐➲✉ ♥➔② ❝á♥ ✤ó♥❣ ✈î✐ ∀ε > 0,


lim Fn (y) = F (y).

n→∞

❚r♦♥❣ ♣❤➛♥ t✐➳♣ t❤❡♦ t❛ ❞ò♥❣ X ∗ ✤➸ ❦➼ ❤✐➺✉ ❝❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝❤✉➞♥ ❤â❛❀ tù❝ ❧➔ X ∗ ❧➔
✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ✈î✐ ❤➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t
1
FX ∗ (x) = √
2π

✂x

u2

e− 2 du
−∞

❚❛ ♥❤➢❝ ❧↕✐ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ t➢❝✿
✶✳ ◆➳✉ X ∗ ∈ N (0, 1) t❤➻ σ ∈ N (0, σ2). ❚❤➟t ✈➟②✱
x

P (σX ∗ < x) = P r(X ∗ <

1
x
)= √
σ
2π

✂σ


e

−t2
2

dt

0

✣➦t u = σt ⇒ du = σdt
t
✉

0
0

x
σ
x

1
⇒ P (σX ∗ < x) = √
σ 2π
⇒ σX ∗ ∈ N (0; σ 2 )

✷✳ ◆➳✉ = X1∗, X2∗ ∈ N (0,
❚❛ ❝â✿

1)


✂x

−u2

e 2σ2 du
0

✈➔ X1∗, X2∗ ✤ë❝ ❧➟♣ t❤➻ σ1X1∗ + σ2X2∗ ∈ N (0,

σ12 + σ22 ).

E(σ1 X1∗ + σ2 X2∗ ) = E(σ1 X1∗ ) + E(σ2 X2∗ )
= σ1 E(X1∗ ) + σ2 E(X2∗ )
= 0

❱➔
D(σ1 X1∗ + σ2 X2∗ ) = D(σ1 X1∗ ) + D(σ2 X2∗ )
=

E(σ1 X1∗ − (E(σ1 X1∗ ))2 + E(σ2 X2∗ )2 − (E(σ2 X2∗ ))2


✶✳✷✳ ✣à♥❤ ❧þ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠ ❝❤♦ ❝→❝ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ❝â ❝ò♥❣ ♣❤➙♥ ♣❤è✐✿ ✶✺
= σ12 E(X1∗ )2 − (E(X1∗ ))2 + σ22 E(X2∗ )2 − (E(X2∗ ))2
= σ12 D(X1∗ ) + σ22 D(X2∗ )
= σ12 + σ22

❍ì♥ ♥ú❛✱ t❛ ❝ô♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ σ1X1∗ + σ2X2∗ ❝â ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥✳
=⇒ σ1 X1∗ + σ2 X2∗ ∈ N (0, σ12 + σ22 )


❚ø ✤â t❛ ❝â t♦→♥ tû ❧✐➯♥ ❦➳t✿
Vσ1 X1∗ +σ2 X2∗ = Vσ1 X1∗ Vσ2 X2∗ = VσX ∗

tr♦♥❣ ✤â σ = (σ12 + σ22)

1
2

.

✶✳✷✳ ✣à♥❤ ❧þ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠ ❝❤♦ ❝→❝ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝
❧➟♣ ❝â ❝ò♥❣ ♣❤➙♥ ♣❤è✐✿
●✐↔ sû X ❧➔ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ✭✈î✐ ❤➔♠ ♣❤➙♥ ♣❤è✐ FX ✮ ❝â✿
✂

E(X) =

✂

xdFX (x) = 0, D(X) =

x2 dFX (x) = 1

❑❤✐ ✤â✱ ♠ët ❞↕♥❣ ❝õ❛ ✤à♥❤ ❧þ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠ ❝â t❤➸ ✤÷ñ❝ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✿
✣à♥❤ ❧þ ✶✳✶✳ ❈❤♦ X1, X2, . . . ❧➔ ❞➣② ❝→❝ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ❝ò♥❣ ♣❤➙♥ ♣❤è✐ ✈î✐
✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ X ✱ ✈➔ ✤➦t✿
1
Sn = √ (X1 + X2 + . . . + Xn )
n


❑❤✐ ✤â ❝→❝ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ Sn ❤ë✐ tö t❤❡♦ ♣❤➙♥ ♣❤è✐ ✤➳♥ ♠ët ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝❤✉➞♥
❤â❛ X ∗✳
❇ê ✤➲ ✶✳✻✳ ❈❤♦ A, B ❧➔ ❤❛✐ t♦→♥ tû ❝♦ ❣✐❛♦ ❤♦→♥ ✈î✐ ♥❤❛✉✳ ❑❤✐ ✤â✱ ❝❤♦ ❜➜t ❦➻ f ∈ CB (R)
An f − B n f ≤ n Af − Bf

❈❤ù♥❣ ♠✐♥❤✳ ✣➸ t❤➜② ✤÷ñ❝ ✤✐➲✉ ♥➔②✱ t❛ ①➨t✿
n−1

An f − B n f =

An−i−1 (A − B)B i f
i=0
n−1

An−i−1 B i (A − B)f

=
i=0


✶✻

❈❤÷ì♥❣ ✶✳ ❚♦→♥ tû ❚r♦tt❡r tr♦♥❣ ❝→❝ ✤à♥❤ ❧➼ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠
n−1

=⇒ An f − B n f

An−i−1 B i (A − B)f


=
i=0
n−1

An−i−1 B i (A − B)f

≤
i=0
n−1

B i (A − B)f

≤
i=0
n−1

≤

(A − B)f
i=0

= n Af − Bf

❚ø ❜ê ✤➲ ✶✳✺✱ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧þ ✶✳✶ t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤
lim VSn f − VX ∗ f = 0, ∀f ∈ CB2 (R)

✣➦t σ = √1n , t❛ ❝â✿
VSn = Vσ(X1 +X2 +...+Xn )
= VσX1 VσX2 . . . VσXn


❉➣② X1, X2, ..., Xn ✤ë❝ ❧➟♣ ❝ò♥❣ ♣❤➙♥ ♣❤è✐ ♥➯♥✿
VSn = VσX VσX . . . VσX
n
= VσX

❚÷ì♥❣ tü t❛ ❝â✿

n
VX ∗ = VσX
∗

⑩♣ ❞ö♥❣ ❜ê ✤➲ ✶✳✻✿
VSn f − VX ∗ f

=

n
n
VσX
f − VσX
∗f

≤ n VσX f − VσX ∗ f

◆❤÷ ✈➟② ♠✉è♥ ❝❤ù♥❣ ♠✐♥❤
lim VSn f − VX ∗ f = 0, ∀f ∈ CB2 (R)

t❤➻ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤
lim n VσX f − VσX ∗ f = 0, ∀f ∈ CB2 (R)


n→∞


✶✳✷✳ ✣à♥❤ ❧þ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠ ❝❤♦ ❝→❝ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ❝â ❝ò♥❣ ♣❤➙♥ ♣❤è✐✿ ✶✼
❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ f ∈ CB2 (R)✱ t❛ ❝â ❝❤✉é✐ ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r ❞↕♥❣✿
1
f (x + y) = f (y) + xf (y) + x2 f ”(η)
2
1
1
= f (y) + xf (y) + x2 f ”(y) + x2 (f ”(η) − f ”(y))
2
2

✈î✐ y ≤ η ≤ y + x
❉♦

f ∈ CB2 (R)✱ f
|f ”(η) − f ”(y)| < ε✳

❧✐➯♥ tö❝ ✤➲✉✱ ♥➯♥

∀ε > 0, ∃δ > 0

s❛♦ ❝❤♦

∀η

t❤ä❛|η − y|


< δ

t❤➻

❇➙② ❣✐í t❛ t➼♥❤ t♦→♥ ♠ët ①➜♣ ①➾ tî✐ VσX f.
✂

(VσX f )(y) =

f (y + σx)dFX (x)
✂

= f (y)

✂

dFX (x) + f (y)σ

xdFX (x)

✂

✂

1
1
f ”(y)σ 2 x2 dFX (x) + σ 2 (f ”(η) − f ”(y))x2 dFX (x)
+
2
2

✂
1
1
= f (y) + σ 2 f ”(y) + σ 2 (f ”(η) − f ”(y))x2 dFX (x)
2
2
✂
1 2
1 2
= f (y) + σ f ”(y) + σ
(f ”(η) − f ”(y))x2 dFX (x)
2
2
+

1 2
σ
2

|x|< σδ

✂

(f ”(η) − f ”(y))x2 dFX (x)

|x|≥ σδ

1
1 2
⇒ |(VσX f )(y) − f (y) − σ 2 f ”(y)| =

σ
2
2

✂

(f ”(η) − f ”(y))x2 dFX (x)
|x|< σδ

✂

(f ”(η) − f ”(y))x2 dFX (x)

+
|x|≥ σδ

1 2
≤
σ
2

✂

(f ”(η) − f ”(y))x2 dFX (x)
|x|< σδ

1 2
+
σ
2


✂

(f ”(η) − f ”(y))x2 dFX (x)
|x|≥ σδ


≤

1 2
σ 
2

✂

|x|< σδ

✂

εx2 dFX (x) + 2 f ”



x2 dFX (x)

|x|≥ σδ


✶✽


❈❤÷ì♥❣ ✶✳ ❚♦→♥ tû ❚r♦tt❡r tr♦♥❣ ❝→❝ ✤à♥❤ ❧➼ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠
1 2
σ . 2ε = σ 2 . ε
2
= εn−1

≤
δ
✭❑❤✐
✁ |x| < σ ,

|η − y| ≤ |σx| < δ
2
x dFX (x) = 0)

lim

k→∞

✈➔ ∀η,

|f ”(η) − f ”(y)| ≤ 2 f ”

✱

✁

x2 dFX (x)

❧➔ ❤ú✉ ❤↕♥


|x|≥k

❉♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝❤✉➞♥ ❤â❛ ❝ô♥❣ ❝â E(X ∗) = 0,

(VσX ∗ f )(y).

D(X ∗ ) = 1✳

×î❝ ❧÷ñ♥❣ t÷ì♥❣ tü ❝❤♦

1
⇒ |(VσX ∗ f )(y) − f (y) − σ 2 f ”(y)| ≤ εn−1
2

▲➜② ❤✐➺✉ ❣✐ú❛ ❤❛✐ ÷î❝ ❧÷ñ♥❣
1
1
|(VσX f )(y) − (VσX ∗ f )(y)| = |(VσX f )(y) − f (y) − σ 2 f ”(y) − (VσX ∗ f )(y) + f (y) + σ 2 f ”(y)|
2
2
1 2
1
≤ |(VσX f )(y) − f (y) − σ f ”(y)| + |(VσX ∗ f )(y) − f (y) − σ 2 f ”(y)|
2
2
≤ 2εn−1

❤❛②


n VσX f − VσX ∗ f ≤ 2ε

❈❤å♥ ❣✐→ trà n ✤õ ❧î♥✳ ✣✐➲✉ ♥➔② ✤ó♥❣ ❝❤♦ ❜➜t ❦➻ ε > 0 ✈➔ ❞♦ ✤â
lim n VσX f − VσX ∗ f = 0

n→∞

❚ø ✤â s✉② r❛✿

lim VSn f − VX ∗ f = 0, ∀f ∈ CB2 (R)

✶✳✸✳ ✣à♥❤ ❧þ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠ ❝❤♦ ❝→❝ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ❦❤æ♥❣
❝ò♥❣ ♣❤➙♥ ♣❤è✐✿
●✐↔ sû r➡♥❣ X1, X2, . . . ❧➔ ❞➣② ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ✈î✐ Xi ❝â ❤➔♠ ♣❤➙♥ ♣❤è✐ FX
t÷ì♥❣ ù♥❣✳ ❑❤✐ ✤â✱ ♠é✐ Xi ❝â✿

i

✂

E(Xi ) =

✣➦t sn =

n

✂

xdFXi (x) = 0, D(Xi ) =


x2 dFXi (x) = σi2 .

1
2

σi2

i=1

❉➣② X1, X2, . . . t❤♦↔ ✤✐➲✉ ❦✐➺♥ ▲✐♥❞❡❜❡r❣ ♥➳✉✿
n

lim

n→∞

✂

s−2
n

x2 dFXi (x) = 0, ∀δ > 0
i=1

|x|≥δsn


✶✾

✶✳✸✳ ✣à♥❤ ❧þ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠ ❝❤♦ ❝→❝ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ❦❤æ♥❣ ❝ò♥❣ ♣❤➙♥ ♣❤è✐✿


✣à♥❤ ❧þ ✶✳✷✳ ❈❤♦ X , X , . . . ❧➔ ❞➣② ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ♥❤÷ tr➯♥✳
1

2

✣➦t Sn = s−1
n (X1 + X2 + . . . + Xn )
❑❤✐ ✤â✱ ♥➳✉ ✤✐➲✉ ❦✐➺♥ ▲✐♥❞❡❜❡r❣ ✤÷ñ❝ t❤ä❛ t❤➻ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ Sn ❤ë✐ tö t❤❡♦ ♣❤➙♥
♣❤è✐ ✤➳♥ ♠ët ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝❤✉➞♥ ❤â❛ X ∗ ∼ N (0, 1).
❇ê ✤➲ ✶✳✼✳ ❈❤♦ A1, A2, . . . , An; B1, B2, . . . , Bn ❧➔ ♥❤ú♥❣ t♦→♥ tû ❝♦✱ ❣✐❛♦ ❤♦→♥ ✈î✐ ♥❤❛✉✳ ❑❤✐
✤â✱ ❝❤♦ ❜➜t ❦ý f ∈ CB (R)
n

||A1 A2 . . . An f − B1 B2 . . . Bn f || ≤

Ai f − Bi f
i=1

❈❤ù♥❣ ♠✐♥❤✳ ✿ ❚❛ ❝â✿
n

A1 A2 . . . An f − B1 B2 . . . Bn f =

A1 A2 . . . Ai−1 (Ai − Bi )Bi+1 Bi+2 . . . Bn f
i=1
n

A1 A2 . . . Ai−1 .Bi+1 Bi+2 . . . Bn (Ai − Bi )f


=
i=1

n

=⇒ ||A1 A2 . . . An f − B1 B2 . . . Bn f || =

A1 A2 . . . Ai−1 .Bi+1 Bi+2 . . . Bn (Ai − Bi )f
i=1
n

≤

A1 A2 . . . Ai−1 .Bi+1 Bi+2 . . . Bn (Ai − Bi )f
i=1
n

≤

Bi+1 Bi+2 . . . Bn (Ai − Bi )f
i=1
n

≤

(Ai − Bi )f
i=1
n

Ai f − Bi f


=
i=1

❇ê ✤➲ ✶✳✽✳ ●å✐ X ❧➔ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ❝â ❤➔♠ ♣❤➙♥ ♣❤è✐ F ✱ ❝â E(X ) = 0, D(X ) = 1✳
∗

X∗

∗

∗

◆➳✉ ①→❝ ✤à♥❤ Xi∗ = σiX ∗✱ ❦❤✐ ✤â Xi∗ ❝â ♣❤÷ì♥❣ s❛✐ ❣✐è♥❣ ♥❤÷ Xi.
◆➳✉ ❞➣② X1, X2, . . . t❤ä❛ ✤✐➲✉ ❦✐➺♥ ▲✐♥❞❡❜❡r❣✱ t❤➻ ❞➣② X1∗, X2∗, . . . ❝ô♥❣ t❤ä❛ ✤✐➲✉ ❦✐➺♥ ▲✐♥❞❡✲
❜❡r❣✳
❈❤ù♥❣ ♠✐♥❤✳ ✣➦t kn = max
{σi s−1
n }
i≤n

❚❛ ❝❤ù♥❣ ♠✐♥❤ ♥➳✉ X1, X2, . . . t❤ä❛ ✤✐➲✉ ❦✐➺♥ ▲✐♥❞❡❜❡r❣✱ t❤➻ n→∞
lim kn = 0✱ ❦❤✐ ✤â ✤✐➲✉ ❦✐➺♥
❝õ❛ X1∗, X2∗, . . . ❝ô♥❣ ✤÷ñ❝ t❤ä❛✳


✷✵

❈❤÷ì♥❣ ✶✳ ❚♦→♥ tû ❚r♦tt❡r tr♦♥❣ ❝→❝ ✤à♥❤ ❧➼ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠


❱➻ kn = max
{σi s−1
n } ♥➯♥ ∃j ∈ 1,
i≤n
❚❛ ❝â✿

n : σj = kn sn

✂

n

✂

s−2
n

s−2
n

2

x dFXi (x) ≥
i=1

|x|≥δsn

x2 dFXj (x)
✂


|x|≥δsn
2
s−2
n (σj

=

x2 dFXj (x))

−
|x|<δsn

2 2
2
≥ s−2
n (σj − δ sn )
= kn2 − δ 2

●✐↔ t❤✐➳t✱ s−2
n

n

✁

x2 dfi (x) → 0

i=1 |x|≥δsn

❦❤✐ n → ∞, ∀δ > 0


❉♦ ✤â✱
lim sup kn2 − δ 2 ≤ 0

n→∞

⇒ lim sup kn2 ≤ δ 2 , ∀δ > 0
n→∞

❈❤♦ δ → 0
=⇒ lim sup kn2 = 0
n→∞

=⇒ lim kn2 = 0
n→∞

=⇒ lim kn = 0
n→∞

◆➳✉ FX i ❧➔ ❤➔♠ ♣❤➙♥ ♣❤è✐ ❝õ❛
∗

✂

x2i dFX ∗ i

✂

✱ t❛ ❝â✿


✂

σi2 x2 dFX ∗ (x)

(xi ) =

≤

|x|≥δsn σi−1

|xi |≥δsn

▲➜② tê♥❣ tr➯♥ i ✈➔ ❝❤✐❛ s2n =

n
i=1

σi2 ✱

✁

✂

x2 dFX ∗ (x).

(xi ) ≤
−1
|x|≥δkn

|xi |≥δsn


x2 dFX ∗ (x) → 0

−1
|x|≥δkn

n

⇒

x2 dFX ∗ (x).

t❛ ✤÷ñ❝✿

x2i dFX ∗ i
i=1

σi2
−1
|x|≥δkn

✂

n

s−2
n

❈❤♦ ❜➜t ❦ý δ > 0,


Xi∗

❦❤✐ n → ∞ ✈➻ kn → 0

✂

s−2
n

x2i dFX ∗ i (xi ) → 0
i=1

|xi |≥δsn

❚ù❝ ❧➔ ✤✐➲✉ ❦✐➺♥ ▲✐♥❞❡❜❡r❣ ✤÷ñ❝ t❤ä❛✳


✶✳✸✳ ✣à♥❤ ❧þ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠ ❝❤♦ ❝→❝ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ❦❤æ♥❣ ❝ò♥❣ ♣❤➙♥ ♣❤è✐✿

✷✶

❱➔ ❣✐è♥❣ ♥❤÷ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ❝→❝ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ❝ò♥❣ ♣❤➙♥ ♣❤è✐✱ ✤✐➲✉ ♥➔② ✤õ ✤➸
❝❤♦ t❤➜②✿
lim ||VSn f − VX ∗ f || = 0, ∀f ∈ CB2 (R)

n→∞

❚❛ ❝â✿

VSn = VX1 s−1

VX2 s−1
. . . VXn s−1
n
n
n

✈➔

. . . Vσn X ∗ s−1
Vσ2 X ∗ s−1
VX ∗ = Vσ1 X ∗ s−1
n
n
n

❚❤❡♦ ❜ê ✤➲ ✶✳✼✱

=⇒ ||VSn f − VX ∗ f || = ||VX1 s−1
VX2 s−1
. . . VXn s−1
f − Vσ1 X ∗ s−1
Vσ2 X ∗ s−1
. . . Vσn X ∗ s−1
f ||
n
n
n
n
n
n

n

≤

VXi s−1
f − Vσi X ∗ s−1
f
n
n
i=1

✣à♥❤ ❧þ ✶✳✷ s➩ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤ ♥➳✉ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝
n

VXi s−1
f − Vσi X ∗ s−1
f = 0, ∀f ∈ CB2 (R)
n
n

lim

n→∞

i=1

❚ù❝ ❧➔✱ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤
n

|VSi s−1

f − Vσi X ∗ s−1
f | = 0, ∀f ∈ CB2 (R)
n
n

lim sup

n→∞

i=1

❈❤ù♥❣ ♠✐♥❤✳ ❚➼♥❤ t♦→♥ ♥❤÷ ð ❜ê ✤➲ ✶✳✺

✣➦t σ = s−1
n , F = Fi ✈➔

✁

x2 dFi (x) = σi2

❈❤♦ f ∈ CB2 (R) t❛ ❝â ❝❤✉é✐ ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r ❞↕♥❣✿
1
f (xi σ + y) = f (y) + σxi f (y) + σ 2 x2i f ”(η)
2
1
1
= f (y) + σxi f (y) + σ 2 x2i f ”(y) + σ 2 x2i (f ”(η) − f ”(y))
2
2


✈î✐ y ≤ η ≤ y + σxi
❉♦ f

∈ CB2 (R)✱ f
|f ”(η) − f ”(y)| < ε✳

❧✐➯♥ tö❝ ✤➲✉✱ ♥➯♥ ∀ε

> 0, ∃δ > 0

❇➙② ❣✐í t❛ t➼♥❤ t♦→♥ ♠ët ①➜♣ ①➾ tî✐ VσX f.
i

✂

(VσXi f )(y) =

f (y + σx)dFXi (x)

s❛♦ ❝❤♦ ∀η t❤ä❛ |η − y|

< δ

t❤➻


✷✷

❈❤÷ì♥❣ ✶✳ ❚♦→♥ tû ❚r♦tt❡r tr♦♥❣ ❝→❝ ✤à♥❤ ❧➼ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠
✂


= f (y)

✂

dFXi (x) + f (y)σ

xdFXi (x)

✂

✂

1
1
f ”(y)σ 2 x2 dFXi (x) + σ 2 (f ”(η) − f ”(y))x2 dFXi (x)
+
2
2
✂
1 2 2
1 2
= f (y) + σ σi f ”(y) + σ
(f ”(η) − f ”(y))x2 dFXi (x)
2
2
✂
1 2
1 2 2
(f ”(η) − f ”(y))x2 dFXi (x)

= f (y) + σ σi f ”(y) + σ
2
2
+

1 2
σ
2

✂

|x|<δσ

(f ”(η) − f ”(y))x2 dFXi (x)

|x|≥δσ

✂

1 2
1
⇒ |(VσXi f )(y) − f (y) − σ 2 σi2 f ”(y)| =
σ
2
2

(f ”(η) − f ”(y))x2 dFXi (x)
|x|<δσ

✂


(f ”(η) − f ”(y))x2 dFXi (x)

+
|x|≥δσ

✂

1 2
≤
σ
2

(f ”(η) − f ”(y))x2 dFXi (x)
|x|<δσ

✂

1 2
+
σ
2

(f ”(η) − f ”(y))x2 dFXi (x)
|x|≥δσ


≤

1 2

σ 
2

✂



✂

εx2 dFXi (x) + 2 f ”

|x|<δσ

1 2 2
≤
εσ σ + ||f ”||σ 2
2 i


x2 dFXi (x)
|x|≥δσ

✂

x2 dFXi (x)

|x|≥δσ

✭❑❤✐ |x| < δσ,


|η − y| ≤ | σx | < δ

✈➔ ∀η,

|f ”(η) − f ”(y)| ≤ 2 f ” .✮

◆â✐ ❝→❝❤ ❦❤→❝✱ t❛ ❝â✿

✂

1
1 2 −2
−2
|(VXi s−1
f )(y) − f (y) − σi2 s−2
n f ”(y)| ≤ εσi sn + ||f ”||sn
n
2
2

x2 dFXi (x)

|x|≥δs−1
n

×î❝ ❧÷ñ♥❣ t÷ì♥❣ tü ❝❤♦ (Vσ X s
i

∗ −1
n


f )(y).

1
1
−2
⇒ |(Vσi X ∗ s−1
f )(y) − f (y) − σi2 sn−2 f ”(y)| ≤ εσi2 s−2
n + ||f ”||sn
n
2
2

✂

|x|≥δs−1
n

x2 dFX ∗ i (x)


✶✳✸✳ ✣à♥❤ ❧þ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠ ❝❤♦ ❝→❝ ✤↕✐ ❧÷ñ♥❣ ♥❣➝✉ ♥❤✐➯♥ ❦❤æ♥❣ ❝ò♥❣ ♣❤➙♥ ♣❤è✐✿

❚r♦♥❣ ✤â✱ FX i ❧➔ ❤➔♠ ♣❤➙♥ ♣❤è✐ ❝õ❛ σiX ∗.
∗

✣➦t ξi, n = s−2
n

✁


x2 dFXi (x), ξi,

n

= s−2
n

|x|≥δs−1
n

✁

x2 dFX ∗ i (x)

|x|≥δs−1
n

❑❤✐ ✤â✱
1
f )(y) − f (y) − σi2 s−2
(VXi s−1
n f ”(y)
n
2
1
f )(y) − f (y) − σi2 s−2
− (Vσi X ∗ s−1
n f ”(y)
n

2
1
≤ (VXi s−1
f )(y) − f (y) − σi2 s−2
n f ”(y)
n
2
1
f )(y) − f (y) − σi2 s−2
+ (Vσi X ∗ s−1
n f ”(y)
n
2

f )(y)| =
f )(y) − (Vσi X ∗ s−1
=⇒ |(VXi s−1
n
n

≤ εσi2 s−2
n + ||f ”||(ξi,

n

+ ξi, n )

▲➜② tê♥❣ t❛ ✤÷ñ❝✿
n


n

|(VXi s−1
f )(y) − (Vσi X ∗ s−1
f )(y)| ≤ ε + ||f ”||
n
n
i=1

(ξi,

n

+ ξi, n ).

i=1

❚❤❡♦ ❜ê ✤➲ ✶✳✽✱ ❞➣② σ1X1∗, σ2X2∗, . . . ❝ô♥❣ t❤ä❛ ✤✐➲✉ ❦✐➺♥ ▲✐♥❞❡❜❡r❣✳
n

=⇒ lim

n

(ξi, n ) = lim

n→∞

i=1


n→∞

ξi, n ) = 0
i=1

n

|VXi s−1
f − Vσi X ∗ s−1
f | ≤ ε, ∀ε > 0
n
n

=⇒ lim sup
n→∞

i=1

❈❤♦ ε → ∞
n

|VXi s−1
f − Vσi X ∗ s−1
f| = 0
n
n

=⇒ lim sup
n→∞


i=1
n

VXi s−1
f − Vσi X ∗ s−1
f
n
n

=⇒ lim

n→∞

i=1

= 0, ∀f ∈ C 2

✷✸


✷✹

❈❤÷ì♥❣ ✶✳ ❚♦→♥ tû ❚r♦tt❡r tr♦♥❣ ❝→❝ ✤à♥❤ ❧➼ ❣✐î✐ ❤↕♥ tr✉♥❣ t➙♠


ữỡ
ố ở ở tử tr ợ tr t

r ữỡ t s tố ở ở tử tr ợ tr t
ổ ử t tỷ rttr

sỷ ữủ X1,

X2 , . . .

õ 0 < D(Xn) < t

n

n

Tn = s1
n

[Xk E(Xk )], s2n = D
k=1

õ ữủ X1,

[Xk E(Xk )]
k=1

X2 , . . .

ữủ ồ tọ ợ tr t
FTn (x) FX (x), x

ợ (n ) FT (x) ồ ố ừ tờ õ Tn X
õ ố t N (0, 1)
n


ỵ CB (R) ợ tỹ tử tr R
CBr1 (R) = f CB (R) : f (j) CB (R), 1 j r 1


ữủ X ợ ố F

t
tr R õ ổ tt ố r r ổ j à(j) ừ
X ữủ ữ s


r

|x|r dFX (x) < +

r = E(|X| ) =
R



xj d[FX (x) FX (x)] = 0, (0

à(j) =
R

ú ỵ


j < r)


X (x)