Khả tích đều theo nghĩa casàro và luật mạnh số lớn

✶

▲ê✐ ♥ã✐ ➤➬✉

❚r♦♥❣ ❧ý t❤✉②Õt ①➳❝ s✉✃t✱ ❧✉❐t ♠➵♥❤ sè ❧í♥ ➤ã♥❣ ♠ét ✈❛✐ trß q✉❛♥ trä♥❣✳
▲✉❐t ♠➵♥❤ sè ❧í♥ ❝❤♦ ❞➲② ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣✱ ❝ï♥❣ ♣❤➞♥ ố
ợ r tết từ ữ ❝đ❛ t❤Õ ❦û tr➢í❝✳ ◆➝♠ ✶✾✽✶✱
❊t❡♠❛❞✐ ❬✼❪ ➤➲ ♠ë ré♥❣ ❦Õt q✉➯ ♥➭② ❜➺♥❣ ❝➳❝❤ t❤❛② ➤✐Ị✉ ❦✐Ư♥ ➤é❝ ❧❐♣ ❜ë✐ ➤✐Ị✉
❦✐Ư♥ ➤é❝ ❧❐♣ ➤➠✐ ♠ét✳ ➜✐ t❤❡♦ ❤➢í♥❣ ♥➭②✱ ♥➝♠ ✶✾✽✸✱ ❈s♦r❣♦✱ ❚❛♥❞♦r✐ ✈➭ ❚♦t✐❦
❬✻❪ ➤➲ t❤✐Õt ❧❐♣ ➤➢ỵ❝ ❧✉❐t ♠➵♥❤ sè ❧í♥ ❝❤♦ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣ ➤➠✐
♠ét✱ ❦❤➠♥❣ ❝ï♥❣ ♣❤➞♥ ♣❤è✐✱ ➤å♥❣ t❤ê✐ ❊t❡♠❛❞✐ ❬✽❪ ❝ị♥❣ ➤➲ t❤✐Õt ❧❐♣ ➤➢ỵ❝ ❧✉❐t
♠➵♥❤ sè ❧í♥ ❝❤♦ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➠♥❣ ➞♠✳ ●➬♥ ➤➞②✱ ♥❤✐Ò✉ t➳❝ ❣✐➯ q✉❛♥
t➞♠ ➤Õ♥ ✈✐Ư❝ t❤✐Õt ❧❐♣ ❧✉❐t ♠➵♥❤ sè ❧í♥ ❝❤♦ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➯ tÝ❝❤ ➤Ò✉
✭❬✸❪✱ ❬✹❪✮✳ ❚r➟♥ ❝➡ së ➤ã ✈➭ t❤❛♠ ❦❤➯♦ ❜➭✐ ❜➳♦ ❝ñ❛ ❈❤❛♥❞r❛ ✈➭ ●♦s✇❛♠✐ ❬✷❪✱
❝❤ó♥❣ t➠✐ ➤➲ ❧ù❛ ❝❤ä♥ ➤Ị t➭✐ ❧✉❐♥ ✈➝♥

✬✬❑❤➯ tÝ❝❤ ➤Ị✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦ ✈➭

❧✉❐t ♠➵♥❤ sè ❧í♥✧✳
▲✉❐♥ ✈➝♥ ❣å♠ ❤❛✐ ❝❤➢➡♥❣✳ ❚r♦♥❣ ❈❤➢➡♥❣ ✶✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❝➳❝ ❦✐Õ♥
t❤ø❝ ❝➡ së ❝ñ❛ ❧ý t❤✉②Õt ①➳❝ s✉✃t✱ ❝➬♥ t❤✐Õt ➤Ĩ tr×♥❤ ❜➭② ❝➳❝ ✈✃♥ ➤Ị ❝đ❛ ❈❤➢➡♥❣
✷✳ ➜ã ❧➭ ♠ét sè ❦❤➳✐ ♥✐Ö♠ ✈➭ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t ✈➭ ❜✐Õ♥
♥❣➱✉ ♥❤✐➟♥✱ ❦ú ✈ä♥❣ ✈➭ ♣❤➢➡♥❣ s❛✐ ❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✱ ❤ä ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥
➤é❝ ❧❐♣✱ ➤é❝ ❧❐♣ ➤➠✐ ♠ét✱✳✳✳ ➜å♥❣ t❤ê✐ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ư♠ ❦❤➯ tÝ❝❤ ➤Ị✉ ❝đ❛ ♠ét
❤ä ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✱ ❝➳❝ ♠Ư♥❤ ➤Ị✱ tÝ♥❤ ❝❤✃t ❝ã ❧✐➟♥ q✉❛♥✳
❚r♦♥❣ ❈❤➢➡♥❣ ✷✱ tr➢í❝ ❤Õt ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ✈Ị ❧✉❐t ♠➵♥❤ sè ❧í♥ ➤è✐ ✈í✐
❞➲② ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➠♥❣ ➞♠✱ s❛✉ ➤ã ❝❤ó♥❣ t➠✐ t×♠ ❤✐Ĩ✉ ✈Ị ❧✉❐t ♠➵♥❤ sè
❧í♥ ✈í✐ ❞➲② ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣ ➤➠✐ ♠ét✱ ❝✉è✐ ❝ï♥❣ ❝❤ó♥❣ t➠✐ ♥❣❤✐➟♥
❝ø✉ ✈Ị ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➯ tÝ❝❤ ➤Ị✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦✳
▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ trù❝
t✐Õ♣ ❝đ❛ ●❙✳❚❙ ◆❣✉②Ơ♥ ❱➝♥ ◗✉➯♥❣✳ ❚➳❝ ❣✐➯ ①✐♥ ➤➢ỵ❝ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ s➞✉

s➽❝ ➤Õ♥ t❤➬② ✈Ị sù q✉❛♥ t➞♠ ✈➭ ♥❤✐Ưt t×♥❤ ❤➢í♥❣ ❞➱♥ ♠➭ t❤➬② ➤➲ ❞➭♥❤ ❝❤♦ t➳❝


✷

❣✐➯ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ♥❣❤✐➟♥ ❝ø✉ ➤Ị t➭✐✳
❚➳❝ ❣✐➯ ❝ị♥❣ ①✐♥ ❣ư✐ ❧ê✐ ❝➯♠ ➡♥ s➞✉ s➽❝ tí✐ ❝➳❝ t❤➬② ❝➠ tr♦♥❣ ❇é ♠➠♥ ❳➳❝
s✉✃t t❤è♥❣ ❦➟ ✈➭ ❚♦➳♥ ø♥❣ ❞ô♥❣✱ ❑❤♦❛ ❙➢ ♣❤➵♠ ❚♦➳♥ ❤ä❝✱ Pò ọ
ì t t×♥❤ ❣✐ó♣ ➤ì✱ ➤é♥❣ ✈✐➟♥✱ t➵♦ ➤✐Ị✉ ❦✐Ư♥ ❝❤♦ t➳❝ ❣✐➯
tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ t➵✐ tr➢ê♥❣✳
▼➷❝ ❞ï ➤➲ ❝ã ♥❤✐Ò✉ ❝è ❣➽♥❣✱ s♦♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ❝➳❝ tế sót
rt ợ ữ ờ ỉ ❜➯♦✱ ♥❤÷♥❣ ý ❦✐Õ♥ ➤ã♥❣ ❣ã♣ ❝đ❛ q✉ý
t❤➬② ❝➠ ✈➭ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥ ❤➡♥✳

❱✐♥❤✱ t❤➳♥❣ ✽ ♥➝♠ ✷✵✶✻✳
❚➳❝ ❣✐➯


✸

❈❤➢➡♥❣ ✶✳ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ

❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ❦❤➳✐ ♥✐Ư♠ ✈➭ tÝ♥❤ ❝❤✃t ❝➡
❜➯♥ ✈Ò ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t ✈➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✱ ❦ú ✈ä♥❣ ❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✱
❤ä ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣✱ ➤é❝ ❧❐♣ ➤➠✐ ♠ét✱ ❦❤➳✐ ♥✐Ư♠ ❦❤➯ tÝ❝❤ ➤Ị✉ ❝đ❛ ♠ét
❤ä ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✱ ❝➳❝ ♠Ư♥❤ ➤Ị✱ tÝ♥❤ ❝❤✃t ❝ã ❧✐➟♥ q✉❛♥✱✳✳✳ ❈➳❝ ❦Õt q✉➯ ❝đ❛
❝❤➢➡♥❣ ♥➭② sÏ ➤➢ỵ❝ sư ❞ô♥❣ ë ❝❤➢➡♥❣ s❛✉✳

✶✳✶✳ ❑❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t
✶✳✶✳✶✳ ❑❤➠♥❣ ❣✐❛♥ ➤♦ ✈➭ ➤é ➤♦ ①➳❝ s✉✃t

●✐➯ sö

Ω✳

Ω

❧➭ ♠ét t❐♣ tï② ý rỗ

ó

(, F)

0 A F

ột



➤➵✐ sè ❝➳❝ t❐♣ ❝♦♥ ❝đ❛

➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ➤♦✳ ●✐➯ sö

❦❤➠♥❣ ❣✐❛♥ ➤♦✳ ▼ét ➳♥❤ ①➵

(i) P(A)

F

(Ω, F)

❧➭ ♠ét

P : F → R ➤➢ỵ❝ ❣ä✐ ❧➭ ➤é ➤♦ ①➳❝ s✉✃t tr➟♥ F

♥Õ✉

✭tÝ♥❤ ❦❤➠♥❣ ➞♠✮❀

(ii) P(Ω) = 1 ✭tÝ♥❤ ❝❤✉➮♥ ❤ã❛✮❀
(iii)

◆Õ✉
∞

An ∈ F (n = 1, 2, 3, ...)✱ Ai ∩ Aj = Ai Aj = ∅ (i = j)
∞

An ) =

P(

P(An ) ✭tÝ♥❤ ❝é♥❣ tÝ♥❤ ➤Õ♠ ➤➢ỵ❝✮✳

n=1

n=1

❈➳❝ ➤✐Ị✉ ❦✐Ư♥
❜❛

(i) − (iii) ➤➢ỵ❝ ❣ä✐ ❧➭ ❤Ư t✐➟♥ ➤Ị ❑♦❧♠♦❣♦r♦✈ ✈Ị ①➳❝ s✉✃t✳ ❇é

(Ω, F, P) ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t✳
❚❐♣

σ

Ω ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❜✐Õ♥ ố s

số

ỗ

F

AF

ợ ọ



➤➵✐ sè ❝➳❝ ❜✐Õ♥ ❝è✳

➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❜✐Õ♥ ❝è✳

❇✐Õ♥ ❝è

Ω∈F

❇✐Õ♥ ❝è

∅∈F

❇✐Õ♥ ❝è

A = Ω\A ➤➢ỵ❝ ❣ä✐ ❧➭ ❜✐Õ♥ ❝è ➤è✐ ❧❐♣ ❝đ❛ ❜✐Õ♥ ❝è A✳

◆Õ✉

t❤×

❣ä✐ ❧➭ ❜✐Õ♥ ❝è ❝❤➽❝ ❝❤➽♥✳
❣ä✐ ❧➭ ❜✐Õ♥ ❝è ❦❤➠♥❣ t❤Ó ❝ã✳

A ∩ B = AB = tì A B

ợ ọ ế ❝è ①✉♥❣ ❦❤➽❝✳


✹

❑❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t

(Ω, F, P)

❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t ➤➬② ➤ñ ♥Õ✉ ♠ä✐

t❐♣ ❝♦♥ ❝ñ❛ ❜✐Õ♥ ❝è ❝ã ①➳❝ s✉✃t ❦❤➠♥❣ ➤Ị✉ ❧➭ ❜✐Õ♥ ❝è✳ ➜Ĩ ➤➡♥ ❣✐➯♥✱ tõ ♥❛② ✈Ò
s❛✉✱ ❦❤✐ ♥ã✐ ➤Õ♥ ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t

(Ω, F, P)✱

t❛ ❧✉➠♥ ①❡♠ ➤ã ❧➭ ❦❤➠♥❣ ❣✐❛♥

①➳❝ s✉✃t ➤➬② ➤đ✳

❈❤ó ý✳ ➜✐Ị✉ ❦✐Ư♥ (ii) tr♦♥❣ ➤Þ♥❤ ♥❣❤Ü❛ tr➟♥ ➤➯♠ ❜➯♦ r➺♥❣ ❜✐Õ♥ ❝è ❝❤➽❝ ❝❤➽♥ ❝ã
①➳❝ s✉✃t ❜➺♥❣ ✶✳ ❚✉② ♥❤✐➟♥✱ ❝ã ♥❤÷♥❣ ❜✐Õ♥ ❝è ❝ã ①➳❝ s✉✃t ❜➺♥❣ ✶ ♥❤➢♥❣ ❝❤➢❛
❝❤➽❝ ❝❤➽♥ ➤➲ ❧➭ ❜✐Õ♥ ❝è ❝❤➽❝ ❝❤➽♥✳ ◆❤÷♥❣ ❜✐Õ♥ ❝è ♥❤➢ ✈❐② ❣ä✐ ❧➭ ❜✐Õ♥ ❝è ❤➬✉
❝❤➽❝ ❝❤➽♥✳

✶✳✶✳✷✳ ❈➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ①➳❝ s✉✃t
●✐➯ sư

A, B, C, ... ❧➭ ♥❤÷♥❣ ❜✐Õ♥ ❝è✳ ❑❤✐ ➤ã✱ ①➳❝ s✉✃t ❝đ❛ ❝❤ó♥❣ ❝ã ❝➳❝ tÝ♥❤

❝❤✃t s❛✉✿

✶✳

P(∅) = 0✳

✷✳ ◆Õ✉

✸✳

P(A) = 1 − P(A)✳

✹✳ ◆Õ✉

✺✳

AB = ∅ t❤× P(A ∪ B) = P(A) + P(B)✳

A⊂B

t❤×

P(B\A) = P(B) − P(A) ✈➭ ❞♦ ➤ã P(A)

P(B)✳

P(A ∪ B) = P(A) + P(B) − P(AB)✳

✻✳

n

n

P(Ak ) −

Ak ) =

P(
k=1

k=1

1 k

P(Ak Al Am ) − ... + (−1)n−1 P(A1 A2 ...An ).

+
1 k∞
✼✳

P(
n=1

P(Ak Ai )

∞

An )

P(An )✳
n=1

✽✳ ✭❚Ý♥❤ ❧✐➟♥ tơ❝ ❝đ❛ ①➳❝ s✉✃t✮


✺

(i)

◆Õ✉

(An , n

1)

A1 ⊂ A2 ⊂ ... ⊂ An ⊂ ...✱

❧➭ ❞➲② ➤➡♥ ➤✐Ư✉ t➝♥❣✱

tå♥ t➵✐

∞

lim P(An ) = P(

n→∞

(ii)

◆Õ✉

t❤×

(An , n

An ).

n=1

1) ❧➭ ❞➲② ➤➡♥ ➤✐Ö✉ ❣✐➯♠✱ A1 ⊃ A2 ⊃ ... ⊃ An ⊃ ...✱ t❤×

tå♥ t➵✐

∞

lim P(An ) = P(

n→∞

An ).

n=1

✶✳✶✳✸✳ ❳➳❝ s✉✃t ❝ã ➤✐Ị✉ ❦✐Ư♥
➜Þ♥❤ ♥❣❤Ü❛✳

●✐➯ sö

(Ω, F, P)

❧➭ ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t✳

A, B ∈ F ✱ P(A) > 0✳

❑❤✐ ➤ã sè

P(B/A) =

P(AB)
P(A)

➤➢ỵ❝ ❣ä✐ ❧➭ ①➳❝ s✉✃t ❝ã ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ❜✐Õ♥ ❝è

B

➤è✐ ✈í✐ ❜✐Õ♥ ❝è

A✳

❚Ý♥❤ ❝❤✃t✳
✶✳

P(B/A)

0✳

✷✳ ◆Õ✉

B ⊃ A t❤× P(B/A) = 1✱ ➤➷❝ ❜✐Ưt P(Ω/A) = 1✳

✸✳ ◆Õ✉

(Bn ) ❧➭ ❞➲② ❝➳❝ ❜✐Õ♥ ❝è ➤➠✐ ♠ét ①✉♥❣ ❦❤➽❝ t❤×
∞

∞

Bn /A) =

P(
n=1
❚õ ❝➳❝ tÝ♥❤ ❝❤✃t

①➵

P(Bn /A).
n=1

1 − 3 s✉② r❛ r➺♥❣ ♥Õ✉ A ❧➭ ♠ét ❜✐Õ♥ ❝è✱ P(A) > 0 t❤× ➳♥❤

PA : F → R ①➳❝ ➤Þ♥❤ ❜ë✐ ❝➠♥❣ t❤ø❝
PA (B) = P(B/A), (∀B ∈ F)

❝ị♥❣ ❧➭ ①➳❝ s✉✃t tr➟♥
s✉✃t✳

F✳

❉♦ ➤ã

PA

❝ã ➤➬② ➤đ ❝➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ ➤é ➤♦ ①➳❝


✻

✹✳ ✭◗✉② t➽❝ ♥❤➞♥✮✳ ●✐➯ sö

A1 , A2 , ..., An (n

2)✱ ❧➭ ♥ ❜✐Õ♥ ❝è ❜✃t ❦ú s❛♦ ❝❤♦

P(A1 A2 ...An ) > 0✳ ❑❤✐ ➤ã
P(A1 A2 ...An ) = P(A1 )P(A2 /A1 )...P(An /A1 ...An−1 ).

✶✳✶✳✹✳ ❚Ý♥❤ ➤é❝ ❧❐♣ ❝đ❛ ❝➳❝ ❜✐Õ♥ ❝è
●✐➯ sư

(Ω, F, P) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t✳

➜Þ♥❤ ♥❣❤Ü❛ ✶✳ ❍❛✐ ❜✐Õ♥ ❝è A ✈➭ B

➤➢ỵ❝ ❣ä✐ ❧➭ ➤é❝ ❧❐♣ ♥Õ✉

P(AB) = P(A).P(B).

❚Ý♥❤ ❝❤✃t✳
✶✳

A✱ B

➤é❝ ❧❐♣ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐

✷✳ ❍❛✐ ❜✐Õ♥ ❝è

A

✈➭

B

P(A/B) = P(A) ❤♦➷❝ P(B/A) = P(B)✳

➤é❝ ❧❐♣ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ♠ét tr♦♥❣ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉

t❤á❛ ♠➲♥

(i) A✱ B

➤é❝ ❧❐♣❀

(ii) A✱ B ✱ ➤é❝ ❧❐♣❀
(iii) A✱ B

➤é❝ ❧❐♣✳

❉➢í✐ ➤➞② sÏ tr×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ ➤é❝ ❧❐♣ ❝đ❛ ♠ét ❤ä ❜✐Õ♥ ❝è✳

➜Þ♥❤ ♥❣❤Ü❛ ✷✳ ❍ä ❝➳❝ ❜✐Õ♥ ❝è (Ai )i∈I

➤➢ỵ❝ ❣ä✐ ❧➭ ➤é❝ ❧❐♣ ➤➠✐ ♠ét ♥Õ✉ ❤❛✐ ❜✐Õ♥

❝è ❜✃t ❦ú ❝ñ❛ ❤ä ➤Ị✉ ➤é❝ ❧❐♣✳
❍ä ❝➳❝ ❜✐Õ♥ ❝è

(Ai )i∈I

➤➢ỵ❝ ❣ä✐ ❧➭ ➤é❝ ❧❐♣ t♦➭♥ ❝ô❝ ✭❣ä✐ t➽t ❧➭ ➤é❝ ❧❐♣✮ ♥Õ✉

➤è✐ ✈í✐ ♠ä✐ ❤ä ❤÷✉ ❤➵♥ ❝➳❝ ❜✐Õ♥ ❝è

Ai1 , Ai2 , ..., Ain

❝đ❛ ❤ä ➤ã✱ t❛ ➤Ị✉ ❝ã

P(Ai1 Ai2 ...Ain ) = P(Ai1 )P(Ai2 )...P(Ain ).
▼ét ❤ä ➤é❝ ❧❐♣ t❤× ➤é❝ ❧❐♣ ➤➠✐ ♠ét✳ ❚✉② ♥❤✐➟♥ ➤✐Ị✉ ♥❣➢ỵ❝ ❧➵✐ ♥ã✐ ❝❤✉♥❣
❦❤➠♥❣ ➤ó♥❣✳


✼

➜è✐ ✈í✐ ❞➲② ➤é❝ ❧❐♣ ❝➳❝ ❜✐Õ♥ ❝è✱ t❛ ❝ã tÝ♥❤ ❝❤✃t q✉❛♥ trä♥❣ s❛✉ ➤➞②✱ ❣ä✐ ❧➭
❇ỉ ➤Ị ❇♦r❡❧ ✲ ❈❛♥t❡❧❧✐✳

➜Þ♥❤ ❧ý✳

✭❇ỉ ➤Ị ❇♦r❡❧ ✲ ❈❛♥t❡❧❧✐✮✳ ●✐➯ sư

(An , n

1)

❧➭ ❞➲② ❝➳❝ ❜✐Õ♥ ❝è✳ ❑❤✐

➤ã

∞

(i)

P(An ) < ∞ t❤× P(lim sup An ) = 0❀

◆Õ✉

n=1
∞

(ii)

P(An ) = ∞ ✈➭ (An , n

◆Õ✉

1) ➤é❝ ❧❐♣ t❤× P(lim sup An ) = 1,

n=1
tr♦♥❣ ➤ã

∞

∞

Ak .

lim sup An =
n=1 k=n

❚õ ➤Þ♥❤ ❧ý tr➟♥✱ ❝ã t❤Ĩ s✉② r❛ ♥❣❛② ❤Ư q✉➯ s❛✉ ➤➞②

❍Ư q✉➯✳ ✭▲✉❐t ✵ ✲✶ ❇♦r❡❧ ✲ ❈❛♥t❡❧❧✐✮✳ ◆Õ✉ (An , n

1) ❧➭ ❞➲② ❜✐Õ♥ ❝è ➤é❝ ❧❐♣✱

t❤× P(lim sup An ) ❝❤Ø ❝ã t❤Ĩ ♥❤❐♥ ♠ét tr♦♥❣ ❤❛✐ ❣✐➳ trÞ tù t ỗ


P(An ) ộ tụ ♣❤➞♥ ❦ú✳
n=1

✶✳✷✳ ➳♥❤ ①➵ ➤♦ ➤➢ỵ❝ ✈➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥
✶✳✷✳✶✳ ợ
ị ĩ

sử

(1 , F1 )



(2 , F2 )

❧➭ ❤❛✐ ❦❤➠♥❣ ❣✐❛♥ ➤♦✳

➳♥❤ ①➵ X

:

Ω1 → Ω2 ❣ä✐ ❧➭ ➳♥❤ ①➵ F1 /F2 ➤♦ ➤➢ỵ❝ ♥Õ✉ ✈í✐ ♠ä✐ B ∈ F2 t❤× X −1 (B) ∈ F1 ✳

❚Ý♥❤ ❝❤✃t✳ ●✐➯ sö (Ω1 , F1 )✱ (Ω2 , F2 ) ✈➭ (Ω3 , F3 ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ➤♦✳
✶✳ ◆Õ✉

X

F1 ⊂ G1 ✱ G2 ⊂ F2

❧➭ ➳♥❤ ①➵

✷✳ ●✐➯ sö

F2 /F3
✸✳ ●✐➯ sö

G1 /G2

✈➭

X : Ω1 → Ω2

❧➭ ➳♥❤ ①➵

F1 /F2

➤♦ ➤➢ỵ❝✳

X : Ω1 → Ω2

❧➭ ➳♥❤ ①➵

➤♦ ➤➢ỵ❝✳ ❑❤✐ ➤ã

F1 /F2

➤♦ ➤➢ỵ❝✱

Y ◦ X : Ω1 → Ω3

Y : Ω2 → Ω3

❧➭ ➳♥❤ ①➵

F1 /F3

X −1 (C) ∈ F1

✈í✐ ♠ä✐

C ∈ C✳

❧➭ ➳♥❤ ①➵

➤♦ ➤➢ỵ❝✳

F2 = σ(C)✳ ❑❤✐ ➤ã X : (Ω1 , F1 ) → (Ω2 , F2 ) ❧➭ F1 /F2

✈➭ ❝❤Ø ❦❤✐

➤♦ ➤➢ỵ❝ tì

ợ

✽

❍Ư q✉➯✳ ●✐➯ sư (Ω1 , τ1 )✱ (Ω2 , τ2 ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠✱ ➳♥❤ ①➵ X : Ω1 → Ω2
X

❧✐➟♥ tô❝✳ ❑❤✐ ➤ã
t➢➡♥❣ ø♥❣ ❧➭ ❝➳❝

❧➭ ➳♥❤ ①➵

B(Ω1 )/B(Ω2 )

σ ✲➤➵✐ sè ❇♦r❡❧ tr➟♥ Ω1

✈➭

➤♦ ➤➢ỵ❝✱ tr♦♥❣ ➤ã

B(Ω1 ), B(Ω2 )

Ω2 ✳

✶✳✷✳✷✳ ❇✐Õ♥ ♥❣➱✉ ♥❤✐➟♥
➜Þ♥❤ ♥❣❤Ü❛✳
❝đ❛

G

σ

●✐➯ sư

F✳

✲ ➤➵✐ sè

(Ω, F, P)

❑❤✐ ➤ã ➳♥❤ ①➵

✲ ➤♦ ➤➢ỵ❝ ế ó

X 1 (B) G)
ợ tì

X

❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t✱

X : Ω → R

G/B(R)

G

❧➭

σ

➤➢ỵ❝ ❣ä✐ ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

➤♦ ➤➢ỵ❝ ✭tø❝ ❧➭ ✈í✐ ♠ä✐

❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ➤➷❝ ❜✐Öt✱ ❦❤✐

✲ ➤➵✐ sè ❝♦♥

X

B ∈ B(R)

❧➭ ❜✐Õ♥ ♥❣➱✉

F

tì



ợ ọ ột ế ♥❤✐➟♥✳

❍✐Ĩ♥ ♥❤✐➟♥✱ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥
t❤✃② r➺♥❣ ♥Õ✉

X

G

✲ ➤♦ ➤➢ỵ❝ ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✳ ▼➷t ❦❤➳❝✱ ❞Ơ

❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ t❤× ❤ä

σ(X) = {X −1 (B) : B ∈ B(R)}
❧❐♣ t❤➭♥❤ ♠ét
s✐♥❤ ❜ë✐

X✳

σ

✲ ➤➵✐ sè ❝♦♥ ❝ñ❛

➜ã ❧➭

❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

G

σ

σ

✲ ➤➵✐ sè ❜Ð ♥❤✃t ♠➭

X

✲ ➤♦ ➤➢ỵ❝ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐

◆Õ✉ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

X

F✱ σ

✲ ➤➵✐ sè

✲ ➤➵✐ sè ♥➭② ❣ä✐ ❧➭

σ

✲ ➤➵✐ sè

➤♦ ➤➢ỵ❝✳ ❚õ ➤ã s✉② r❛ r➺♥❣

X

❧➭

σ(X) ⊂ G

ỉ ữ trị tì ó ợ ọ ế


ế ò ợ ọ ợ

í t
ị ý X

❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ♠ét tr♦♥❣ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉

➤➞② t❤á❛ ♠➲♥

(i) (X < a) := (ω : X(ω) < a) ∈ F
(ii) (X

a) := (ω : X(ω)

✈í✐ ♠ä✐

a) ∈ F

(iii) (X > a) := (ω : X(ω) > a) ∈ F

a ∈ R✳

✈í✐ ♠ä✐

a ∈ R✳

✈í✐ ♠ä✐

a ∈ R✳


✾

(iv) (X

a) ∈ F

a) := (ω : X(ω)

➜Þ♥❤ ❧ý ✷✳

●✐➯ sö

X1 , X2 , ..., Xn

(Ω, F, P)✱ f : Rn → R

✈í✐ ♠ä✐

a ∈ R✳

❧➭ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ù ị tr

ợ ó ĩ

f



B(Rn )/B(R)

➤➢ỵ❝✮✳

❑❤✐ ➤ã

Y = f (X1 , ..., Xn ) :Ω → R

ω → f (X1 (ω), ..., Xn (ω))
❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✳

➜Þ♥❤ ❧ý ✸✳

●✐➯ sư

(Ω, F, P)✳

❑❤✐ ➤ã✱ ♥Õ✉

limXn ✱ lim Xn
n→∞

(Xn , n

1)

❧➭ ❞➲② ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❝ï♥❣ ị tr

inf Xn , sup Xn
n

ữ tì

n

inf Xn , sup Xn ✱ limXn ✱
n

n

✭♥Õ✉ tå♥ t➵✐✮ ➤Ò✉ ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✳

➜Þ♥❤ ❧ý ✹✳ ◆Õ✉ X

❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➠♥❣ ➞♠ t❤× tå♥ t➵✐ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

➤➡♥ ❣✐➯♥✱ ❦❤➠♥❣ ➞♠

1) s❛♦ ❝❤♦ Xn ↑ X

(Xn , n

✭❦❤✐

n → ∞✮✳

❈❤ó ý r➺♥❣ ❝➳❝ tÝ♥❤ ❝❤✃t tr➟♥ ❝đ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❝ã t❤Ĩ ♠ë ré♥❣ ❝❤♦ ❜✐Õ♥
♥❣➱✉ ♥❤✐➟♥

G

✲ ➤♦ ➤➢ỵ❝ ❜✃t ❦ú✳

✶✳✷✳✸✳ P❤➞♥ ♣❤è✐ ①➳❝ s✉✃t
➜Þ♥❤ ♥❣❤Ü❛✳ ●✐➯ sư (Ω, F, P) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t✱ X : Ω → R ❧➭ ❜✐Õ♥ ♥❣➱✉
♥❤✐➟♥✳ ❑❤✐ ➤ã ❤➭♠ t❐♣

PX : B(R) → R

B → PX (B) = P(X −1 (B))
➤➢ỵ❝ ❣ä✐ ❧➭ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛

X✳

❚Ý♥❤ ❝❤✃t✳
✶✳

PX

❧➭ ➤é ➤♦ ①➳❝ s✉✃t tr➟♥

✷✳ ◆Õ✉

Q

B(R)✳

❧➭ ➤é ➤♦ ①➳❝ s✉✃t tr➟♥

B(R)

t❤×

Q

❧➭ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝đ❛ ♠ét

❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ♥➭♦ ➤ã✳

❈❤ó ý✳ ❚➢➡♥❣ ø♥❣ ❣✐÷❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ✈➭ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝đ❛ ❝❤ó♥❣ ❦❤➠♥❣
♣❤➯✐ ❧➭ t➢➡♥❣ ø♥❣ ✶✲✶✳ ◆❤÷♥❣ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❝ã ❝ï♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ➤➢ỵ❝


✶✵

❣ä✐ ❧➭ ♥❤÷♥❣ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❝ï♥❣ ♣❤➞♥ ♣❤è✐✳

✶✳✷✳✹✳ ❍➭♠ ♣❤➞♥ ♣❤è✐
➜Þ♥❤ ♥❣❤Ü❛✳ ●✐➯ sư (Ω, F, P) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t✱ X : Ω −→ R ❧➭ ❜✐Õ♥
♥❣➱✉ ♥❤✐➟♥✳ ❑❤✐ ➤ã✱ ❤➭♠ sè
❧➭ ❤➭♠ ♣❤➞♥ ♣❤è✐ ❝ñ❛

FX (x) = P(X < x) = P(ω : X(ω) < x) ➤➢ỵ❝ ❣ä✐

X✳

◆❤❐♥ ①Ðt✳ FX (x) = P X −1 (−∞, x) = PX [(−∞, x)]✳
❚Ý♥❤ ❝❤✃t✳
✶✳

0

✷✳ ◆Õ✉

FX

1✳

a

t❤×

F (b) − F (a) = P(a

X < b)❀

❞♦ ➤ã

F (x)

❧➭ ❤➭♠ ❦❤➠♥❣

❣✐➯♠✳

✸✳

✹✳

lim F (x) = 1❀ lim F (x) = 0✳

x→+∞

x→−∞

lim F (x) = F (a)
x↑a

✈➭

lim F (x) = P(X
x↓a

a)✳

❉♦ ➤ã

F (x)

❧✐➟♥ tô❝ tr➳✐ t➵✐

F (x) ❧✐➟♥ tô❝ t➵✐ a ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ P(a) = 0✳

♠ä✐ ➤✐Ĩ♠✱

✶✳✷✳✺✳ ❑ú ✈ä♥❣
➜Þ♥❤ ♥❣❤Ü❛✳

●✐➯ sư

X : (Ω, F, P) −→ (R, B(R))

tÝ❝❤ ♣❤➞♥ ▲❡❜❡s❣✉❡ ❝ñ❛

X

✈➭ ❦ý ❤✐Ö✉ ❧➭

X

t❤❡♦ ➤é ➤♦

P

❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✳ ❑❤✐ ➤ã

✭♥Õ✉ tå♥ t➵✐✮ ➤➢ỵ❝ ❣ä✐ ❧➭ ❦ú ✈ä♥❣ ❝đ❛

EX ✳

❱❐②

EX =

XdP.
Ω

◆Õ✉ tå♥ t➵✐

E|X|p < ∞ ✭p > 0✮✱ t❤× t❛ ♥ã✐ X

E|X| < tì X

ợ ọ ế ❦❤➯ tÝ❝❤✳

❚Ý♥❤ ❝❤✃t✳ ❑ú ✈ä♥❣ ❝ã ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉ ➤➞②
✶✳ ◆Õ✉

X

✷✳ ◆Õ✉

X=C

❦❤➯ tÝ❝❤ ❜❐❝ ♣✳ ➜➷❝ ❜✐Ưt✱ ♥Õ✉

0 t❤× EX
t❤×

0✳

EX = C ✳


✶✶

✸✳ ◆Õ✉ tå♥ t➵✐

EX

t❤× ✈í✐ ♠ä✐

✹✳ ◆Õ✉ tå♥ t➵✐

EX

✈➭

✺✳ ◆Õ✉

✻✳

X

EX =

EY

t❤×

C ∈ R✱ t❛ ❝ã E(CX) = CEX ✳

E(X ± Y ) = EX ± EY ✳

0 ✈➭ EX = 0 t❤× X = 0✳








i xi pi ♥Õ✉

X

rê✐ r➵❝ ♥❤❐♥ ❝➳❝ ❣✐➳ trÞ

x1 , x2 , ...

✈í✐ P(X = xi ) = pi ✳




+∞

 −∞
xp(x)dx ♥Õ✉ X ❧✐➟♥ tô❝ ❝ã ❤➭♠ ♠❐t ➤é p✳

f : R → R ❧➭ ❤➭♠ ợ Y = f (X) tì

ổ qt ế

EY =









i f (xi )pi ♥Õ✉

X

rê✐ r➵❝ ♥❤❐♥ ❝➳❝ ❣✐➳ trÞ

✈í✐ P(X = xi ) = pi ✳





 +∞ f (x)p(x)dx ♥Õ✉ X

−∞

✼✳ ✭❇✃t ➤➻♥❣ t❤ø❝ ▼❛r❦♦✈✮✳ ●✐➯ sư
✈í✐ ♠ä✐

X

x1 , x2 , ...

❧✐➟♥ tô❝ ❝ã ❤➭♠ ♠❐t ➤é

p✳

❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➠♥❣ ➞♠✳ ❑❤✐ ➤ã

ε > 0 t❛ ❝ã
P(X

ε)

EX

.
ε

✶✳✷✳✻✳ P❤➢➡♥❣ s❛✐
➜Þ♥❤ ♥❣❤Ü❛✳

●✐➯ sư

X

❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✳ ❑❤✐ ➤ã✱ sè

✭♥Õ✉ tå♥ t➵✐✮ ➤➢ỵ❝ ❣ä✐ ❧➭ ♣❤➢➡♥❣ s❛✐ ❝đ❛
P❤➢➡♥❣ s❛✐ ❝đ❛ ❜✐Õ♥

X

DX = E(X EX)2

X

ò ợ ý ệ

varX

í ❝❤✃t✳ P❤➢➡♥❣ s❛✐ ❝ã ♥❤÷♥❣ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ s❛✉ ➤➞②✿
✶✳

DX = EX 2 − (EX)2 ✳

✷✳

DX

✸✳

DX = 0 ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ X = EX = C h.c.c✳

✹✳

D(CX) = C 2 DX ✳

0✳


✶✷

✺✳ ✭❇✃t ➤➻♥❣ t❤ø❝ ❈❤❡❜②s❤❡✈✮ ●✐➯ sư
♥Õ✉ tå♥ t➵✐

DX

t❤× ✈í✐ ♠ä✐

X

ε > 0✱ t❛ ❝ã

P(|X − EX|
tr♦♥❣ ➤ã

C

❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❜✃t ❦ú✳ ❑❤✐ ➤ã

DX
,
ε2

ε)

❧➭ ❤➺♥❣ sè✱ ❤➬✉ ❝❤➽❝ ❝❤➽♥ ợ ết ọ ở

h.c.c

ế ộ
ị ♥❣❤Ü❛✳

●✐➯ sö

(Ω, F, P)

{Ci : i ∈ I, Ci ⊂ F}

❧➭ ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t✳ ❍ä ❝➳❝ ❧í♣ ❜✐Õ♥ ❝è

➤➢ỵ❝ ❣ä✐ ❧➭ ➤é❝ ❧❐♣

(➤é❝

❧❐♣ ➤➠✐ ♠ét) ♥Õ✉ ✈í✐ ♠ä✐

Ai ∈ Ci ✱ ❤ä ❜✐Õ♥ ❝è {Ai , i ∈ I} ➤é❝ ❧❐♣ ✭t➢➡♥❣ ø♥❣✱ ➤é❝ ❧❐♣ ➤➠✐ ♠ét✮✳
❍ä ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥
❤ä

{Xi , i ∈ I} ➤➢ỵ❝ ❣ä✐ ❧➭ ➤é❝ ❧❐♣ (➤é❝ ❧❐♣ ➤➠✐ ♠ét) ♥Õ✉

σ ✲➤➵✐ sè {σ(Xi ), i ∈ I} ➤é❝ ❧❐♣ ✭t➢➡♥❣ ø♥❣✱ ➤é❝ ❧❐♣ ➤➠✐ ♠ét✮✳

❚Ý♥❤ ❝❤✃t✳
✶✳ ●✐➯ sö

{Xi , i ∈ I}

❧➭ ❤ä ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣✱

❤➭♠ ➤♦ ➤➢ỵ❝✳ ❑❤✐ ➤ã ❤ä

✷✳ ◆Õ✉

X1 , X2 , . . . , Xn

fi : R → R (i ∈ I)

❧➭

{fi (Xi ), i ∈ I} ➤é❝ ❧❐♣✳

❧➭ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣ t❤×

E(X1 X2 . . . Xn ) = EX1 EX2 . . . EXn .
✸✳ ◆Õ✉

X1 , X2 , ..., Xn

❧➭ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣ ➤➠✐ ♠ét t❤×

D(X1 + · · · + Xn ) = DX1 + · · · + DXn .

✶✳✷✳✽✳ ❈➳❝ ❞➵♥❣ ❤é✐ tơ
➜Þ♥❤ ♥❣❤Ü❛✳ ●✐➯ sư {X, Xn , n
❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t

•

❉➲②

N ∈F

{Xn }n

s❛♦ ❝❤♦

1} ❧➭ ❤ä ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❝ï♥❣ ①➳❝ ➤Þ♥❤ tr➟♥

(Ω, F, P)✳ ❚❛ ♥ã✐✿

1 ❤é✐ tô ❤➬✉ ❝❤➽❝ ❝❤➽♥ ➤Õ♥

X

❦❤✐

n→∞

♥Õ✉ tå♥ t➵✐ t❐♣

P(N ) = 0 ✈➭ Xn (ω) → X(ω) ❦❤✐ n → ∞ ✈í✐ ♠ä✐ ω ∈ Ω\N ✳


✶✸

❑ý ❤✐Ö✉

h. c. c.

Xn → X h.c.c ❤♦➷❝ Xn −−−→ X

• ❉➲② {Xn }n

1 ❤é✐ tơ ➤➬② ➤đ ➤Õ♥

X

❦❤✐

n → ∞✳

❦❤✐

n → ∞ ♥Õ✉ ✈í✐ ♠ä✐ ε > 0 t❤×

∞

P(|Xn − X| > ε) < ∞.
n=1
❑ý ❤✐Ư✉

c

Xn →
− X

• ❉➲② {Xn }n

❦❤✐

n → ∞✳

1 ❤é✐ tô t❤❡♦ ①➳❝ s✉✃t ➤Õ♥

X

❦❤✐

n → ∞ ♥Õ✉ ✈í✐ ♠ä✐ ε > 0

t❤×

lim P(|Xn − X| > ε) = 0.

n→∞
❑ý ❤✐Ư✉

•

❉➲②

P

Xn −→ X

{Xn }n

n →



1 ộ tụ t tr ì

ế

Lp

Xn X

ã {Xn }n

❦❤✐

1 ❤é✐ tô t❤❡♦ ♣❤➞♥ ♣❤è✐ (❤é✐ tô ②Õ✉) ➤Õ♥

lim Fn (x) = F (x)

Fn

✈➭

F

✈í✐ ♠ä✐

X

❦❤✐

n → ∞ ♥Õ✉

♥Õ✉

❑ý ❤✐Ư✉

x ∈ C(F ).

t➢➡♥❣ ø♥❣ ❧➭ ❤➭♠ ♣❤➞♥ ♣❤è✐ ❝đ❛ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

X ✱ C(F ) ❧➭ t❐♣ ❤ỵ♣ ❝➳❝ ➤✐Ĩ♠ ♠➭ t➵✐ ➤ã F

❧✐➟♥ tơ❝✳

D

Xn −
→ X✳

❚Ý♥❤ ❝❤✃t✳
h. c. c

Xn −−−→ X

❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✈í✐ ♠ä✐

ε > 0✱

lim P(sup |Xm − X| > ε) = 0.

n→∞

✸✳ ◆Õ✉

n→∞

n → ∞✳

n→∞

✷✳ ◆Õ✉

❦❤✐

n→∞

❑ý ❤✐Ö✉

✶✳

X

1) ❦❤➯ tÝ❝❤ ❜❐❝ p ✈➭ lim E|Xn − X|p = 0✳

X, Xn (n

❚r♦♥❣ ➤ã

p>0

c

Xn →
− X

t❤×

m n

h. c. c

Xn −−−→ X ✳
∞

E|Xn − X|p < ∞
n=1

Xn

✈➭


✶✹

✈í✐

h. c. c

p > 0 ♥➭♦ ➤ã t❤× Xn −−−→ X.
{Xn , n

✹✳ ●✐➯ sö

1}

❧➭ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣✳ ◆Õ✉

h. c. c

Xn −−−→ C

t❤×

c

Xn →
− C✳
h. c. c

✺✳ ◆Õ✉

Xn −−−→ X

✻✳ ◆Õ✉

Xn −→ X

✼✳ ◆Õ✉

Xn −
→X

P

D

✽✳ ●✐➯ sö

{Xn }n

❤♦➷❝

Lp

Xn −→ X

✈í✐

P

p > 0 ♥➭♦ ➤ã t❤× Xn −→ X.

D

Xn −
→ X.

t❤×

P

P(X = C) = 1 t❤× Xn −→ X.

✈➭

1 ✈➭

{Yn }n

1 ❧➭ ❝➳❝ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥
∞

P(Xn = Yn ) < ∞.

n=1
❑❤✐ ➤ã
✐✮ ◆Õ✉

h. c. c

Xn −−−→ X

t❤×

h. c. c

Yn −−−→ X

∞

❦❤✐

n → ∞;

∞

Xn

✐✐✮ ế

ộ tụ

h.c.c tì

n=1

Yn

ộ tụ

h.c.c

n=1

tí ề
ị ĩ
ề ế ✈í✐ ♠ä✐

❍ä ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

{Xi , i ∈ I}

➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➯ tÝ❝❤

> 0✱ tå♥ t➵✐ δ > 0 s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ❜✐Õ♥ ❝è A ♠➭ P(A) < δ

t❤×

|Xi |dP <

sup
i∈I
❍ä ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

✈➭

sup E|Xi | < ∞.
i∈I

A

{Xi , i ∈ I} ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➯ tÝ❝❤ ➤Ị✉ tr➟♥ ✭❤ä ❦❤➯ tÝ❝❤

➤Ị✉ ❞➢í✐✮ ♥Õ✉ ❤ä ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

{Xi+ , i ∈ I}( t➢➡♥❣ ø♥❣{Xi− , i ∈ I})

❧➭ ❤ä

❦❤➯ tÝ❝❤ ➤Ị✉✳

❚Ý♥❤ ❝❤✃t ✶✳✸✳✷✳ ❚õ ➤Þ♥❤ ♥❣❤Ü❛ t❛ s✉② r❛ ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉ ➤➞②✿
(i) {Xi }i∈I
(ii)
(iii)

◆Õ✉

◆Õ✉

❦❤➯ tÝ❝❤ ➤Ò✉ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐

{Xi }i∈I
{Xi }i∈I

✈➭

{Yi }i∈I

{|Xi |}i∈I

❦❤➯ tÝ❝❤ ➤Ò✉ t❤×

❦❤➯ tÝ❝❤ ➤Ị✉❀

{Xi + Yi }i∈I

❦❤➯ tÝ❝❤ ➤Ị✉❀

❦❤➯ tÝ❝❤ ➤Ị✉ t❤× ♠ä✐ ❤ä ❝♦♥ ❝đ❛ ♥ã ❝ị♥❣ ❦❤➯ tÝ❝❤ ➤Ị✉❀


✶✺

(iv) {Xi }i∈I

❦❤➯ tÝ❝❤ ➤Ò✉ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ♥ã ❦❤➯ tÝ❝❤ ➤Ị✉ tr➟♥ ✈➭ ❦❤➯ tÝ❝❤ ➤Ị✉

❞➢í✐❀

(v)

◆Õ✉

|Xi |

Y

✈í✐ ♠ä✐

i∈I

EY < tì {Xi }iI



tí ề

ị í ❦❤➯ tÝ❝❤ ➤Ò✉✮ ❍ä ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ {Xi }i∈I

❦❤➯ tÝ❝❤

➤Ò✉ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐

|Xi |dP = 0.

lim sup

a→∞ i∈I

|Xi |>a

➜Þ♥❤ ❧Ý ✶✳✸✳✹✳
✭✐✮ ◆Õ✉ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

❦❤✐

n → ∞ t❤× X Lp

ợ ế

P

Xn X



ị í



{Xn }n

{Xn }n

1 ❦❤➯ tÝ❝❤ ➤Ị✉ ✈í✐ p

Lp

Xn −→ X
1

⊂ Lp

❦❤✐

✈➭

n → ∞✳
Lp

Xn −→ X

❦❤✐

n → ∞

t❤×

X ∈ Lp ✱

n → ∞ ✈➭ {|Xn |p , n ∈ N} ❦❤➯ tÝ❝❤ ➤Ò✉✳
✭❱❛❧❧❡Ð P♦✉ss✐♥✮ ❍ä ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

✈➭ ❝❤Ø ❦❤✐ tå♥ t➵✐ ❤➭♠ ❧å✐✱ t➝♥❣✱ ❦❤➠♥❣ ➞♠

sup E(G(|Xi |)) < ∞.
i∈I

P

> 0 ♥➭♦ ➤ã ✈➭ Xn → X

G(x)

{Xi }i∈I

s❛♦ ❝❤♦

❦❤➯ tÝ❝❤ ➤Ò✉ ❦❤✐

G(x)
=∞
x→∞ x
lim

✈➭


✶✻

❈❤➢➡♥❣ ✷✳ ❑❤➯ tÝ❝❤ ➤Ò✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦ ✈➭ ❧✉❐t ♠➵♥❤
sè ❧í♥

❚r♦♥❣ s✉èt ❝❤➢➡♥❣ ♥➭②✱ t❛ ❧✉➠♥ ❣✐➯ sư r➺♥❣

{Xn }n

1 ❧➭ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✱

n

Sn =

Xj

✈➭

{f (n)}n

1 ❧➭ ❞➲② sè t➝♥❣

f (n) > 0 ✈➭ f (n) → ∞✳

j=1

✷✳✶✳ ▲✉❐t ♠➵♥❤ sè ❧í♥ ➤è✐ ✈í✐ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➠♥❣ ➞♠
➜Þ♥❤ ❧ý s❛✉ ♠ë ré♥❣ ➜Þ♥❤ ❧ý ✶ tr♦♥❣ ❬✽❪✳

➜Þ♥❤ ❧ý ✷✳✶✳✶✳ ❈❤♦ {Xn }n

1 ❧➭ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➠♥❣ ➞♠ ✈í✐ ♣❤➢➡♥❣ s❛✐

❤÷✉ ❤➵♥✳ ●✐➯ sư r➺♥❣

n

E(|Xk |/f (n))] = A < ∞❀

(i) sup[
n 1

(ii)

k=1

tå♥ t➵✐ ♠ét ❞➲② ❦Ð♣

{ρij } ❝➳❝ sè t❤ù❝ ❦❤➠♥❣ ➞♠ s❛♦ ❝❤♦
n

n

ρij

var(Sn )

n

1;

i=1 j=1
∞

∞

ρij /(f (i ∨ j))2 < ∞, (i ∨ j = max(i, j)).

(iii)
i=1 j=1
❑❤✐ ➤ã

[S(n) − E(S(n))]/f (n) → 0 h.c.c n .

ứ

ớ ỗ

ớ

> 1, > 0

l

T = [A/ ]

n ❝è ➤Þ♥❤ ❝❤ó♥❣ t❛ ➤Þ♥❤ ♥❣❤Ü❛ ❞➲② {ml }l
m1 = inf{m

✈➭ ✈í✐

✈➭ ➤➷t

0 : αm

❧➭ ♣❤➬♥ ♥❣✉②➟♥ ❝đ❛

A/

✳

1 ♥❤➢ s❛✉✿

f (n) < αm+1 ,

✈í✐ ♥ ♥➭♦ ➤ã},

2✱
ml = inf{m > ml−1 : αm

❈❤ó ý r➺♥❣ tõ

f (n) ↑ ∞✱

❞➢➡♥❣ t❤♦➯ ♠➲♥

0

s✉② r❛ r➺♥❣

f (n) < αm+1 ,
{ml }l

✈í✐ ♥ ♥➭♦ ➤ã}.

1 ❧➭ ♠ét ❞➲② ❝♦♥ ❝➳❝ sè ♥❣✉②➟♥

m1 < m2 < m3 < ... ↑

ớ ỗ

l = 1, 2, ...

ớ

✶✼

s = 0, 1, 2, ..., T ✱ ➤➷t
Al (s) = {k : αml
❚❛ ➤Þ♥❤ ♥❣❤Ü❛

f (k) < αml +1 }✳

❱í✐

❚õ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ❞➲②

{ml }l

Al (s)

❦❤➠♥❣ ♣❤➯✐ ❧➭

1 ✱ s✉② r❛ r➺♥❣✱

f (kl± (s))

α ml

f (kl± (s)) < αml +1 ✳

s = 0, 1, 2, ..., T ✱ t❛ ❝ã
(f (kl± (s)))−2

(f (kl± (s)))−2 var(S(kl± (s)))

∞

∞

∞

∞

l:kl± (s) i∨j
∞

i=1 j=1
∞

∞

∞

∞

∞

ρij
i=1 j=1

α4
α2 − 1
❚❛ ➤Þ♥❤ ♥❣❤Ü❛

∞

α

l:kl± (s) i∨j
∞

i=1 j=1

α2
α2 − 1

α

m=ml(i,j)
∞

α−2ml

ρij
∞

.

l=l(i,j)
∞

ρij α−2ml(i,j)
i=1 j=1

ρij (f (i ∨ j)−2 < ∞.

i=1 j=1

Z(n) = [S(n)−ES(n)]/f (n)✳ ❙ư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ❈❤❡❜②s❤❡✈
> 0✱

t❛ ❝ã✱ ✈í✐ ♠ä✐

∞

∞

P(|Z(kl± (s))|

(f (kl± (s)))−2 var(S(kl± (s))) < ∞.

> )

l=1

l=1

❙✉② r ớ ỗ

s = 0, 1, 2, ..., T, Z(kl (s)) → 0

s = 0, 1, 2, ..., T, Z(kl± (s)) → 0

sè tù ♥❤✐➟♥
s❛♦ ❝❤♦✱

∞

−2ml

l:αml +1 >f (i∨j)
∞
−2m

α−2ml

ρij
i=1 j=1

ρij
i=1 j=1



(f (kl (s)))2

ij

=

ij
i=1 j=1


l=1

l=1

ớ ỗ

kl (s) kl (s)





ủ

ế

Al (s) t rỗ tì t t kl+ = kl = inf{k : αml

kl± (s) ✈í✐ f (n) ↑ ∞ t❤× f (i)

i

1
ES(k) [s , (s + 1) )}.
f (k)

kl+ (s) = sup Al (s), kl (s) = inf Al (s)

t rỗ ◆❣➢ỵ❝ ❧➵✐✱ ♥Õ✉

✈➭ ♥Õ✉

f (k) < αml +1 ,

n✱

α ml

tå♥ t➵✐

l = l(n)

✈➭

f (n) < αml +1 ✈➭

kl± (s)✱ t❛ ❝ã kl−

n

kl+ ✳

➤➬② ➤ñ✱ ❦❤✐

❤➬✉ ❝❤➽❝ ❝❤➽♥✱ ❦❤✐

l → ∞✳

l → ∞✳

❉♦ ➤ã✱

❱í✐ ♠ä✐

s = s(n) ✈í✐ lim l(n) = ∞, 0 s(n) T ✱
n→∞
1
ES(n) ∈ [s , (s + 1) ). ❚❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛
f (n)


✶✽

1
ES(kl± (s)) ∈ [s , (s + 1) )
±
f (kl (s))
➤♦➵♥ [s , (s + 1) ) ❝ã ➤é ❞➭✐ ❜➺♥❣ ♥➟♥

❱×

✈➭

1
ES(n) ∈ [s , (s + 1) ).
f (n)

▼➭

1
1
ES(kl± (s)) −

ES(n) < .
±
f (n)
f (kl (s)
❚õ ➤ã s✉② r❛ ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳

❍Ư q✉➯ ✷✳✶✳✷✳ ❈❤♦ {Xn }n

1 ❧➭ ♠ét ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➠♥❣ ➞♠ ✈í✐ ♣❤➢➡♥❣

s❛✐ ❤÷✉ ❤➵♥✳ ●✐➯ sö

n

E(|Xk − E(Xk )|)/f (n)] < ∞;

(i) sup[
n>1

k=1

(ii) E(Xi Xj )

E(Xi )E(Xj ), ∀i = j;

∞

(f (n))−2 var(Xn ) < ∞✳

(iii)

n=1

❑❤✐ ➤ã

[S(n) − E(S(n))]/f (n) → 0 h.c.c ❦❤✐ n → ∞.

❈❤ø♥❣ ♠✐♥❤✳

➜➷t

ρij = max E(Xi Xj ) − E(Xi )E(Xj ), 0

✳ ❑❤✐ ➤ã✱ tõ ➤✐Ị✉

n
❦✐Ư♥

(ii)

s✉② r❛

❦✐Ư♥

(ii)

✈➭

varSn

(iii)

t➵✐ ♠ét ❞➲②

✈➭

ρij = 0

❈❤♦

{Bn }n

{Xn }n

n

E(Xi I(Xi ∈ Bnc ) = o(f (n))❀
i=1
n
n 1

E(|Xk |I(Xk ∈ Bn ))/f (n)] < ∞;
k=1

❝đ❛ ❤Ư q✉➯✱ t❛

1 ❝➳❝ t❐♣ ❇♦r❡❧ t❤♦➯ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➞②✿

n=1

(c) sup[

(i)

❞♦ ➤ã ❝➳❝ ➤✐Ị✉

1 ❧➭ ♠ét ❞➲② ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ❦❤➠♥❣ ➞♠✱ s❛♦ ❝❤♦ tå♥

P(Xn ∈ Bnc ) < ∞;

(b)

i = j✱

(i) ❝đ❛ ➤Þ♥❤ ❧ý✳ ❉♦ ó ệ q ợ ứ



(a)

ớ ọ

i=1
ủ ị ý ợ t❤á❛ ♠➲♥✳ ❚õ ➤✐Ị✉ ❦✐Ư♥

s✉② r❛ ➤➢ỵ❝ ➤✐Ị✉ ❦✐Ư♥

❍Ư q✉➯ ✷✳✶✳✸✳

varXi

✶✾

(d)

1✱ tå♥ t➵✐ ❞➲② ❦Ð♣ {ρij } ❝➳❝ sè t❤ù❝ s

n

ớ ỗ

n

n

2

E |Sn E(Sn )|

ij
i=1 j=1







i=1 j=1
tr ➤ã

Bnc

ρij
< ∞, i ∨ j = max(i, j);
(f (i ∨ j))2
n

❧➭ ♣❤➬♥ ❜ï ❝ñ❛

Bn

✈➭

Xk I(Xk ∈ Bk )✳

Sn =
k=1

❑❤✐ ➤ã✱

(f (n))−1 [S(n) − E(S(n))] → 0 h.c.c ❦❤✐ n → ∞✳

❈❤ø♥❣ ♠✐♥❤✳

➜➷t

Yn = Xn I(Xn ∈ Bn ), n

1✳

❚õ ✭❝✮ ✈➭ ✭❞✮ s✉② r❛

{Yn }n

1

t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➤Þ♥❤ ❧ý✳ ❉♦ ➤ã

n
−1

(Yi − E(Yi )) → 0 h.c.c ❦❤✐ n → ∞.

(f (n))

i=1
❑Õt ❤ỵ♣ ➤✐Ị✉ ♥➭② ✈í✐ ✭❜✮✱ t❛ ❝ã

n
−1

(Yi − E(Xi )) → 0 h.c.c ❦❤✐ n → ∞.

(f (n))

i=1
❈✉è✐ ❝ï♥❣✱ ❞♦ ✭❛✮ ♥➟♥

∞

∞
−1

−1

P(Xn ∈ Bnc ) < ∞.

P(f (n) [S(n)−E(S(n) = f (n) [S(n)−E(S(n))
n=1

n=1

❙✉② r❛

(f (n))−1 [S(n) − E(S(n))] → 0 h.c.c ❦❤✐ n → ∞.
➜ã ❧➭ ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳

✷✳✷✳ ▲✉❐t ♠➵♥❤ sè ❧í♥ ➤è✐ ✈í✐ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣ ➤➠✐ ♠ét
❈❤ó♥❣ t❛ ❦ý ❤✐Ư✉
❝đ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

X+

✈➭

X−

❧➬♥ ❧➢ỵt ❧➭ ♣❤➬♥ ❞➢➡♥❣ ✈➭ ♣❤➬♥ ➞♠ t➢➡♥❣ ø♥❣

X✳

❇➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ➜Þ♥❤ ❧Ý ✷✳✶✳✶✱ t tết ợ ị ý s ề t
sè ❧í♥ ❝❤♦ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣ ➤➠✐ ♠ét✱ ❦❤➠♥❣ ❝ï♥❣ ♣❤➞♥ ♣❤è✐✳


✷✵

➜Þ♥❤ ❧Ý ✷✳✷✳✶✳

❈❤♦

{Xn }n

1 ❧➭ ♠ét ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣ ➤➠✐ ♠ét ✈í✐

♣❤➢➡♥❣ s❛✐ ❤÷✉ ❤➵♥✳ ●✐➯ sư

n

E(|Xk − E(Xk )|)/f (n)] < ∞;

(i) sup[
n 1

k=1

∞

(f (n))−2 var(Xn ) < ∞✳

(ii)
n=1

❑❤✐ ➤ã

[S(n) − E(S(n))]/f (n) → 0 h.c.c ❦❤✐ n → ∞.

❈❤ø♥❣ ♠✐♥❤✳ ➜➷t Yn = (Xn − E(Xn ))+ ✈➭ Zn = (Xn − E(Xn ))− (n
❝ã

E(Yn2 )

var(Yn )

var(Xn )

➤ã✱ tõ ❣✐➯ t❤✐Õt s✉② r❛ r➺♥❣ ❞➲②
❧ý ✷✳✶✳✶ ✭✈í✐

✈➭

E(Yn )

{Yn }n

E(|Xn − E(Xn )|)(n

1)✳ ❚❛
1)✳

❉♦

1 t❤á❛ ♠➲♥ t✃t ❝➯ ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ➜Þ♥❤

ρij = max E(Yi Yj ) − E(Yi )E(Yj ), 0 = 0✮✳ ❙✉② r❛
n
−1

(Yi − E(Yi )) → 0 h.c.c✱ ❦❤✐ n → ∞.

(f (n))

i=1
❇➺♥❣ ❝➳❝❤ t❤❛②

Xn

❜ë✐

−Xn

t❛ ➤➢ỵ❝

n
−1

(Zi − E(Zi )) → 0 h.c.c✱ ❦❤✐ n → ∞.

(f (n))

i=1
▲➵✐ ❝ã

n
−1

n
−1

E(Yi ) − (f (n))

(f (n))

i=1

n
−1

E(Xi − E(Xi )) = 0.

E(Zi ) = (f (n))
i=1

i=1

❚õ ➤ã✱ s✉② r❛ ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳
➜Þ♥❤ ❧ý s❛✉ ➤➞② ❧➭ ❝➠♥❣ ❝ơ q✉❛♥ trä♥❣ ➤Ĩ t❛ t❤✉ ➤➢ỵ❝ ❝➳❝ ❦Õt q✉➯ t✐Õ♣ t❤❡♦✳

➜Þ♥❤ ❧ý ✷✳✷✳✷✳

❈❤♦

{Xn }n

1 ❧➭ ♠ét ❞➲② ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➯ tÝ❝❤ ✈➭ ➤é❝

❧❐♣ ➤➠✐ ♠ét✱ s❛♦ ❝❤♦ tå♥ t➵✐ ♠ét ❞➲②
❦✐Ö♥✿

∞

P(Xn ∈ Bnc ) < ∞;

(a)
n=1

{Bn }

❝➳❝ t❐♣ ❇♦r❡❧ t❤♦➯ ♠➲♥ ❝➳❝ ➤✐Ò✉


✷✶

n

E(Xi I(Xi ∈ Bnc )) = o(f (n))❀

(b)
i=1

∞

(f (n)−2 )var(Xn I(Xn ∈ Bn )) < ∞;

(c)
n=1

n

E(|Xk |I(Xk ∈ Bn ))/f (n)] < ∞,

(d) sup[
n 1

k=1
tr♦♥❣ ➤ã
❑❤✐ ➤ã✱

Bnc

❧➭ ♣❤➬♥ ❜ï ❝ñ❛

Bn ✳

(f (n))−1 [S(n) − E(S(n))] → 0 h.c.c ❦❤✐ n → ∞✳

❈❤ø♥❣ ♠✐♥❤✳

➜➷t

Yn = Xn I(Xn ∈ Bn ), n

1✳

❚õ ✭❝✮ ✈➭ ✭❞✮ s✉② r❛

{Yn }n

1

t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✶✳ ❉♦ ➤ã✱

n
−1

(Yi − E(Yi )) → 0 h.c.c✱ ❦❤✐ n → ∞.

(f (n))

i=1
❑Õt ❤ỵ♣ ➤✐Ị✉ ♥➭② ✈í✐ ✭❜✮✱ t❛ ❝ã

n
−1

(Yi − E(Xi )) → 0 h.c.c✱ ❦❤✐ n → ∞.

(f (n))

i=1

❈✉è✐ ❝ï♥❣✱ ❞♦ ✭❛✮ ♥➟♥

∞

P(f (n))−1 [S(n) − E(S(n))] = f (n))−1 [S(n) − E(S(n))]
n=1
∞

P(Xn ∈ Bnc ) < ∞.
n=1
❙✉② r❛

(f (n))−1 [S(n) − E(S(n))] → 0 h.c.c✱ ❦❤✐ n → ∞.
➜ã ❧➭ ➤✐Ò✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳
❉➢í✐ ➤➞② ❧➭ ♠ét ♠ë ré♥❣ ❝đ❛ ❧✉❐t ♠➵♥❤ sè ❧í♥ ❊t❡♠❛❞✐✳

➜Þ♥❤ ❧Ý ✷✳✷✳✸✳
✈➭ ➤➷t

❈❤♦

{Xn }n

G(x) = sup P(|Xn |
n 1

1 ❧➭ ♠ét ❞➲② ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣ ➤➠✐ ♠ét

x) ✈í✐ x

0✳ ◆Õ✉

∞

G(x)dx < ∞
0


✷✷

n
−1

ci (Xi − E(Xi )) → 0 ❤✳❝✳❝ ❦❤✐ n ớ ỗ {cn }n

n

tì

1 ị

i=1

ứ ❚❛ ❝ã
∞

E(|Xn |) =

+∞

P(|Xn |

G(x)dx < ∞.

x)dx

0

0

❙✉② r❛

+∞

sup E(|Xn |)

G(x)dx < ∞.

n 1

Yi = ci Xi ✱

❉♦ ➤ã✱ ❜➺♥❣ ❝➳❝❤ ➤➷t
♠✐♥❤ r➺♥❣ ❞➲②

{Xn }n

0

∞

P(|Xn | > n)

∞

n=1

∞

n+1

G(x)dx < ∞.

G(x)dx
n=1

Bn =

G(n) ♥➟♥

G(n)

n=1

❚❛ sÏ ♣❤➯✐ ❝❤ø♥❣

1 t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧Ý ✷✳✷✳✷ ✈í✐

[−n, n], f (n) = n✳ ❚❛ ❝ã P(|Xn | > n)
∞

ci = 1✳

❝ã t❤Ó ①❡♠

n

1

❉♦ ➤ã ➤✐Ị✉ ❦✐Ư♥ ✭❛✮ ➤➢ỵ❝ t❤♦➯ ♠➲♥✳ ➜Ĩ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ị✉ ❦✐Ư♥ ✭❜✮✱ ❧➢✉ ý r➺♥❣
➤è✐ ✈í✐ ♠ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

Z

❦❤➠♥❣ ➞♠ ❜✃t ❦× ✈➭ ✈í✐

α

0✱ t❛ ❝ã

∞

E(ZI(Z

α)) =

P(ZI(Z
0

=

α) > x)dx
∞

α

P(ZI(Z

α) > x)dx +

0

P(ZI(Z
α

∞

= αP(Z

α) +

α) > x)dx

x)dx.

P(Z
α

❉♦ ➤ã

∞

E(|Xn |I(|Xn | > n)) = nP(|Xn |

G(x)dx → 0,

n) +
n

❦❤✐

n → ∞✱ s✉② r❛ ➤✐Ị✉ ❦✐Ư♥ ✭❜✮ ➤ó♥❣✳ ▲➵✐ ❝ã
∞

E (|Xk |I(|Xk | < k)) =

P (|Xk |I(|Xk | < k) > x) dx
0

k

k

P (|Xk | > x) dx
0

G(x)dx.
0

✷✸

❙✉② r❛

n

1
n

n

1
n

E(|Xk |I(|Xk | < k))
k=1

k

G(x)dx
0

k=1

∞

n

G(x)dx < ∞.

G(x)dx
0

0

❉♦ ➤ã✱ ➤✐Ò✉ ❦✐Ư♥ ✭❞✮ ❝ị♥❣ ➤ó♥❣✳
❇➞② ❣✐ê t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ị✉ ❦✐Ư♥ ✭❝✮✳ ❈❤ó ý r➺♥❣ ➤è✐ ✈í✐
♥❤✐➟♥ ❦❤➠♥❣ ➞♠ ✈➭

Z

❧➭ ♠ét ❜✐Õ♥ ♥❣➱✉

α > 0 t❤×
α

α)) =

E(ZI(Z

0

α) > x)dx

P(x < Z

α)dx

α

=
0

P(ZI(Z

α

x)dx.

P(Z
0
❉♦ ➤ã✱

∞

∞

n−2 E(Xn2 I(|Xn |

n2

n−2

n))

n=1

x1/2 )dx

0

n=1
∞

P(|Xn |

n2
−2

G(x1/2 )dy

n

0

n=1
∞

=2

n
n=1
∞

∞

n
−2

yG(y)dy = 2

0

n=1
∞

j

=2
j=1
∞

j=1

j−1

j

j=1

j−1

yG(y)dy

∞
−2

yG(y)dy
j−1

n

n
−2

n
n=j

4

j

j
j=1

−1

yG(y)dy
j−1

j

G(y)dy < ∞.

4

❚✐Õ♣ t❤❡♦✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét ♠ë ré♥❣ ❝đ❛ ❧✉❐t ♠➵♥❤ sè ❧í♥
❝đ❛ ❈❤✉♥❣ ❬✺❪✳

➜Þ♥❤ ❧Ý ✷✳✷✳✹✳ ❈❤♦ {Xn }n

1 ❧➭ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣ ➤➠✐ ♠ét ✈➭

n

E(|Xk |I(|Xk |

sup[
n 1

k=1

ak ))/f (n)] < ∞.


✷✹

gn : (0, ∞) → (0, ∞)

❈❤♦

❧➭ ❤➭♠ sè t➝♥❣ t❤❡♦

x

✈í✐ ♠ä✐

n

1, gn (0)

❝ã t❤Ĩ

♥❤❐♥ ❣✐➳ trÞ tï② ý✳ ●✐➯ sö r➺♥❣✿

x/gn (x) ✈➭

gn (x)
x2

❣✐➯♠ t❤❡♦

x

∞

E(gn (|Xn |))/gn (an ) < ∞ ✈➭ ❞➲② {an /f (n)}n

1 ❜Þ ❝❤➷♥✳ ❑❤✐ ➤ã

n=1

(f (n))−1 [S(n) − E(S(n))] → 0 h.c.c ❦❤✐ n → ∞.

❈❤ø♥❣ ♠✐♥❤✳

❚❛ sư ❞ơ♥❣ ➜Þ♥❤ ❧Ý ✷✳✷✳✷ ✈í✐

♠✐♥❤ r➺♥❣ ❞➲②

{Xn } t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧Ý ✷✳✷✳✷✳ ❉♦ ❝➳❝ ❤➭♠ sè

gn

t➝♥❣✱ ♥➟♥ t❛ ❝ã

∞

Bn = [−an , an ]✳

{|Xn | > an } ⊂ {gn (|Xn |)

gn (an )}✳ ❉♦ ➤ã

∞

∞

P(|Xn | > an )
n=1

P(gn (|Xn |)

E(gn (|Xn |))/gn (an ) < ∞.

gn (an ))

n=1

n=1

❉♦ ➤ã✱ ➤✐Ị✉ ❦✐Ư♥ ✭❛✮ ➤➢ỵ❝ t❤♦➯ ♠➲♥✳

gn (x)
an ✱ ❦❤✐ x
gn (an )

❚❛ sÏ ♣❤➯✐ ❝❤ø♥❣

❱×

x/gn (x)

❣✐➯♠ t❤❡♦

x✱

♥➟♥

x

an ✳ ❉♦ ➤ã

E(|Xn |I(|Xn | > an ))

an E(gn (|Xn |))/gn (an ).

❙✉② r❛

∞

∞
−1

(f (n)) E(|Xn |I(|Xn | > an ))
n=1

an E(gn (|Xn |))/(f (n)gn (an )) < ∞.
n=1

❇➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ❇ỉ ➤Ị ❑r♦♥❡❝❦❡r✱ t❛ s✉② r❛ ➤✐Ị✉ ❦✐Ư♥
sÏ ❝❤ø♥❣ ♠✐♥❤

0

x

{Xn }n

1 t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥

❚❛ ❝ã

t❤♦➯ ♠➲♥✳ ❚❛

x2

gn (x) 2
a
gn (an ) n

❦❤✐

an ✱ ♥➟♥

∞

∞
−2

(f (n))
n=1

(c)✳

(b)

E(|Xn2 |I(|Xn |

a2n E(gn (|Xn |)/(f (n)gn (an )) < ∞.

an ))
n=1

❚❤❡♦ ❣✐➯ t❤✐Õt ❤✐Ó♥ ♥❤✐➟♥ t❛ ❝ã ➤✐Ị✉ ❦✐Ư♥

(d)✳ ❱❐②

(f (n))−1 [S(n) − E(S(n))] → 0 h.c.c ❦❤✐ n → ∞.

✷✺

❍Ö q✉➯ ✷✳✷✳✺✳ ❈❤♦ {Xn } ❧➭ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣ ➤➠✐ ♠ét ✈➭
n

E(|Xk |I(|Xk |

sup[
n 1

ak ))/n] < .

k=1


ế

1

pn

2 ớ ỗ n

npn E(|Xn |) < tì

1 ✈➭
n=1

n−1 [S(n) − E(S(n))] → 0 h.c.c ❦❤✐ n → ∞.

❈❤ø♥❣ ♠✐♥❤✳ ❙ư ❞ơ♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✹ ✈í✐ gn (x) = xpn , an = n ✈➭ f (n) = n t
ợ ề ứ

ị í {Xn }n

1 ❧➭ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣ ➤➠✐ ♠ét ✈➭

n

E(|Xk |I(|Xk |

sup[
n 1
❈❤♦

ak ))/f (n)] < ∞.

k=1

gn : (0, ∞) → (0, ∞)

❧➭ ❤➭♠ sè t➝♥❣ t❤❡♦

x

✈í✐ ♠ä✐

n

1, gn (0)

❝ã t❤Ĩ

♥❤❐♥ ❣✐➳ trÞ tï② ý✳ ●✐➯ sư r➺♥❣✿

x/gn (x) t➝♥❣ t❤❡♦ x ✈➭ E(Xn ) = 0,
∞

E(gn (|Xn |))/gn (an ) < ∞ ✈➭ {an /f (n)}n

1 ❜Þ ❝❤➷♥✳ ❑❤✐ ➤ã

n=1

(f (n))−1 [S(n) − E(S(n))] → 0 h.c.c ❦❤✐ n → ∞.

❈❤ø♥❣ ♠✐♥❤✳

❚❛ ❝ò♥❣ sÏ ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ➜Þ♥❤ ❧Ý ✷✳✷✳✷ ✈í✐

Bn = [−an , an ]✳
∞

∞

P(|Xn | > an )
n=1
s✉② r❛ ➤✐Ị✉ ❦✐Ư♥
❣✐➯ t❤✐Õt

P(gn (|Xn |)

gn (an )) < ∞.

n=1

(a) t❤♦➯ ♠➲♥✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ị✉ ❦✐Ư♥ (b) ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣

E(Xn ) = 0✳ ❚❛ ❝ã

∞

∞
−1

(f (n))−1 E(|Xn |I(|Xn |

(f (n)) E(|Xn |I(|Xn | > an )) =
n=1

n=1
∞

an E(gn (|Xn |))/(f (n)gn (an )) < ∞.
n=1

an ))