Mở rộng luật yếu số lớn kolmogorov feller

✶

▼Ö❈ ▲Ö❈

▼ð ✤➛✉

✷

✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✹

✶✳✶

❑❤→✐ ♥✐➺♠ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✷

❈→❝ sè ✤➦❝ tr÷♥❣ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✸

▼ët sè ❦❤→✐ ♥✐➺♠ ❤ë✐ tö ❝õ❛ ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✳ ✳ ✳ ✳

✺

✶✳✹

❚➼♥❤ ✤ë❝ ❧➟♣✱ ✤ë❝ ❧➟♣ ✤æ✐ ♠ët✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✺

▼ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ▲✉➟t sè ❧î♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✻

▼ët sè ❜➜t ✤➥♥❣ t❤ù❝ ❝ì ❜↔♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✷ ▼ð rë♥❣ ❧✉➟t ②➳✉ sè ❧î♥ ❑♦❧♠♦❣♦r♦✈✲ ❋❡❧❧❡r✳

✶✸

✷✳✶

▲✉➟t ②➳✉ sè ❧î♥ ❑♦❧♠♦❣♦r♦✈✲ ❋❡❧❧❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✷✳✷

▼ð rë♥❣ ❧✉➟t ②➳✉ sè ❧î♥ ❑♦❧♠♦❣♦r♦✈✲❋❡❧❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✷

❑➳t ❧✉➟♥

✷✾

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✵




é
t số ợ õ ởt trỏ rt q trồ tr ỵ tt st
t số ợ tr số ồ ừ
ở tử t st t số ợ tr
số ồ ừ ở tử t số ợ
t ừ ữủ ổ ố s t q ữủ
Pss s r rở
t số ợ ợ ữủ r t t q ữủ
r t
t số ợ rr ữ r ừ tỹ
t số ợ trữớ ủ ở ũ
ố ổ ỏ ọ ý ồ tỗ t r ỡ s ồ t t
t ú tổ ự t

rr

rở t số ợ

ỗ ữỡ

ữỡ tự
r ữỡ ú tổ ữ r
số trữ ừ ởt số ở tử ừ
t ở ở ổ ởt t số ợ ỗ
tớ ú tổ tr ởt số t tự ờ ử ử ự
ỵ tr ữỡ

ữỡ rở t số ợ rr
ở ừ ỗ tt t ú tổ

tr tt ỵ

t số ợ r




r ỵ tr t ú tổ tr ỵ
ỵ t q rở t số ợ r r
ữủ tỹ t trữớ ồ ữợ sỹ ữợ t
t ừ P t
tọ ớ ỡ s s t tợ P ữớ t
t t ữợ t tr sốt q tr ồ t t
ỷ ớ ỡ tợ P P ự

r ỏ P r ỗ tớ t t
ỡ ừ t ổ tr t
t ố ũ t ỡ trữớ P
ý ỗ t tr ợ ồ
st tố ồ ở t t ú ù t
tr sốt tớ ồ t t
ũ õ ố ữ ổ t tr ọ ỳ
t sõt rt ữủ ỳ ớ qỵ ừ
t ổ ồ

t




✹

❈❍×❒◆● ✶

❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
❚r♦♥❣ t♦➔♥ ❜ë ❧✉➟♥ ✈➠♥ t❛ ❧✉æ♥ ❣✐↔ sû ( Ω, F, P ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t
❝è ✤à♥❤✳

✶✳✶ ❑❤→✐ ♥✐➺♠ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû G ❧➔ σ ✲ ✤↕✐ sè ❝♦♥ ❝õ❛ σ✲ ✤↕✐ sè F ✳ ❑❤✐ ✤â →♥❤
①↕ X : Ω → R ✤÷ñ❝ ❣å✐ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ G ✲ ✤♦ ✤÷ñ❝ ♥➳✉ ♥â ❧➔ →♥❤ ①↕

G/B(R) ✤♦ ✤÷ñ❝ ✭tù❝ ❧➔ ✈î✐ ♠å✐ B ∈ B(R)) t❤➻ X −1 (B) ∈ G ✮✳
◆➳✉ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ❝❤➾ ♥❤➟♥ ❤ú✉ ❤↕♥ ❣✐→ trà✱ t❤➻ ♥â ✤÷ñ❝ ❣å✐ ❧➔ ❜✐➳♥
♥❣➝✉ ♥❤✐➯♥ ✤ì♥ ❣✐↔♥✳

✶✳✶✳✷ ❱➼ ❞ö✳ ✳●✐↔ sû A ∈ F ✳ ✣➦t
IA (ω) =

1 ♥➳✉ ω ∈ A✱
0 ♥➳✉ ω ∈ A.

❑❤✐ ✤â IA ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ì♥ ❣✐↔♥✳

✶✳✷ ❈→❝ sè ✤➦❝ tr÷♥❣ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû X : (Ω, F, P) → (R, B(R)) ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✳
❑❤✐ ✤â t➼❝❤ ♣❤➙♥ ▲❡❜❡s❣✉❡ ❝õ❛ X t❤❡♦ ✤ë ✤♦ P ✭♥➳✉ tç♥ t↕✐✮ ✤÷ñ❝ ❣å✐ ❧➔ ❦➻

✈å♥❣ ❝õ❛ X ✈➔ ❦➼ ❤✐➺✉ ❧➔ EX ✳ ❱➟②

EX =
✳

XdP
Ω




t
EX =

xi pi X rớ r tr x1 , x2 , ... ợ P(X = xi ) = pi
i
+




xp(x)dx X tử õ t ở



ờ qt f : R R ữủ Y = f (X) t


i f (x)i pi X rớ r tr x1 , x2 , ... ợ P(X = xi ) = pi
EY = +

f (x)p(x)dx X tử õ t ở


sỷ X õ
số DX := E(X EX)2 tỗ t ữủ ồ ữỡ s ừ X

t ứ tr t t ừ ồ s r r
ữỡ s DX ừ X õ t tỗ t ổ tỗ t
tỗ t t õ t ữủ t t ổ tự

DX =

(xi EX)2 pi X rớ r ợ P(X = xi ) = pi
+

(x EX)2 p(x)dx X tử õ t ở





ởt số ở tử ừ
sỷ {Xn , n 1} ũ tr ổ
st (, F, P )

{Xn, n 1} ữủ ồ ở
tử t st X n ợ ồ > 0 t
õ

lim P (|Xn X| > ) = 0.

n
P

ỵ Xn X.


✻

✶✳✸✳✷ ✣à♥❤ ♥❣❤➽❛✳ ❉➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ {Xn, n ≥ 1} ✤÷ñ❝ ❣å✐ ❧➔ ❤ë✐
tö ❤➛✉ ❝❤➢❝ ❝❤➢♥ ✤➳♥ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ✭❦❤✐ n → ∞✮ ♥➳✉
P

ω : lim Xn (ω) = X(ω)
n→∞

= 1.

h.c.c

❑þ ❤✐➺✉ Xn −→ X ❤♦➦❝ lim Xn = X h.c.c.
n→∞

✶✳✸✳✸ ✣à♥❤ ♥❣❤➽❛✳ ❚❛ ♥â✐ ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ {Xn, n ≥} ❤ë✐ tö t❤❡♦
tr✉♥❣ ❜➻♥❤ ❝➜♣ p (p > 0) ✤➳♥ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ✭❦❤✐ n → ∞) ♥➳✉
lim E|Xn − X|p = 0.

n→∞

❑þ ❤✐➺✉

Lp

Xn −→ X (n → ∞)✳

✶✳✸✳✹ ✣à♥❤ ❧þ✳ ❉➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ {Xn, n ≥ 1} ❤ë✐ tö ❤➛✉ ❝❤➢❝
❝❤➢♥ ✤➳♥ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈î✐ ε > 0 ❜➜t ❦ý✱
lim P

n→∞

sup |Xk − X| > ε

= 0.

k≥n

❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ ♠é✐ ε > 0 ✈➔ ♠é✐ n = 1, 2, ... ✤➦t
∞

Dn (ε) =

sup |Xm − X| > ε =
m≥n

n t➠♥❣✮ ✈➔ Ω \ Dn (ε) = Dn (ε) =

|Xm − X| > ε ✳ ❑❤✐ ✤â Dn (ε) ↓ ✭❦❤✐

m=n
∞

|Xm − X| ≤ ε .
m=n

❉♦ ✤â

ω∈

lim |Xn − X| = 0

n→∞

⇔ lim |Xn (ω) − X(ω)| = 0
n→∞

⇔ ∀ε > 0, ∃n : |Xm (ω) − X(ω)| ≤ ε, ∀m ≥ n
⇔ ∀k > 0, ∃n : |Xm (ω) − X(ω)| ≤ 1/k, ∀m ≥ n
⇔ ∀k > 0, ∃n : ω ∈ Dn (1/k)
∞

∞

⇔ω∈

Dn (1/k).
k=1 n=1


✼

❙✉② r❛

∞

∞

lim |Xn − X| = 0 =

n→∞

Dn (1/k).
k=1 n=1

◆➯♥

Xn → Xh.c.c
∞

∞

⇔P

Dn (1/k)) = 1
k=1 n=1
∞

Dn (1/k)) = 1 ∀k = 1, 2...

⇔P
n=1
∞

⇔P

Dn (1/k) = 0 ∀k = 1, 2...
n=1

⇔ lim P Dn (1/k) = 0 ∀k = 1, 2...

n→∞

⇔ lim P Dn (ε) = 0, (✈➻ Dn (ε) ↓).
n→∞

✣à♥❤ ❧þ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳

✶✳✹ ❚➼♥❤ ✤ë❝ ❧➟♣✱ ✤ë❝ ❧➟♣ ✤æ✐ ♠ët✳
✶✳✹✳✶ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✱ ❦❤✐ ✤â
F(X) = {X −1 (B) : B ∈ B(R)}
✤÷ñ❝ ❣å✐ ❧➔ σ ✲✤↕✐ sè s✐♥❤ ❜ð✐ X ✳
❍å ❤ú✉ ❤↕♥ {Fi , 1 ≤ i ≤ n} ❝→❝ σ ✲✤↕✐ sè ❝♦♥ ❝õ❛ F ✤÷ñ❝ ❣å✐ ❧➔ ✤ë❝ ❧➟♣ ♥➳✉
n

P

n

Ai
i=1

=

P (Ai ),
i=1

✤è✐ ✈î✐ ♠å✐ Ai ∈ Fi (1 ≤ i ≤ n) ❜➜t ❦ý✳ ❍å ✈æ ❤↕♥ {Fi , i ∈ I} ❝→❝ σ ✲✤↕✐ sè
❝♦♥ ❝õ❛ F ✤÷ñ❝ ❣å✐ ❧➔ ✤ë❝ ❧➟♣ ♥➳✉ ♠å✐ ❤å ❝♦♥ ❤ú✉ ❤↕♥ ❝õ❛ ♥â ✤ë❝ ❧➟♣✳ ❍å
❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ {Xi , i ∈ I} ✤÷ñ❝ ❣å✐ ❧➔ ✤ë❝ ❧➟♣ ♥➳✉ ❝→❝ σ ✲✤↕✐ sè s✐♥❤
❜ð✐ ❝❤ó♥❣ {F(Xi ), i ∈ I} ✤ë❝ ❧➟♣✳ ❍å ❝→❝ ❜✐➳♥ ❝è {Ai , i ∈ I} ✤÷ñ❝ ❣å✐ ❧➔

✤ë❝ ❧➟♣ ♥➳✉ ❤å ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ {IAi , i ∈ I} ✤ë❝ ❧➟♣✳


✽

✶✳✹✳✷ ✣à♥❤ ♥❣❤➽❛✳ ❍å ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ {Xi, i ∈ I} ✤÷ñ❝ ❣å✐ ❧➔ ✤ë❝
❧➟♣ ✤æ✐ ♠ët ♥➳✉ Xi ✈➔ Xj ✤ë❝ ❧➟♣ ✈î✐ ♠å✐ i = j, i, j ∈ I ✳

✶✳✺ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ▲✉➟t sè ❧î♥
●✐↔ sû {Xn , n ≥ 1} ❧➔ ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝ò♥❣ ①→❝ ✤à♥❤ tr➯♥ ❦❤æ♥❣
❣✐❛♥ ①→❝ s✉➜t ( Ω, F, P )✳ ✣➦t Sn = X1 + X2 + · · · + Xn .

✶✳✺✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❉➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ {Xn, n ≥ 1} ✤÷ñ❝ ❣å✐ ❧➔ t✉➙♥
t❤❡♦ ▲✉➟t ②➳✉ sè ❧î♥ ♥➳✉

Sn − ESn P
−→ 0.
n
❉➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ {Xn , n ≥ 1} ✤÷ñ❝ ❣å✐ ❧➔ t✉➙♥ t❤❡♦ ▲✉➟t ②➳✉ sè

❧î♥ tê♥❣ q✉→t ♥➳✉ tç♥ t↕✐ ❤❛✐ ❞➣② sè an , n ≥ 1, bn , n ≥ 1, 0 < bn ↑ ∞ s❛♦
❝❤♦

Sn − an P
−→ 0.
bn

✶✳✺✳✷ ✣à♥❤ ♥❣❤➽❛✳ ❉➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ {Xn, n ≥ 1} ✤÷ñ❝ ❣å✐ ❧➔ t✉➙♥
t❤❡♦ ▲✉➟t ♠↕♥❤ sè ❧î♥ ♥➳✉

Sn − ESn h.c.c
−→ 0.
n
❉➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ {Xn , n ≥ 1} ✤÷ñ❝ ❣å✐ ❧➔ t✉➙♥ t❤❡♦ ▲✉➟t ♠↕♥❤ sè

❧î♥ tê♥❣ q✉→t ♥➳✉ tç♥ t↕✐ ❤❛✐ ❞➣② sè (an ), (bn ), 0 < bn ↑ ∞ s❛♦ ❝❤♦
Sn − an h.c.c
−→ 0.
bn

✶✳✻ ▼ët sè ❜➜t ✤➥♥❣ t❤ù❝ ❝ì ❜↔♥✳
✶✳✻✳✶ ❇ê ✤➲ ✭❇➜t ✤➥♥❣ t❤ù❝ ▼❛r❦♦✈✮✳ ●✐↔ sû ❳ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
❦❤æ♥❣ ➙♠✳ ❑❤✐ ✤â ♥➳✉ tç♥ t↕✐ EX t❤➻ ✈î✐ ♠å✐ ε > 0, t❛ ❝â

P(X ≥ ε) ≤

EX
.
ε


✾

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â

Ω

XdP

XdP +

XdP =

EX =

(X≥ε)

(0≤X<ε)

dP = εP(X ≥ ε).

≥ε
(X≥ε)

❙✉② r❛

P(X ≥ ε) ≤

EX
.
ε

✶✳✻✳✷ ❇ê ✤➲ ✭❇➜t ✤➥♥❣ t❤ù❝ ❈❤❡❜②s❤❡✈✮✳ ●✐↔ sû X ❧➔ ❜✐➳♥ ♥❣➝✉
♥❤✐➯♥ ❜➜t ❦ý✳ ❑❤✐ ✤â ♥➳✉ tç♥ t↕✐ DX t❤➻ ✈î✐ ♠å✐ ε > 0✱ t❛ ❝â

P(|X − EX| ≥ ε) ≤

DX
.

ε2

❈❤ù♥❣ ♠✐♥❤✳ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ▼❛r❦♦✈ ❝❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
Y = |X − EX|2 ≥ 0
t❛ ✤÷ñ❝

P(|X − EX| ≥ ε) = P(Y ≥ ε2 ) ≤

EY
DX
= 2 .
2
ε
ε

✶✳✻✳✸ ❇ê ✤➲ ✭❇➜t ✤➥♥❣ t❤ù❝ ❑♦❧♠♦❣♦r♦✈✮✳ ●✐↔ sû X1, X2, ..., Xn ❧➔
❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ EXi = 0, DXi = σi2 (∀i = 1, 2, ..., n). ✣➦t
k

Sk = X1 + X2 + · · · + Xk =

Xi
i=1

❑❤✐ ✤â ✈î✐ ♠å✐ ε > 0, t❛ ❝â
✭✐✮ P( max |Sk | ≥ ε) ≤
1≤k≤n

1
ε2

n

i=1

σi2 .

(1 ≤ k ≤ n).


✶✵

✭✐✐✮ ◆➳✉ P( max |Xk | ≤ c) = 1 t❤➻
1≤k≤n

P( max |Sk | ≥ ε) ≥ 1 −

(ε + c)2

1≤k≤n

n

.

σi2

i=1

❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮✳ ✣➦t

A1 = (|S1 | ≥ ε)
A2 = (|S1 | < ε; |S2 | ≥ ε)
...
Ak = ( max |Si | < ε; |Sk | ≥ ε)
1≤i≤k−1

...
An = ( max |Si | < ε; |Sn | ≥ ε)
1≤i≤n−1

A = ( max |Sk | ≥ ε).
1≤k≤n

❱➟② t❤➻

n

A=

AiAj = ∅ (i = j).

Ak ;
k=1

❚❛ ❝â
n

σi2 = DSn = ESn2
i=1
n

≥

ESn2 IA

n

ESn2 IAk

=
k=1

n

k=1
n

ESk2 IAk

=

n
2

E(Sn − Sk ) IAk + 2

+

k=1
n

k=1
n

ESk2 IAk +

=
k=1
n

k=1

ESk (Sn − Sk )IAk
k=1
n

E(Sn − Sk )2 IAk ≥
k=1

ESk2 IAk
k=1

n

ε2 EIAk = ε2

≥

E[Sk + (Sn − Sk )]2 IAk

=

P(Ak ) = ε2 P(A).
k=1


✶✶

✣â ❧➔ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳
✭✐✐✮✳ ❚❛ ❝â

ESn2 IA = ESn2 − ESn2 IA ≥ ESn2 − ε2 P(A)
= ESn2 − ε2 + ε2 P(A).

✭✶✳✶✮

❚r➯♥ Ak ✱ t❛ ❝â |Sk−1 | ≤ ε, |Sk | ≤ |Sk−1 | + |Xk | ≤ ε + c ♥➯♥
n

ESn2 IA

n

ESn2 IAk

=

n

ESk2 IAk

=

k=1

k=1

k=1
n

n

≤ (c + ε)2

E(Sn − Sk )2 IAk

+

P(Ak ) + D(Sn )
k=1

(Ak )
k=1

2

✭✶✳✷✮

≤ P(A)[(c + ε) + D(Sn )].
❚ø ✭✶✳✶✮ ✈➔ ✭✶✳✷✮ s✉② r❛

D(Sn ) − ε2
P(A) ≥
(c + ε)2 + D(Sn ) − ε2
(c + ε)2
(c + ε)2
=1−
≥1−
.
(c + ε)2 + D(Sn ) − ε2
D(Sn )

✶✳✻✳✹ ❍➺ q✉↔✳ ●✐↔ sû {Xn, n ≥ 1} ❧➔ ❞➣② ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱
EXn = 0 (∀n = 1, 2, ...). ❑❤✐ ✤â ✈î✐ ♠å✐ ε > 0✱ t❛ ❝â
✭✐✮
✭✐✐✮

P( max |Sm − Sn | > ε) ≤
n≤m≤k

P( sup |Sm − Sn | > ε) ≤
m≥n

1
ε2

1
ε2

k

DXm ✳

m=n+1
∞

DXm .
m=n+1

❈❤ù♥❣ ♠✐♥❤✳ ✣➦t Y1 = 0, ..., Yn = 0, Yn+1 = 0, Yn+1 = Xn+1 , ..., Yk = Xk .
❑❤✐ ✤â✱ Y1 , ..., Yk t❤ä❛ ♠➣♥ ❣✐↔ t❤✐➳t ❝õ❛ ✤à♥❤ ❧➼ ✈➔

Y1 + · · · + Yn + Yn+1 + · · · + Ym = Xn+1 + · · · + Xm = Sm − Sn ;
k

k

DYm =
m=1

DXm .
m=n+1


✶✷

❚ø ✤â s✉② r❛ ✭✐✮✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ✭✐✐✮✱ t❛ ✤➦t

Bk = ( max |Sm − Sn | > ε).
n≤m≤k

❑❤✐ ✤â (Bk , k > n) ❧➔ ❞➣② t➠♥❣ ❝→❝ ❜✐➳♥ ❝è ✈➔
∞

Bk = ( sup |Sm − Sn | > ε).
k=n+1

m≥n

❉♦ ✤â
∞

P( sup |Sm − Sn | > ε) = P(
m≥n

Bk )

k=n+1

= lim P(Bk )
k→∞

1
≤ 2 lim
ε k→∞
1
= 2
ε

k

DXm
m=n+1

∞

DXm .
m=n+1


✶✸

❈❍×❒◆● ✷

▼Ð ❘❐◆● ▲❯❾❚ ❨➌❯ ❙➮ ▲❰◆ ❑❖▲▼❖●❖❘❖❱✲ ❋❊▲▲❊❘✳
✷✳✶ ▲✉➟t ②➳✉ sè ❧î♥ ❑♦❧♠♦❣♦r♦✈✲ ❋❡❧❧❡r✳
✷✳✶✳✶ ✣è✐ ①ù♥❣ ❤â❛✳ ◆➳✉ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ❝â ❤➔♠ ♣❤➙♥ ♣❤è✐ F ❝❤ó♥❣
tæ✐ ❦➼ ❤✐➺✉ ❤➔♠ ♣❤➙♥ ♣❤è✐ ❝õ❛ −X ❧➔ −F ✳ ❚↕✐ ❝→❝ ✤✐➸♠ ❧✐➯♥ tö❝ ❝õ❛ F ❝❤ó♥❣
t❛ ❝â

−F (x) = 1 − F (−x)
✈➔ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ −F ♥❤÷ t❤➳ ❧➔ ❞✉② ♥❤➜t✳ ❍➔♠ ♣❤➙♥ ♣❤è✐ F ✤÷ñ❝ ❣å✐ ❧➔
✤è✐ ①ù♥❣ ♥➳✉ −F = F ✭ ❑❤✐ ❤➔♠ ♠➟t ✤ë f tç♥ t↕✐ t❛ ❝ô♥❣ ❝â f (−x) = f (x)✮✳
●✐↔ sû X1 ✈➔ X2 ❧➔ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ❝ò♥❣ ❝â ❤➔♠ ♣❤➙♥ ♣❤è✐ F ✳
❑❤✐ ✤â X1 − X2 ❝â ❤➔♠ ♣❤➙♥ ♣❤è✐ ✤è✐ ①ù♥❣ F 0 ❝❤♦ ❜ð✐

F 0 = F ∗ −F.
❙û ❞ö♥❣ t➼♥❤ ❝❤➜t ✤è✐ ①ù♥❣ t❛ ❝â F 0 (x) = 1 − F 0 (−x) ❞➵ t❤➜②
+∞

F 0 (x) =

F (x + y)F (dy).
−∞

❈❤ó♥❣ t❛ ♥â✐ r➡♥❣ F 0 ♥❤➟♥ ✤÷ñ❝ ❜➡♥❣ ❝→❝❤ ❧➔♠ ✤è✐ ①ù♥❣ F ✳

✷✳✶✳✷ ❇ê ✤➲ ✭❇➜t ✤➥♥❣ t❤ù❝ ✤è✐ ①ù♥❣✮✳ ●✐↔ sû X1 ✈➔ X2 ❧➔ ❝→❝ ❜✐➳♥
♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ❝ò♥❣ ♣❤➙♥ ♣❤è✐✱ ✈î✐ t > 0, t❛ ❝â
1
P(|X1 − X2 | > t) ≤ 2P(|X1 | > t)
2

✭✷✳✶✮

◆➳✉ ❝❤å♥ a ≥ 0 s❛♦ ❝❤♦ P(Xi ≤ a) ≥ p ✈➔ P(Xi ≥ −a) ≥ p✱ t❤➻

P(|X1 − X2 | > t) ≥ pP(|X1 | > t + a).

✭✷✳✷✮




t 0 ởt ừ Xi t
1
P(|X1 X2 | > t) P(|X1 | > t)
2

ự ố tr ừ ổ t r trứ |X1 | > 12 t
|X2 | > 21 t õ ú ố tr ổ t
r X1 > t + a, X2 a, ụ ữ X1 < t a X2 a
t õ

ờ X1, X2, ..., Xn ở õ ố ố ự t
Sn = X1 + X2 + ã ã ã + Xn õ ố ố ự
1
P(|X1 + X2 + ã ã ã + Xn | > t) P(max |Xi | > t).
2



Xi õ ố F t
1
P(|X1 + X2 + ã ã ã + Xn | t) (1 en[1F (t)+F (t)] ).
2



ự sỷ M t
tr X1 , ..., Xn õ tr tt ố ợ t t

T = Sn M (M, T ) ố ố ự tờ ủ tr ố
(M, T ) ụ õ ũ ố ó r

P(M > t) P(M > t, T 0) + P(M > t, T 0).

tự t st ụ ữ

1
P(S > t) = P(M + T > t) P(M > t, T 0) P(M > t)
2



tữỡ tỹ
ự tọ ữ ỵ t F tử

P(max |Xi | t) = (F (t) F (t))n en[1F (t)+F (t)]



1 x < ex 0 < x < 1.

ờ sỷ X1, X2, ..., Xn ở t
Sn = X1 + X2 + ã ã ã + Xn ,

Yk,n = Xk I{|Xk | n}, k = 1, 2, ..., n


✶✺

Sn = Y1,n +Y2,n +· · ·+Yn,n ,

n ≥ 1,

k = 1, 2, ..., n,

mn = E(Sn ) = nE(Y1,n ).

❑❤✐ ✤â t❛ ❝â

P(|Sn − mn | > t) ≤ P(|Sn − mn | > t) + P(Sn = Sn )
1
1
2
P
Sn − E(Y1,n ) > x ≤
E(Y1,n
) + nP(|X1 | > n)
n
nx2
❈❤ù♥❣ ♠✐♥❤✳ ✶✮❚❛ ❝â
(|Sn − mn | > t) ⊂ (|Sn − mn | > t) ∪ (Sn = Sn )
❉♦ ✤â

P(|Sn − mn | > t) ≤ P (|Sn − mn | > t) ∪ (Sn = Sn )
≤ P(|Sn − mn | > t) + P(Sn = Sn ).
❱➟② t❛ ❝â ✭✷✳✾✮✳
✷✮✣➦t t = nx →♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ▼❛r❦♦✈ t❛ ❝â

1
E|Sn − mn |2
2
2
n x

n
1
= 2 2E
(Yk,n − EYk,n )
n x

P(|Sn − mn | > nx) ≤

k=1

=

1
n2 x2

n

E(Yk,n − EYk,n )2
k=1

1
E(Y1,n − EY1 , n)2
2
nx
1
≤
E(Y1,n )2 .
2
nx

=

▼➦t ❦❤→❝

n

(

n

Yk,n ) ⊂

Xk =
k=1

n

k=1

(Xk = Yk,n )
k=1

2

✭✷✳✾✮
✭✷✳✶✵✮


✶✻

❉♦ ✤â
n

P(Sn = Sn ) = P(

n

Xk =
k=1

Yk,n )
k=1

n

≤

P(Xk = Yk,n )
i=1
n

P(|Xk | > n)

=
k=1

= nP(|X1 | > n).
❑➳t ❤ñ♣ ✭✷✳✾✮ t❛ ❝â ✭✷✳✶✵✮✳

✷✳✶✳✺ ✣à♥❤ ❧þ ✭▲✉➟t ②➳✉ sè ❧î♥ ❑♦❧♠♦❣♦r♦✈ ✲ ❋❡❧❧❡r✮✳ ●✐↔ sû

{Xn , n ≥ 1} ❧➔ ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ❝ò♥❣ ♣❤➙♥ ♣❤è✐

✣➦t Sn = X1 + X2 + · · · + Xn ,

Yk,n = Xk I{|Xk | ≤ n},

k = 1, 2, ..., n.

µ = EXk I(|Xk | ≤ n) = EYk,n ,

❑❤✐ ✤â ✈î✐ ♠å✐ ε > 0, t❛ ❝â
Sn
− µ > ε → 0 ❦❤✐
n

P

n → ∞.

❦❤✐ ✈➔ ❝❤➾ ❦❤✐
tP(|Xk | > t) → 0 ❦❤✐

t → ∞.

❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ✤õ✿ ❙û ❞ö♥❣ ✭✷✳✶✵✮ t❛ ❝â

P

Sn
1

− µ > ε ≤ 2 E(Y1,n )2 + nP(|X1 | > n)
n
nε

▼➔
n

1
1
E(Y1,n )2 =
n
n

x2 dF (x)
−n
n

=

2
n

x2 dF (x)
0


✶✼

n

2
=
− n2 (1 − F (n)) + 2
n

x(1 − F (x)dx
0

n

= 2(−nP(X > n)) +

4
n

xP(X > x)dx.
0

❚❛ ❝â

tP(|X| > t) ≥ t(P(X > t)) ≥ 0
s✉② r❛

t(P(X > t)) → 0 khi t → ∞
❉♦ ✤â

1
2
n E(Y1,n )

→ 0 ✳ ❱➟② P

Sn
n

− µ > ε → 0 ❦❤✐ n → ∞.

✣✐➲✉ ❦✐➺♥ ❝➛♥✿ ◆❤÷ ❝❤ó♥❣ tæ✐ ✤➣ ❣✐î✐ t❤✐➺✉ ð ♠ö❝ ✷✳✶✳✶ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ Xk0
t❤✉ ✤÷ñ❝ trü❝ t✐➳♣ ❜ð✐ ✤è✐ ①ù♥❣ ❤â❛ tø Xk ✳ ❚ê♥❣ ❝õ❛ ❝❤ó♥❣ Sn0 ❝â t❤➸ ♥❤➟♥
✤÷ñ❝ ❜ð✐ ✤è✐ ①ù♥❣ ❤â❛ tø Sn − nµ. ●✐↔ sû a ❧➔ ♠ët ♠❡❞✐❛♥ ❝õ❛ ❜✐➳♥ ♥❣➝✉
♥❤✐➯♥ Xk . ❙û ❞ö♥❣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✮✱ ✭✷✳✷✮ ✈➔ ✭✷✳✺✮ ❝❤ó♥❣ t❛ ❝â

2P(|Sn − nµ| > nε) ≥ P(|Sn0 | > 2nµ)
0
1
≥ [1 − e(−nP(|X1 |>2nε)) ]
2
1
1
≥ [1 − e− 2 nP(|X1 |>2nε+|a|) ].
2
❚ø ❣✐↔ t❤✐➳t ❝õ❛ ✤✐➲✉ ❦✐➺♥ ❝➛♥

P
◆➯♥ t❛ ❝â

Sn
− µ > ε → 0 ❦❤✐ n → ∞
n

1
− nP(|X1 | > 2nε + |a|) → 0 ❦❤✐ n → ∞
2

❤❛②

tP(|Xk | > t) → 0 ❦❤✐ t → ∞.
✣à♥❤ ❧➼ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳




ỵ sỷ {Xn, n 1} ở
ũ ố {bn , n 1} số tỹ ữỡ t + ợ
k = 1, 2, .., n, n 1 t
Sn = X1 + X2 + ã ã ã + Xn , Yk,n = Xk I{|Xk | bn },
n

n

Yk,n ,

Sn =

àn =

k=1

EYk,n .
k=n

n

P(|Xk | > bn ) 0 n ,



k=1

1
b2n

t

n

t

n ,



k=1

Sn àn p
0
bn

r



D(Yk,n ) 0

àn
0
bn





n

n ,

Sn p
0
bn





n .

ự õ
1

P
bn

n

n

(Yk,n EYk,n ) > = P
k=1

(Yk,n EYk,n ) > bn
k=1
n

n

Yk,n E(

=P
k=1



=

1
D(
b2n 2
1
b2n 2

Yk,n ) > bn
k=1

n

Yk,n ) t tự s
k=1

n

DYk,n 0 n
k=1


✶✾

❙✉② r❛

Sn − µn p
−→ 0 ❦❤✐ n → ∞
bn

❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤

Sn − µn p
−→ 0 ❦❤✐ n → ∞.
bn
❚❤➟t ✈➟②✱ t❛ ❝â

P(

S − µn
Sn − µn
= n
) = P(Sn = Sn )
bn
bn
n

n

Xk =

= P(
k=1
n

≤ P(

Yk,n )
k=1

(Xk = Yk,n ))

k=1
n

≤

P(Xk = Yk,n )
k=1
n

P(|Xk | > bn ) → 0 khi n → ∞(❉♦ ✭✷✳✶✶✮)

≤
k=1

❙✉② r❛

Sn − µn
S − µn
= n
)=0
n→∞
bn
bn
Sn − µn p
Sn − µn p
▼➔
−→ 0 ♥➯♥
−→ 0
bn
bn
lim P(

✭⑩♣ ❞ö♥❣ ❦➳t q✉↔

lim P(Xn = Yn ) = 0

n→0

t❤➻

p

p

Xn −→ 0 ⇔ Yn −→ 0✮

◆❣♦➔✐ r❛✱ ♥➳✉

µn
→ 0 tø
bn

Sn − µn p
−→ 0 s✉② r❛
bn

Sn p
−→ 0.
bn

✣à♥❤ ❧➼ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳

✷✳✶✳✼ ❱➼ ❞ö✳ ●✐↔ sû {Xn, n ≥ 1} ❧➔ ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ❝ò♥❣
❝â ❤➔♠ ♠➟t ✤ë

f (x) =

c
x2 ln|x|

♥➳✉ |x| > 2✱

0

♥➳✉ |x| ≤ 2✳


✷✵

tr♦♥❣ ✤â c ❧➔ ♠ët ❤➡♥❣ sè✳
❚❛ ❝â

+∞

EX1 =

xf (x)dx =
−∞

|x|>2

lim 2c(ln(lnx))

n→+∞

n
2

c
dx
xln|x|

+∞

= 2c
2

dx
xlnx

n

=

lim 2c

n→+∞

2

dx
xlnx

=

= ∞✳ ◆❤÷ ✈➟② ❦➻ ✈å♥❣ ❦❤æ♥❣ tç♥ t↕✐✱ ❜➙② ❣✐í t❛ ❦✐➸♠

tr❛ ✤✐➲✉ ❦✐➺♥ ✤õ ❝õ❛ ✤à♥❤ ❧➼ ✷✳✶✳✺

nP(|X1 | > n) = n[P(X1 > n) + P(X1 < −n)]
−n

= n[1 − P(X1 ≤ n) +
−∞
−n

n

= n[1 −

f (x)dx +

−∞
+∞

= n[

f (x)dx]

f (x)dx]

−∞
−n

f (x)dx +

f (x)dx]

−∞

n
+∞

f (x)dx (❉♦ f (x) ❧➔ ❤➔♠ ❝❤➤♥)

= 2n
n

+∞

1
dx (❉♦ x ≥ n ♥➯♥ lnx ≥ lnn)
x2

2c
n
≤
lnn
n

2c
→ 0 ❦❤✐ n → ∞.
lnn
❱➟② ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ {Xn , n ≥ 1} t✉➙♥ t❤❡♦ ❧✉➟t ②➳✉ sè ❧î♥
=

❑♦❧♠♦❣♦r♦✈✲❋❡❧❧❡r✳

✷✳✶✳✽ ❱➼ ❞ö✳ ●✐↔ sû {Xn, n ≥ 1} ❧➔ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ❝ò♥❣ ❝â
❤➔♠ ♠➟t ✤ë

f (x) =

1
2x2

0

♥➳✉ |x| > 1✱
♥➳✉ |x| ≤ 1✳

❚❛ ❝â
+∞

EX1 =

−1

xf (x) =
−∞

+∞

1
dx +

2x
−∞

1
dx
2x
1


✷✶

+∞

1

1
dx +
2x

=
+∞
+∞

1
dx
2x
1
+∞

1

dx −
2x

=
1

1
dx.
2x
1

❚➼❝❤ ♣❤➙♥ tr➯♥ ❧➔ ❦❤æ♥❣ tç♥ t↕✐✱ ✈➻ ❝❤å♥ ❞➣② xn = ean ✈➔

yn = ean +1

(an → +∞)✳
❚❛ ❝â

an

an +1

1
dx −
2x

lim (

n→+∞

1

✈➔

1
1
dx) = −
2x
2
1

an +1

an

1
1
dx) =
2x
2

1
dx −
2x

lim (

n→+∞

1

1

◆❤÷ ✈➟② ❦➻ ✈å♥❣ ❦❤æ♥❣ tç♥ t↕✐✱ ✈➔ ✤✐➲✉ ❦✐➺♥ ✤õ ❝õ❛ ✤à♥❤ ❧➼ ✷✳✶✳✺✳
+∞

1
dx = 1
x2

nP(|X1 | > n) = n
n

❦❤æ♥❣ t❤ä❛ ♠➣♥✳ ❇➙② ❣✐í t❛ ❦✐➸♠ tr❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭✷✳✶✶✮ ✈➔ ✭✷✳✶✷✮ ❝õ❛ ✤à♥❤
❧➼ ✷✳✶✳✻
n

✐✮

P(|Xk | > bn ) = nP(|X1 |) > bn ) = nP(|X1 | > nlnn)(❞➣②bn = nlnn, n = 1, 2...)
k=1
+∞

1
−1
dx
=
n(
)
x2

x

=n

+∞

=
nlnn

1
→ 0 ❦❤✐ n → ∞.
lnn

nlnn

1
✐✐✮ 2
bn

n

DYk,n
k=1

1
= 2
bn
≤

1

b2n

n

E(Yk,n − EYk,n )2 (❞➣② bn = nlnn, n = 1, 2...)
k=1
n
2
EYk,n
k=1


✷✷

1
= 2
bn

n

E(Xk2 I(|Xk | ≤ bn ))
k=1
n

1
E(X 2 I(|X1 | ≤ bn ))
2
bn
k=1
n

= 2 E(X 2 I(|X1 | ≤ bn )
bn
=

bn

=

n
b2n

=2

=

x2 f (x)dx
−bn
bn

n
b2n

n
b2n

x2

1
dx
2x2

0
bn

dx =

1
n
=
→ 0 ❦❤✐ n → ∞.
bn
lnn

0

❚❤❡♦ ✤à♥❤ ❧➼ ✷✳✶✳✻ t❛ ❝â

Sn p
−→ 0 ❦❤✐ n → ∞.
nlnn
◆â✐ ❝→❝❤ ❦❤→❝✱ ❧✉➟t ②➳✉ ✈➝♥ ✤ó♥❣ ✈î✐ ❞➣② {bn = nlnn, n ≥ 1}✳ ◆❤÷ ✈➟② q✉❛
❝→❝ ✈➼ ❞ö tr➯♥ t❛ t❤➜② r➡♥❣✱ ♥➳✉ ❞➣② {Xn , n ≥ 1} ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣
❝ò♥❣ ♣❤➙♥ ♣❤è✐ ❦❤æ♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ tP(|Xk | > t) → 0 ❦❤✐ t → ∞
t❤➻ ❝â t❤➸ ❧✉➟t ②➳✉ sè ❧î♥ ✈➝♥ ✤ó♥❣✳ ✣✐➲✉ ♥➔② ❣ñ✐ þ ❝❤ó♥❣ t❛ ✤✐ ✤➳♥ t❤✐➳t ❧➟♣
❝→❝ ❧✉➟t ②➳✉ sè ❧î♥ tê♥❣ q✉→t ❤ì♥ ❧✉➟t ②➳✉ sè ❧î♥ ❑♦❧♠♦❣♦r♦✈ ✲ ❋❡❧❧❡r✳

✷✳✷ ▼ð rë♥❣ ❧✉➟t ②➳✉ sè ❧î♥ ❑♦❧♠♦❣♦r♦✈✲❋❡❧❧❡r
❚r♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❝❤ó♥❣ tæ✐ ❝â sû ❞ö♥❣ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ❤➔♠
❜✐➳♥ ✤ê✐ ✤➲✉ ✈➔ ❝❤➟♠✱ ❝❤ó♥❣ tæ✐ ✤÷❛ r❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ♥➔② ✤➸ sû ❞ö♥❣ ❝❤♦
❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧➼✳

✷✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû a > 0✳ ▼ët ❤➔♠ ❞÷ì♥❣✱ ✤♦ ✤÷ñ❝ u tr➯♥ [a, ∞) ❣å✐
❧➔ ❜✐➳♥ ✤ê✐ ✤➲✉ ✤➳♥ ✈æ ❝ò♥❣ ✈î✐ sè ♠ô ρ, −∞ < ρ < ∞, ❦➼ ❤✐➺✉ u ∈ RV(ρ),


✷✸

♥➳✉

u(tx)
→ xρ ❦❤✐ t → ∞ ✈î✐ ♠å✐ x > 0.
u(t)

◆➳✉ ρ = 0 t❤➻ u ✤÷ì❝ ❣å✐ ❧➔ ❤➔♠ ❜✐➳♥ ✤ê✐ ❝❤➟♠ ✤➳♥ ✈æ ❝ò♥❣✱ ❦➼ ❤✐➺✉ u ∈ SV.
❈❤➥♥❣ ❤↕♥ ❝→❝ ❤➔♠ s❛✉ ✤➙② ❧➔ ❝→❝ ❤➔♠ ❜✐➳♥ ✤ê✐ ✤➲✉ ✤➳♥ ✈æ ❝ò♥❣

xρ , xρ lnx, xρ lnlnx, xρ

✷✳✷✳✷ ❇ê ✤➲✳ ❈❤ó♥❣ t❛ ❝â ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝

lnx
lnlnx

ex ≤ 1 + x + x2 ✈î✐ |x| ≤ 1,
−

1
x < ln(1 − x) < −x ✈î✐ 0 < x < δ < 1.
1−δ

✷✳✷✳✸ ❇ê ✤➲✳ ●✐↔ sû 0 ≤ an < δ < 1, n ≥ 1

✭✷✳✶✺✮
✭✷✳✶✻✮

t❛ ❝â

(1 − an )n → 1 ❦❤✐ n → ∞ ⇐⇒ nan → 0 ❦❤✐ n → ∞

✭✷✳✶✼✮

❍ì♥ ♥ú❛✱ δ ∈ (0; 1), nan < δ(1 − δ) < 1 ✈î✐ n ✤õ ❧î♥ t❤➻
(1 − δ)nan ≤ 1 − (1 − an )n ≤

nan
.
(1 − δ)

✭✷✳✶✽✮

❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ✤õ ❧➔ ❤✐➸♥ ♥❤✐➯♥✳
✣✐➲✉ ❦✐➺♥ ❝➛♥✱ ❣✐↔ sû (1 − an )n → 1 ❦❤✐ n → ∞✳ ❚ø ✭✷✳✶✻✮ ❝❤ó♥❣ t❛ ❝â ✱
❦❤✐ n → ∞,
n

nln(1−an )

1 ← (1 − an ) = e

≤ e−nan ,

−nan
≥ e (1−δ) ,

✤✐➲✉ ✤â ❝❤➾ r❛ r➡♥❣

(1 − an )n → 1 ❦❤✐ n → ∞ =⇒ nan → 0 ❦❤✐ n → ∞.
❈❤å♥ n ✤õ ❧î♥ s❛♦ ❝❤♦ nan < δ(1 − δ) < 1✱ sû ❞ö♥❣ ✭✷✳✶✺✮ t❛ ❝â

(1 − an )n ≤ e−nan ≤ 1 − nan + (nan )2 ≤ 1 − nan (1 − δ).
✈➔
−nan

(1 − an )n ≥ e (1−δ) ≥ 1 −
❞♦ ✤â t❛ ❝â ✭✷✳✶✽✮✳ ❇ê ✤➲ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳

nan
.
(1 − δ)


✷✹

✷✳✷✳✹ ❇ê ✤➲✳ ●✐↔ sû X1, X2, ..., Xn ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ❝ò♥❣
♣❤➙♥ ♣❤è✐ ✱ EXn = 0 ✱ ✈➔ ❤➡♥❣ sè A > 0✱ s❛♦ ❝❤♦ sup |Xn | ≤ A. ❑❤✐ ✤â✱
n

t❛ ❝â
n

DXk ≤ x2 +

(x + A)2 P( max |Sk | > x)

k=1

1≤k≤n

P( max |Sk | ≤ x)

.

1≤k≤n

◆➳✉ P( max |Sk | > x) < δ, δ ∈ (0, 1), t❤➻
1≤k≤n

n

DXk ≤ x2 + (x + A)2
k=1

δ
.
1−δ

✷✳✷✳✺ ✣à♥❤ ❧þ✳ ●✐↔ sû {Xn, n ≥ 1} ❞➣② ❧➔ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣
❝ò♥❣ ♣❤➙♥ ♣❤è✐✳ ✣➦t Yn = max |Xk |❀ {bn , n ≥ 1} ❧➔ ❞➣② sè t❤ü❝ ❞÷ì♥❣
1≤k≤n

t➠♥❣ ✤➳♥ +∞✱ ✈î✐ ♠å✐ ε > 0✱ t❛ ❝â

Yn p
−→ 0 ⇐⇒ nP(|X1 | > bn ε) → 0 ❦❤✐ n → ∞
bn

❈❤ù♥❣ ♠✐♥❤✳ ❱➻

P(Yn > bn ε) = 1 − (P(|X1 | ≤ bn ε))n = 1 − (1 − P(|X1 | > bn ε))n ,
⑩♣ ❞ö♥❣ ✭✷✳✶✽✮ ✈î✐ δ =

1
2

t❛ ❝â

1
nP(|X1 | > bn ε) ≤ P(Yn > bn ε) ≤ 2nP(|X1 | > bn ε),
2
❝❤♦ n → ∞ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

✷✳✷✳✻ ✣à♥❤ ❧þ✳ ●✐↔ sû {Xn, n ≥ 1} ❧➔ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ❝ò♥❣
♣❤➙♥ ♣❤è✐✱ Sn = X1 + X2 + · · · + Xn , n ≥ 1✱ ✈î✐ x > 0, b ∈ RV( ρ1 )✱ ✈î✐ ♠é✐
1

ρ ∈ (0, 1] ❣✐↔ sû b(x) = x ρ l(x)✱ tr♦♥❣ ✤â l ∈ SV ✳ ✣➦t bn = b(n), n ≥ 1✳

❑❤✐ ✤â

Sn − nEXI{|X1 | ≤ bn } p
−→ 0 ❦❤✐ n → ∞
bn

❦❤✐ ✈➔ ❝❤➾ ❦❤✐
nP(|X1 | > bn ) → 0 ❦❤✐ n → ∞.

✭✷✳✶✾✮


✷✺

❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ✤õ✳ ❚❛ ❦✐➸♠ tr❛ ✤✐➲✉ ❦✐➺♥ ✭✷✳✶✶✮ ✈➔ ✭✷✳✶✷✮ ❝õ❛ ✤à♥❤
❧➼ ✷✳✶✳✻✳ ❉♦ ❣✐↔ t❤✐➳t ❝õ❛ ✤✐➲✉ ❦✐➺♥ ✤õ t❛ ❝â ♥❣❛② ✭✷✳✶✶✮✱ ❜➙② ❣✐í t❛ ❦✐➸♠
tr❛ ✭✷✳✶✷✮

1
nEX12 I{|X1 | ≤ bn }
2
bn
n
= 2
bn
≤

=

n
b2n
n
b2n

n

EX12 I{bk−1 < |X1 | ≤ bk }
k=1
n

b2k P(bk−1 < |X1 | ≤ bk )
k=1
n
2

k ρ (l(k))2 P(bk−1 < |X1 | ≤ bk )
k=1
n

n
≤C 2
bn
=C

=C

≤C

n
b2n
n
b2n
n
b2n

k
2

j ( ρ )−1 (l(j))2 P(bk−1 < |X1 | ≤ bk )
j=1

k=1
n

2

j ( ρ )−1 (l(j))2
j=1
n

k=j
2

j ( ρ )−1 (l(j))2 P(bj−1 < |X1 | ≤ bn )
j=1
n
2

j ( ρ )−1 (l(j))2 P(|X1 | > bj−1 )
j=1
n−1

1

≤C

nP(bk−1 < |X1 | ≤ bk )

( ρ2 )−1

n

2

(l(n))2

j ( ρ )−2 (l(j))2 j P(|X1 | > bj ) → 0 ❦❤✐ n → ∞.
j=0

n
2

j ( ρ )−2 (l(j))2 ∼

(❉♦
j=1

2
ρ
n( ) − 1(l(n))2 ❦❤✐ n → ∞).
2−ρ ρ

✣✐➲✉ ❦✐➺♥ ❝➛♥✳ ●✐↔ sû ❝â ✭✷✳✶✸✮✱ tø ❜➜t ✤➥♥❣ t❤ù❝ ✤è✐ ①ù♥❣ ❝❤ó♥❣ t❛ ❜✐➳t
r➡♥❣

Sns p
−→ 0 ❦❤✐ n → ∞.
bn

❱➻
s
s
|Xns | = |Sns − Sn−1
| ≤ |Sns | + |Sn−1
|,