Luật mạnh số lớn đối với dãy trên không banach khoá luận tốt nghiệp đại học

ỵ tt st tr ổ ❧➔ ♠ët ❧➽♥❤ ✈ü❝ ✤➣ ✈➔ ✤❛♥❣ ✤÷đ❝ ♥❤✐➲✉ t→❝ ❣✐↔
q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ✈➔ t❤✉ ✤÷đ❝ ♥❤✐➲✉ ❦➳t q✉↔ s s r ỵ tt st ỵ
ợ ❤↕♥ ♥â✐ ❝❤✉♥❣✱ ❧✉➟t sè ❧ỵ♥ ♥â✐ r✐➯♥❣ ❧➔ ✈➜♥ ✤➲ ✈ø❛ ❝ì ❜↔♥ ❧↕✐ ✈ø❛ ❝â ù♥❣ ❞ư♥❣ rë♥❣ r
r ữớ ỹ ỵ tt s✉➜t ❞ü❛ ✈➔♦ ❤➺ t✐➯♥ ✤➲ ✈➔ ✤➣ t❤✐➳t ❧➟♣
❧✉➟t sè ❧ỵ♥ ♥ê✐ t✐➳♥❣ ♠❛♥❣ t➯♥ ỉ♥❣✳ ▲✉➟t sè ❧ỵ♥ ố ợ t tử ữủ
rt ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉✱ ♠ët sè t số ợ ữủ t t
ợ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ♥❤÷ ❏✳ ▼❛r❝✐♥❦✐❡✇✐❝③✱ ❆✳ ❩②❣♠✉♥❞✱ ❍✳ ❉✳ ❇r✉♥❦✱ ❨✳ ❱✳ Pr♦❦❤♦r♦✈✱ ❑✳
▲✳ ❈❤✉♥❣✱ ❲✳ ❋❡❧❧❡r✱ ✳✳✳ ❈❤♦ tỵ✐ ♥❛②✱ ♥❣❤✐➯♥ ❝ù✉ ♣❤→t tr✐➸♥ ❦➳t q✉↔ ❝õ❛ ❆✳ ◆✳ ❑♦❧♠♦❣♦r♦✈ ✈➔
♠ët sè ❞↕♥❣ ❧✉➟t sè ❧ỵ♥ ❦❤→❝ ✈➝♥ ❧➔ ♠ët ✈➜♥ ✤➲ ❝â t➼♥❤ t❤í✐ sü ❝õ❛ ỵ tt st
t số ợ ờ ừ ②➳✉ t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ở
tr tỹ ởt ữợ t tr✐➸♥ ❝→❝ ❧✉➟t sè ❧ỵ♥ ❝ê ✤✐➸♥ ♥➔② ❧➔ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❧✉➟t
sè ❧ỵ♥ ✤è✐ ✈ỵ✐ ❞➣② ♥❤➟♥ ❣✐→ trà tr ổ t q t ữợ ự
tữớ õ ố t ợ ỵ t❤✉②➳t ❤➻♥❤ ❤å❝ ❇❛♥❛❝❤ ✈➔ t↕♦ r❛ sü ❣✐❛♦ t❤♦❛ ỳ
ỵ tt st t
õ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ ▲✉➟t ♠↕♥❤ sè ❧ỵ♥ ✤è✐ ✈ỵ✐ ❞➣② ♣❤➛♥ tû ♥❣➝✉
♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà tr ổ
õ ỗ ữỡ r ữỡ ✶✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥
✈➲ ①→❝ s✉➜t tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❈→❝ ❦✐➳♥ t❤ù❝ ♥➔② ❧➔ ❝ì sð ✤➸ tr➻♥❤ ❜➔② ❝→❝ ✈➜♥ ✤➲ ừ
ữỡ
ữỡ tr t số ợ ố ✈ỵ✐ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà tr➯♥
❦❤ỉ♥❣ õ trữợ t ú tổ tr➻♥❤ ❜➔② ✈➲ ❝→❝ ❞↕♥❣ ❤ë✐ tö✿ ❍ë✐ tö
❤➛✉ ❝❤➢❝ ❝❤➢♥✱ ❤ë✐ tö t❤❡♦ ①→❝ s✉➜t✱ ❤ë✐ tö ✤➛② ✤õ✱ ❤ë✐ tư t❤❡♦ tr✉♥❣ ❜➻♥❤✳ ❚✐➳♣ t❤❡♦ ❝❤ó♥❣
tỉ✐ ✤➲ ❝➟♣ ✤➳♥ ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ ❝ü❝ ✤↕✐ ✈➔ sü ❤ë✐ tư ❝õ❛ ❝❤✉é✐ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✳
❈✉è✐ ❝ị♥❣✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ✈➲ ❧✉➟t ♠↕♥❤ sè ❧ỵ♥ ✤è✐ ✈ỵ✐ ❞➣② ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ tr♦♥❣
❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❜➜t ❦ý✱ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❘❛❞❡♠❛❝❤❡r ❞↕♥❣ ♣ ✈➔ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♣✲❦❤↔
trì♥ ✭1 ≤ p ≤ 2✮✳
❑❤â❛ ❧✉➟♥ ♥➔② ữủ t ợ sỹ ữợ ừ P ❚❙✳ ◆❣✉②➵♥ ❱➠♥ ◗✉↔♥❣✳
◆❤➙♥ ❞à♣ ♥➔②✱ t→❝ ❣✐↔ ①✐♥ tä ỏ t ỡ sỹ ữợ t t ❝õ❛ ❚❤➛②✳ ❚→❝ ❣✐↔ ❝ơ♥❣
①✐♥ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝ỉ ð ❇ë ♠æ♥ ❳→❝ s✉➜t t❤è♥❣ ❦➯ ✈➔ ❚♦→♥ ù♥❣ ❞ư♥❣✱ ❑❤♦❛ ❚♦→♥✱tr÷í♥❣
✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ q✉❛♥ t➙♠ ✤ë♥❣ ✈✐➯♥ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳

▼➦❝ ❞ò t→❝ ❣✐↔ ✤➣ r➜t ❝è ❣➢♥❣✱ ♥❤÷♥❣ ❝❤➢❝ ❝❤➢♥ ❦❤â❛ ❧✉➟♥ ✈➝♥ ỏ t
ữủ sỹ tr ờ õ ỵ ❝õ❛ ❝→❝ t❤➛②✱ ❝æ ✈➔ ❝→❝ ❜↕♥✳
❱✐♥❤✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✷

❚→❝ ❣✐↔

✶


❈❍×❒◆● ✶

❈⑩❈ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
✶✳✶✳ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❑❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì E ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ tỗ t
Ã

: E R t❤♦↔ ♠➣♥
✭✐✮ x 0, ∀x ∈ E❀
✭✐✐✮ x = 0 ⇔ x = 0❀
✭✐✐✐✮ kx = |k|. x , ∀k ∈ R, ∀x ∈ E❀
✭✐✈✮ x + y
x + y , ∀x, y ∈ E✳

◆➳✉ ✤➦t d(x, y) = x − y (x, y ∈ E) t❤➻ (E, d) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ❑❤✐ ✤â d ✤÷đ❝ ❣å✐ ❧➔
♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ · ✳
◆➳✉ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì t❤ü❝ t❤➻ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ E, · ✤÷đ❝ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥
✤à♥❤ ❝❤✉➞♥ t❤ü❝✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ E, · ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♥➳✉ (E, d)
❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤➛② ✤õ✱ tr♦♥❣ ✤â d ❧➔ ♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ · ✳ ◆➳✉ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì t❤ü❝
✈➔ E, · ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ t❤➻ E, · ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝✳

●✐↔ sû E ổ tỹ ỵ
E = {f : E → R|f ❧➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤✱ ❧✐➯♥ tö❝}.
❚❛ ❣å✐ E∗ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ ❝õ❛ E✳
❱ỵ✐ f ∈ E∗✱ ❝❤✉➞♥ ❝õ❛ f ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝
f = sup |f (x)|,
x

♥➯♥ |f (x)|

1

✈ỵ✐ ♠å✐ x ∈ E
ỵ sỷ E ổ ❇❛♥❛❝❤ t❤ü❝ ✈➔ F
f . x

❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥
✤â♥❣ ❝õ❛ E✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ f ∈ F ∗ ✱ tỗ t f E s f |F = f ✈➔ f = f ✳

❍➺ q✉↔ ✶✳✶✳✹✳ ●✐↔ sû E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ x E, x = 0 tỗ t

f E s❛♦ ❝❤♦ f (x) = x ✈➔ f = 1✳

❍➺ q✉↔ ✶✳✶✳✺✳ ●✐↔ sû E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝✳ ❑❤✐ ✤â ✈ỵ✐ ♠å✐ x, y ∈ E, x = y tỗ t
f E s f (x) = f (y)✳

❍➺ q✉↔ ✶✳✶✳✻✳ ●✐↔ sû E ❧➔ ❦❤æ♥❣ ❣✐❛♥ tỹ õ tỗ t {f , n
n

s❛♦ ❝❤♦ x = supn |fn (x)| ✈ỵ✐ ♠å✐ x ∈ E✳

✷

1} ⊂ E∗


❈❤ó♥❣ t❛ ❝❤✉②➸♥ s❛♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❝→❝ t➟♣ ❇♦r❡❧ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✼✳ ●✐↔ sû E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
t❤ü❝ ✈➔ E∗ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ ❝õ❛ E✳
∗
❚➟♣ A ⊂ E ✤÷đ❝ ❣å✐ ❧➔ t➟♣ trư tỗ t n N ; f1, f2, . . . , fn ∈ E∗; A ∈ B(Rn) s❛♦ ❝❤♦
A = {x ∈ E : f1 (x), . . . , fn (x) A}.

ỵ t t trö ❧➔ F(E)✳

❱➼ ❞ö ✶✳✶✳✽✳

✶✳ ▲➜② f ∈ E∗ ✈➔ a ∈ R t❤➻ A = {x ∈ E : f (x) = a} ❧➔ t➟♣ trö✳
✷✳ ▲➜② f1, f2 ∈ E∗ ✈➔ a, b ∈ R t❤➻ A = {x ∈ E : f1(x) = a, f2(x) = b} t trử

ỵ

F(E) số
E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ❦❤↔ ❧② t❤➻ σ

F(E) = B(E)✱ ✈ỵ✐ B(E) ❧➔ σ ✲✤↕✐ sè ❝→❝

t➟♣ ❇♦r❡❧ ❝õ❛ E✳

✶✳✷✳ P❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
❚r♦♥❣ ♠ư❝ ♥➔② ❝❤ó♥❣ t❛ s➩ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❝ò♥❣ ❝→❝ t➼♥❤ ❝❤➜t ❝ì

❜↔♥ ❝õ❛ ❝❤ó♥❣✳❈❤ó♥❣ t❛ ❧✉ỉ♥ ❣✐↔ sû (Ω, F, P) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t ✤➛② ✤õ✱ E ❧➔ ❦❤æ♥❣ ❣✐❛♥
❇❛♥❛❝❤ t❤ü❝ ❦❤↔ ❧②✱ G ❧➔ σ✲✤↕✐ sè ❝♦♥ ❝õ❛ F ✈➔ B(E) ❧➔ σ✲✤↕✐ sè ❝→❝ t➟♣ ❇♦r❡❧ ❝õ❛ E✳
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ❚❛ ♥â✐ →♥❤ ①↕ X : Ω −→ E ❧➔ ♣❤➛♥ tû−1 ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝✱ X
G/B(E) ữủ ợ ♠å✐ B ∈ B(E) t❤➻ X (B) ∈ G ✮✳
P❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ F ✲✤♦ ✤÷đ❝ s➩ ✤÷đ❝ ❣å✐ ♠ët ❝→❝❤ ✤ì♥ ❣✐↔♥ ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✳ ❍✐➸♥
♥❤✐➯♥✱ ♥➳✉ X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝ t❤➻ X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✳ ▼➦t ❦❤→❝✱ ❞➵
❞➔♥❣ t❤➜② r➡♥❣ ♥➳✉ X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ t❤➻ ❤å
σ(X) = {X −1 (B) : B ∈ B(E)}

❧➟♣ t❤➔♥❤ ♠ët σ✲✤↕✐ sè ❝♦♥ ❝õ❛ σ✲✤↕✐ sè F ✳ σ✲✤↕✐ sè ♥➔② ✤÷đ❝ ❣å✐ ❧➔ σ✲✤↕✐ sè s✐♥❤ ❜ð✐ X ✳ ❍ì♥
♥ú❛✱ σ(X) ❧➔ σ✲✤↕✐ sè ❜➨ ♥❤➜t ♠➔ X ✤♦ ✤÷đ❝✳ ❉♦ ✤â X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝ ❦❤✐
✈➔ ❝❤➾ ❦❤✐ σ(X) ⊂ G ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✷✳ P❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ X : Ω −→ E ✤÷đ❝ ❣å✐ ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ rí✐ r↕❝
♥➳✉ |X(Ω)| ❦❤ỉ♥❣ q✉→ ✤➳♠ ✤÷đ❝✳ ✣➦❝ ❜✐➺t✱ ♥➳✉ |X(Ω)| ❤ú✉ ❤↕♥ t❤➻ X ✤÷đ❝ ❣å✐ ❧➔ ♣❤➛♥ tû ♥❣➝✉
♥❤✐➯♥ ✤ì♥ ❣✐↔♥ ✭tr♦♥❣ ✤â |X(Ω)| ❧➔ ❧ü❝ ❧÷đ♥❣ ❝õ❛ t➟♣ ❤đ♣ X(Ω)✮✳
❱➼ ❞ö ✶✳✷✳✸✳ ●✐↔ sû A ∈ F ✱ a ∈ E✱ a = 0✳ ✣➦t
a ♥➳✉ ω ∈ A;
X(ω) =
0 ♥➳✉ ω ∈
/ A.
❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ B ∈ B(E)✱

∅, ♥➳✉ 0 ∈
/ B, a ∈
/ B;



A, ♥➳✉ 0 ∈
/ B, a ∈ B;

X −1 (B) =

A, ♥➳✉ 0 ∈ B, a ∈
/ B;



Ω, ♥➳✉ 0 ∈ B, a ∈ B
♥➯♥ X −1(B) ∈ F ✳ ❉♦ ✤â X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✳ ❍ì♥ ♥ú❛✱ ✈➻ |X(Ω)| 2 ♥➯♥ X ❧➔ ♣❤➛♥ tû
♥❣➝✉ ♥❤✐➯♥ ✤ì♥ ❣✐↔♥✳
✸


❈❤ó♥❣ t❛ ❝❤✉②➸♥ ✤➳♥ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ tỷ

ỵ

sỷ E1 , E2 ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ❦❤↔ ❧②✱ T : E1 → E2 ❧➔ →♥❤
①↕ B(E1 )/B(E2 ) ✤♦ ✤÷đ❝ ✈➔ X : Ω → E1 ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝✳ ❑❤✐ ✤â →♥❤ ①↕
T ◦X : Ω → E2 ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝✳

❍➺ q✉↔ ✶✳✷✳✺✳ ●✐↔ sû →♥❤ ①↕ X : Ω → E ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝✳ ❑❤✐ ✤â✱ →♥❤ ①↕
X : Ω → R ❧➔ ❜✐➳♥ G ữủ

ỵ s r ởt trữ q trồ ừ tỷ

ỵ ✶✳✷✳✻✳ ⑩♥❤ ①↕ X : Ω → E ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈ỵ✐ ♠å✐
f ∈ E∗ t❤➻ f (X) ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝✳

❍➺ q✉↔ ✶✳✷✳✼✳ ●✐↔ sû X, Y

❧➔ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝✱ a, b ∈ R ✈➔ ξ : Ω → R ❧➔
❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝✳ ❑❤✐ ✤â aX + bY ✈➔ ξX ❧➔ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝✳

❍➺ q✉↔ ✶✳✷✳✽✳ ◆➳✉ {X , n

1} ❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝ ✈➔ Xn → X ❦❤✐ n → ∞
t❤➻ X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝✳
n

❚✐➳♣ t❤❡♦ ❝❤ó♥❣ t❛ ♥❣❤✐➯♥ ❝ù✉ ❝➜✉ tró❝ ❝õ❛ t tỷ

ỵ X : Ω → E ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ữủ X ợ

✤➲✉ ❝õ❛ ♠ët ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ rí✐ r↕❝ G ữủ tỗ t tỷ ♥❣➝✉
♥❤✐➯♥ rí✐ r↕❝ G ✲✤♦ ✤÷đ❝ {Xn , n 1}✱ s❛♦ ❝❤♦

lim sup Xn (ω) − X(ω) = 0.

n→∞ ω∈Ω

✣à♥❤ ỵ X : E ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ X ❧➔ ❣✐ỵ✐

❤↕♥ ✭t❤❡♦ ❝❤✉➞♥ ✮ ❝õ❛ ♠ët ❞➣② ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ì♥ ❣✐↔♥ G ✲✤♦ ✤÷đ❝ {Xn , n 1}✱ s❛♦
❝❤♦ Xn (ω)
2 X(ω) ✈ỵ✐ ♠å✐ n 1 ồ tỗ t↕✐ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥
✤ì♥ ❣✐↔♥ G ✲✤♦ ✤÷đ❝ {Xn , n 1} t❤♦↔ ♠➣♥ lim Xn (ω) − X(ω) = 0 ✈➔ Xn (ω)
2 X(ω)
✈ỵ✐ ♠å✐ n

n→∞

1 ✈➔ ♠å✐ ω ∈ Ω✳

❈❤ó♥❣ t❛ ❦➳t t❤ó❝ ♠ư❝ ♥➔② ✈ỵ✐ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✳
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✶✳ ●✐↔ sû {Xt, t ∈ ∆} ❧➔ ❤å ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❝ò♥❣ ①→❝ ✤à♥❤ tr➯♥
(Ω, F, P)✱ ♥❤➟♥ ❣✐→ trà tr➯♥ (E, B(E))✳ ❑❤✐ ✤â✱ ❤å {Xt , t ∈ ∆} ✤÷đ❝ ❣å✐ ❧➔ ✤ë❝ ❧➟♣ ✤æ✐ ♠ët ✭✤ë❝
❧➟♣ ✮ ♥➳✉ ❤å σ ✲✤↕✐ sè {σ(Xt ), t ∈ ∆} ✤ë❝ ❧➟♣ ✤æ✐ ♠ët ✭✤ë❝ ❧➟♣✮✳
❚ø tr t s r ỵ s

ỵ

sỷ E1 , E2 ổ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ❦❤↔ ❧② ✈➔ {Xt , t ∈ ∆} ❧➔ ❤å
♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ♥❤➟♥ ❣✐→ trà tr♦♥❣ E1 ✳ ❑❤✐ ✤â✱ ♥➳✉ ✈ỵ✐ ♠é✐ t ∈ ∆, Tt : E1 → E2
❧➔ →♥❤ ①↕ B(E1 )/B(E2 ) ✤♦ ✤÷đ❝ t❤➻ ❤å {Tt (Xt ), t ∈ ∆} ❧➔ ❤å ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣
tr tr E2

ỵ sỷ X , X , . . . , X

❧➔ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❝ò♥❣ ①→❝ ✤à♥❤ tr➯♥ (Ω, F, P)✱
♥❤➟♥ ❣✐→ trà tr➯♥ (E, B(E))✳ ❑❤✐ ✤â✱ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ X1 , X2 , . . . , Xn ✤ë❝ ❧➟♣ ❧➔ ✈ỵ✐ ♠å✐
f1 , f2 , . . . , fn ∈ E∗ ✱ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ f (X1 ), f (X2 ), . . . , f (Xn ) ✤ë❝ ❧➟♣✳
1

2

n

✹

✶✳✸✳ ❑ý ✈å♥❣ ❝õ❛ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥
✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳ ●✐↔ sû X : Ω → E ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✳ P❤➛♥ tû m ∈ E ✤÷đ❝ ❣å✐ ❧➔ ❦ý
✈å♥❣

❝õ❛ X ♥➳✉ ✈ỵ✐ ♠å✐ f ∈ E∗ t❛ ❝â

f (m) = E(f (X)).

ỵ m = EX

ử ✶✳✸✳✷✳ ●✐↔ sû a ∈ E✱ A ∈ F ✈➔ X = aI ✱ tù❝ ❧➔
A

X(ω) =

❑❤✐ ✤â✱ ✈➻ ✈ỵ✐ ♠å✐ f ∈ E∗✱
✈➔
♥➯♥ EX = P(A)a✳

♥➳✉ ω ∈ A;
♥➳✉ ω ∈/ A✳

a
0

f (P(A)a) = P(A)f (a)

E(f (X)) = E(f (a)IA ) = f (a)EIA = f (a)P(A),

ỵ ●✐↔ sû X ✱ Y

❧➔ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✱ ξ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝ò♥❣ ①→❝ ✤à♥❤
tr➯♥ (Ω, F, P), a ∈ R✱ α ∈ E✳ ❑❤✐ ✤â✱ ♥➳✉ tỗ t EX, EY, E t
ỗ t E(X + Y ) ✈➔ E(X + Y ) = EX + EY
ỗ t E(aX) E(aX) = aEX
ỗ t E() E() = E
P(X = α) = 1 t❤➻ EX = α❀
✺✳ ◆➳✉ ξ ✈➔ f (X) ✤ë❝ ❧➟♣ ✈ỵ✐ ♠å✐ f ∈ E∗ t tỗ t E(X) E(X) = EEX
ợ ♠å✐ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ T : E → E ✭E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ❦❤↔ ❧②
t tỗ t E(T (X)) E(T (X)) = T (E(X)).

ỵ E X

< t tỗ t EX ✈➔
E X .

EX

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✺✳ ●✐↔ sû X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✈➔ p > 0✳ ◆➳✉ E X
❦❤↔ t➼❝❤ ❜➟❝ p✳

◆➳✉ X ❦❤↔ t➼❝❤ ❜➟❝ 1✱ t❤➻ ✤➸ ✤ì♥ ❣✐↔♥✱ t❛ ♥â✐ X ❦❤↔ t➼❝❤✳

p

< ∞✱

t❤➻ t❛ ♥â✐ X

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✻✳ ❚❛ ♥â✐ →♥❤ ①↕ ϕ : E R ỗ
(ax + (1 a)y)

aϕ(x) + (1 − a)ϕ(y)

✈ỵ✐ ♠å✐ a ∈ [0, 1], x, y E

ỵ t tự s : E R ỗ tö❝✱ X ✈➔ ϕ(X) ❦❤↔

t➼❝❤✱ t❤➻

ϕ(EX)

E ϕ(X) .

✺


✶✳✹✳ ❑ý ✈å♥❣ ❝â ✤✐➲✉ ❦✐➺♥ ✈➔ ♠❛rt✐♥❣❛❧❡
✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✶✳ ●✐↔ sû ✭Ω, F, P✮ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t✱ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ❦❤↔

❧②✱ B(E) ❧➔ σ✲✤↕✐ sè ❇♦r❡❧✳ X : Ω → E ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✱ G ❧➔ σ✲✤↕✐ sè ❝♦♥ ❝õ❛ σ✲✤↕✐ sè
❑❤✐ ✤â ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❨✿ Ω → E ❣å✐ ❧➔ ❦ý ✈å♥❣ ❝â ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❳ ✤è✐ ✈ỵ✐ G ♥➳✉
✭✐✮ Y ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ữủ
E(Y IA) = E(XIA) ợ ồ A G ✳
❑➼ ❤✐➺✉ Y = E(X|G ✮✳
❱➼ ❞ö ✶✳✹✳✷✳ ◆➳✉ G = {, } tỗ t EX E t

F

E(X|G) = EX.

❚❤➟t ✈➟②✱ ✤➦t Y = EX ✱ ❦❤✐ ✤â
✭✐✮ ❱ỵ✐ ♠å✐ B ∈ B(E) t❛ ❝â
Y −1 (B) =

♥➳✉ EX ∈/ B;
♥➳✉ EX ∈ B,

∅
Ω

♥➯♥ Y −1(B) ∈ G ✳ ❱➟② Y ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ữủ
ợ ồ A G t A = ∅ ❤♦➦❝ A = Ω✳
◆➳✉ A = ∅ t❤➻ Y IA = XIA ♥➯♥ E(Y IA) = E(XIA)✳
◆➳✉ A = Ω t❤➻ Y IA = Y ✱ XIA = X ♥➯♥
E(Y IA ) = EY = E(EX) = EX = E(XIA ).

❱➟② E(Y IA) = E(XIA) ✈ỵ✐ ♠å✐ A ∈ G ✳ ❉♦ ✤â Y = E(X|G) ❤❛② E(X|G) = EX ✳
❱➼ ❞ö ✶✳✹✳✸✳ ●✐↔ sû A ∈ F, a ∈ E, X = aIA✳ ❑❤✐ ✤â
E(X|G) = aE(IA |G).

❚❤➟t ✈➟②✱ ✤➦t Y = aE(IA|G)✳
✭✐✮ E(IA|G) ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ G ữủ Y
ữủ
ợ ồ B G t❛ ❝â

= aE(IA |G)

❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦

E(Y IB ) = E(aE(IA |G)IB ) = aE(E(IAB |G))
= aE(IAB ) = aP(AB),

✭✷✳✶✮

E(XIB ) = E(aIAB ) = aP(AB).

✭✷✳✷✮

❚ø (2.1) ✈➔ (2.2) s✉② r❛✱ ✈ỵ✐ ♠å✐ B ∈ G ✱ t❛ ❝â E(Y IB ) = E(XIB )✳
❱➟② Y = E(X|G)
ỵ s t ởt ữỡ ✤➸ ✤à♥❤ ♥❣❤➽❛ ❦ý ✈å♥❣ ❝â ✤✐➲✉ ❦✐➺♥✱ t÷ì♥❣ tü
♥❤÷ ý ồ

ỵ sỷ X, Y

❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✳ ❑❤✐ ✤â Y = E(X|G) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
f (Y ) = E(f (X)|G) ✈ỵ✐ ồ f E

ử ỵ tr ❝→❝ t➼♥❤ ❝❤➜t ❦ý ✈å♥❣ ❝â ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❜✐➳♥ t ự
ữủ ỵ s



ỵ sỷ X, Y
f E ❑❤✐ ✤â

❧➔ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✱ ξ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✱ α ∈ E✱ a ∈ R✱

✶✳ ◆➳✉ ξ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝ t❤ä❛ ♠➣♥ E|ξ| < ∞ ✈➔ E ξX < ∞✱ t❤➻

E(ξX|G) = ξE(X|G).
✷✳ ◆➳✉ X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝ t❤➻ E(X|G) = X.
✸✳ E(X + Y |G) = E(X|G) + E(Y |G)✳
✹✳ E(aX|G) = aE(X|G)✳
✺✳ E(αξ|G) = αE(ξ|G)✳
✻✳ ◆➳✉ G1 ⊂ G2 t❤➻ E E(X|G1 )|G2 = E(X|G1 ) = E E(X|G2 )|G1 ✳
✼✳ ◆➳✉ σ(X) ✤ë❝ ❧➟♣ ✈ỵ✐ G t❤➻ E(X|G) = EX.

❈❤ó♥❣ t❛ ❝❤✉②➸♥ s❛♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❦❤→✐ ♥✐➺♠ ♠❛rt✐♥❣❛❧❡ ✈➔ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡✳
❑❤→✐ ♥✐➺♠ s❛✉ ✤➙② s➩ t✐➳♣ tư❝ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❈❤÷ì♥❣ ✷✳
✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✻✳ ●✐↔ sû {Xn, n 1} ❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✱ {Fn, n 1} ❧➔ ❞➣② t➠♥❣
❝→❝ σ✲✤↕✐ sè ❝♦♥ ❝õ❛ σ✲✤↕✐ sè F ✳ ❑❤✐ ✤â ❞➣② {Xn, Fn, n 1} ❣å✐ ❧➔ ❞➣② ♣❤ị ❤đ♣ ♥➳✉ ✈ỵ✐ ♠å✐
n 1✱ Xn ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ Fn ✲✤♦ ✤÷đ❝✳
❈❤➥♥❣ ❤↕♥✱ ♥➳✉ {Xn, n 1} ❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❜➜t ❦ý ✈➔ Fn = σ(X1, · · · , Xn) ✭❧➔
σ ✲✤↕✐ sè s✐♥❤ ❜ð✐ X1 , · · · , Xn ✮ t❤➻ ❞➣② {Xn , Fn , n 1} ❧➔ ❞➣② ♣❤ị ❤đ♣✳
✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✼✳ ●✐↔ sû {Xn, n 1} ❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✱ {Fn, n 1} ❧➔ ❞➣② t➠♥❣
❝→❝ σ✲✤↕✐ sè ❝♦♥ ❝õ❛ σ✲✤↕✐ sè F ✳ ❑❤✐ ✤â ❞➣② {Xn, Fn, n 1} ✤÷đ❝ ❣å✐ ❧➔ ♠❛rt✐♥❣❛❧❡ ♥➳✉
✭✐✮ {Xn, Fn, n 1} ❧➔ ❞➣② ♣❤ị ❤đ♣ ✈➔ Xn ❦❤↔ t➼❝❤ ✈ỵ✐ ♠å✐ n 1✱
✭✐✐✮ ✈ỵ✐ ♠å✐ m > n t❤➻ E(Xm|Fn) = Xn✳
❉➣② {Xn, Fn, n 1} ✤÷đ❝ ❣å✐ ❧➔ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡ ♥➳✉ ♥â t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✐✮ ✈➔
✭✐✐✮✬ ✈ỵ✐ ♠å✐ m > n t❤➻ E(Xm|F n) = 0✳
❱➼ ❞ö ✶✳✹✳✽✳ ●✐↔ sû {Xn, n 1} ❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ❦❤↔ t➼❝❤✱ EXn = 0
✈ỵ✐n ♠å✐ n 1 ✈➔ Fn = σ(X1, · · · , Xn)✳ ❑❤✐ ✤â {Xn, Fn, n 1} ❧➔ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡ ✈➔ {Sn =
1} ❧➔ ♠❛rt✐♥❣❛❧❡✳
k=1 Xk , Fn , n
❚❤➟t ✈➟②✱ ❞♦ {Xn, , n 1} ❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ♥➯♥ ❝→❝ σ✲✤↕✐ sè σ(X1, · · · , Xn)
✈➔ σ(Xn+1, Xn+2, · · · ) ✤ë❝ ❧➟♣ ✈ỵ✐ ♠å✐ n 1✳ ❉♦ ✤â✱ ✈ỵ✐ ♠å✐ m > n✱ E(Xm|Fn) = EXm = 0 ✈➔
E(Sm |Fn ) = E(Sn |Fn ) + E(Xn+1 |Fn ) + · · · + E(Xm |Fn )

= Sn + EXn+1 + · · · + EXm = Sn .

❚ø ✤â s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

◆❤➟♥ ①➨t ✶✳✹✳✾✳ ◆➳✉ {X , F , n

1} ❧➔ ❞➣② ♣❤ị ❤đ♣✱ Xn ❦❤↔ t➼❝❤ ✈➔ E(Xn+1 |Fn ) = Xn ✈ỵ✐
n
n
♠å✐ n ∈ N✱ t❤➻ {Xn, Fn, n 1} ❧➔ ♠❛rt✐♥❣❛❧❡✳
❚❤➟t ✈➟②✱ ✈ỵ✐ m n✱ t❤❡♦ t➼♥❤ ❝❤➜t ❤ót ❝õ❛ ❦ý ✈å♥❣ ❝â ✤✐➲✉ ❦✐➺♥ t❛ ❝â
Xn = E(Xn+1 |Fn ) = E(E(Xn+2 |Fn+1 )|Fn ) = E(Xm+2 |Fn )

t✐➳♣ tư❝ ♥❤÷ ✈➟②✱ t❛ t❤✉ ✤÷đ❝ Xn = E(Xm|Fn). ❉♦ ✤â {Xn, Fn, n
t tữỡ tỹ ụ ú ố ợ rt


1}

♠❛rt✐♥❣❛❧❡✳


❚ø ✤à♥❤ ♥❣❤➽❛ ♠❛rt✐♥❣❛❧❡ ✈➔ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡✱ ❝â t❤➸ s✉② r❛ ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ✤➙②✳
✶✳ ◆➳✉ {Fn, n 1} ❧➔ ❞➣② t➠♥❣ ❝→❝ σ✲✤↕✐ sè ❝♦♥ ❝õ❛ σ✲✤↕✐ sè F ✱ X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥

❦❤↔ t➼❝❤✱ Xn = E(X|F n )✱ t❤➻ {Xn , Fn , n 1} ❧➔ ♠❛rt✐♥❣❛❧❡✳
✷✳ ◆➳✉ {fn, Fn, n n 1} ❧➔ ❞➣② ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ t❤ü❝ ❧➟♣ t❤➔♥❤ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡ ✈➔ {xn, n
1} ⊂ E✱ t❤➻ {Xn = k=1 xk fk , Fn , n 1} ❧➔ ♠❛rt✐♥❣❛❧❡✳
✸✳ ◆➳✉ {Xn, Fn, n 1} ✈➔ {Yn, Fn, n 1} ❧➔ ♠❛rt✐♥❣❛❧❡ t❤➻ {aXn ± bYn, Fn, n 1}
(a, b ∈ R) ❝ô♥❣ ❧➔ ♠❛rt✐♥❣❛❧❡✳

✹✳ ◆➳✉ {Xn, Fn, n 1} ❧➔ ♠❛rt✐♥❣❛❧❡ t❤➻ {EXn, n 1} ❦❤æ♥❣ ✤ê✐✳
✺✳ ◆➳✉ {Xn, Fn, n 1} ❧➔ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡ t❤➻ EXm = 0 ✈ỵ✐ ♠å✐ m > 1✳

✽


ì


r sốt ữỡ ú tổ ❧✉æ♥ ❣✐↔ sû r➡♥❣ (Ω, F, P) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t ✤➛② ✤õ✱
E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ❦❤↔ ❧②✱ B(E) ❧➔ σ ✲ ✤↕✐ sè ❝→❝ t➟♣ ❇♦r❡❧ tr E ỵ C s
ữủ ũ sè ❞÷ì♥❣ ✈➔ ❣✐→ trà ❝õ❛ ♥â ❝â t❤➸ ❦❤→❝ ♥❤❛✉ ❣✐ú❛ ❝→❝ ❧➛♥ ①✉➜t ❤✐➺♥✳

✷✳✶✳ ❈→❝ ❞↕♥❣ ❤ë✐ tö
✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✳ ●✐↔ sû {X, X , n

1} ❧➔ ❤å ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❝ò♥❣ ①→❝ ✤à♥❤ tr➯♥ Ω ✈➔
n
♥❤➟♥ ❣✐→ trà tr♦♥❣ E✳ ❚❛ ♥â✐✿
❉➣② {Xn, n 1} ❤ë✐ tö ❤➛✉ ❝❤➢❝ ❝❤➢♥ ✤➳♥ X ✭❦❤✐ n → ∞✮✱ tỗ t t N F s
P(N ) = 0 ✈➔ Xn (ω) → X(ω) ✭t❤❡♦ ❝❤✉➞♥✱ ❦❤✐ n ợ ồ \N
ỵ Xn → X ❤✳❝✳❝✳✱ ❤♦➦❝ Xn −h.c.c.
−−→ X ✭❦❤✐ n → ∞✮✳
❉➣② {Xn, n 1} ❤ë✐ tö ✤➛② ✤õ ✤➳♥ X ✭❦❤✐ n → ∞✮✱ ♥➳✉ ✈ỵ✐ ♠å✐ ε > 0 t


P( Xn X > ) < .
n=1

ỵ Xn →

− X ✭❦❤✐ n → ∞✮✳
❉➣② {Xn, n 1} ❤ë✐ tö t❤❡♦ ①→❝ s✉➜t ✤➳♥ X ✭❦❤✐ n → ∞✮✱ ♥➳✉ ✈ỵ✐ ♠å✐ ε > 0 t❤➻
c

lim P( Xn X > ) = 0.

n

P
ỵ Xn
X ✭❦❤✐ n → ∞✮✳
❉➣② {Xn, n 1} ❤ë✐ tö t❤❡♦ tr✉♥❣ ❜➻♥❤ ❝➜♣ p > 0 ✤➳♥ X ✭❦❤✐ n → ∞✮✱ ♥➳✉ X, Xn (n 1)
❦❤↔ t➼❝❤ ❜➟❝ p n
lim E Xn X p = 0
ỵ Xn −L→ X ✭❦❤✐ n → ∞✮✳
w
❉➣② {Xn, n 1} ❤ë✐ tö ②➳✉ ✭t❤❡♦ ♣❤➙♥ ♣❤è✐✮ ✤➳♥ X ✭❦❤✐ n → ∞✮✱ ♥➳✉ PX −→
PX ✱ tr♦♥❣
✤â
p

n

PX : B(E) → R
B P X 1 (B) .
w
ỵ Xn
X ✭❦❤✐ n → ∞✮✳
❱➼ ❞ö ✷✳✶✳✷✳ ❈❤♦ Xn → X ❤✳❝✳❝✳ ✈➔ Yn
n → ∞✮✳

❚❤➟t ✈➟②✱ ✤➦t

→ Y

❤✳❝✳❝✳ ❑❤✐ ✤â Xn + Yn

Ω1 = ω : lim Xn (ω) − X(ω) = 0 ,
n→∞

Ω2 = ω : lim Yn (ω) − Y (ω) = 0 .
n→∞

✾

→ X+Y

❤✳❝✳❝✳ ✭❦❤✐


❚❤❡♦ ❣✐↔ t❤✐➳t Xn → X ❤✳❝✳❝✳ ✈➔ Yn → Y ❤✳❝✳❝✳ ♥➯♥ P(Ω1) = P(Ω2) = 1✱ s✉② r❛ P(Ω1 ∩ Ω2) = 1✳
❑❤✐ ✤â ✈ỵ✐ ♠å✐ ω ∈ Ω1 ∩ Ω2 t❤➻
lim Xn (ω) − X(ω) = 0,

lim Yn (ω) − Y (ω) = 0

n→∞

❤❛②

n→∞

·

·

Xn (ω) → X(ω),

Yn (ω) → Y (ω).

❱➻ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♥➯♥
·

Xn (ω) + Yn (ω) → X(ω) + Y (ω).

✣✐➲✉ ♥➔② ❝❤ù♥❣ tä r➡♥❣
ω ∈ ω : lim Xn + Yn − X − Y (ω) = 0 .
n→∞

❉♦ ✤â

Ω1 ∩ Ω2 ⊂ {ω : lim Xn + Yn − X − Y (ω) = 0},
n→∞

♥➯♥

P lim Xn + Yn − X − Y (ω) = 0 = 1.
n→∞

−−→ X + Y ✭❦❤✐ n → ∞✮✳
❱➟② Xn + Yn −h.c.c.

❱➼ ❞ö ✷✳✶✳✸✳ ❈❤♦ a ∈ E ✈➔ X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ì♥ ❣✐↔♥ ♥❤➟♥ ❝→❝ ❣✐→ trà 0 ✈➔ a ợ
n

L
st tữỡ ự 1 1/n ✈➔ 1/n✳ ❑❤✐ ✤â Xn →P 0 ✈➔ Xn →
0 ✭❦❤✐ n → ∞✮✳
❚❤➟t ✈➟②✱ t❛ ❝â Xn : Ω → R+ ①→❝ ✤à♥❤ ❜ð✐
2

Xn (ω) =

♥➳✉ Xn(ω) = 0,
♥➳✉ Xn(ω) = a.

0
a

❑❤✐ ✤â ✈ỵ✐ ♠å✐ ε > 0✱
0

P Xn − 0 > ε = P Xn > ε
P Xn = a = P(Xn = a)
1
= → 0 ❦❤✐ n → ∞.
n

✣✐➲✉ ♥➔② ✤↔♠ ❜↔♦ r➡♥❣ Xn →P 0 ✭❦❤✐ n → ∞✮✳
L
▼➦t ❦❤→❝✱ ✈➻ 0 E Xn − 0 2 = a 2 · 1/n → 0 ❦❤✐ n → ∞ ♥➯♥ Xn →

0 ✭❦❤✐ n → ∞✮✳
2

◆❤➟♥ ①➨t ✷✳✶✳✹✳ ❚ø ✤à♥❤ ♥❣❤➽❛ tr➯♥ s✉② r❛ r➡♥❣ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ {X , n
n

1} ❤ë✐ tö
♥❤✐➯♥ X ✭❦❤✐

❤➛✉ ❝❤➢❝ ❝❤➢♥ ✭✤➛② ✤õ✱ t❤❡♦ ①→❝ s✉➜t✱ t❤❡♦ tr✉♥❣ ❜➻♥❤ ❝➜♣ p✮ ✤➳♥ ♣❤➛♥ tû ♥❣➝✉
n → ∞✮ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❞➣② ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✭t❤ü❝✮ { Xn − X , n 1} ❤ë✐ tö ❤➛✉ ❝❤➢❝ ❝❤➢♥
✭✤➛② ✤õ✱ t❤❡♦ ①→❝ s✉➜t✱ t❤❡♦ tr✉♥❣ ❜➻♥❤ ❝➜♣ p✮ ✤➳♥ 0 ✭❦❤✐ n → ∞✮✳ ❉♦ ✤â✱ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣
❝→❝ t➼♥❤ ❝❤➜t t÷ì♥❣ ù♥❣ ❝õ❛ ❞➣② ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ t❤ü❝ ✭①❡♠ ❬❄❪✮✱ t❛ ❝â ♥❣❛② ❝→❝ t➼♥❤ ❝❤➜t s❛✉
✤➙② ❝õ❛ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✳
✶✳ Xn → X ❤✳❝✳❝✳ (❦❤✐ n → ∞) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈ỵ✐ ♠å✐ ε > 0✱
lim P sup Xm − X > ε = 0.

n→∞

m n

✶✵


c
h.c.c.
✷✳ ◆➳✉ Xn →
− X t❤➻ Xn −−−→ X ✭❦❤✐ n → ∞✮✳
c
✸✳ ◆➳✉ {Xn, n 1} ❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ✈➔ Xn −h.c.c.

−−→ C ∈ E t❤➻ Xn →
− C ✭❦❤✐
n → ∞✮✳
L
P
− X ✭❦❤✐ n → ∞✮✳
✹✳ ◆➳✉ Xn −h.c.c.
−−→ X ❤♦➦❝ Xn −→ X t❤➻ Xn →
✺✳ ◆➳✉ ❞➣② {Xn, n ≥ 1} ở tử t st t tỗ t {Xn ; k 1) ⊂ (Xn, n ≥ 1}
p

s❛♦ ❝❤♦ {Xnk ; k

k

1} ❤ë✐ tö ❤✳❝✳❝✳

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✺✳ ❚❛ ♥â✐ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ {X , n
❤➛✉ ❝❤➢❝ ❝❤➢♥ ✭❤✳❝✳❝✳✮

♥➳✉ P(m,n→∞
lim

n

Xm − Xn = 0) = 1❀

1}

❧➔ ❞➣② ❝ì ❜↔♥✿

♥➳✉ m,n→∞
lim P( Xm − Xn > ε) = 0 ✈ỵ✐ ♠å✐ ε > 0❀
t❤❡♦ tr✉♥❣ ❜➻♥❤ ❝➜♣ p > 0 ♥➳✉ lim E Xm − Xn p = 0✳
m,n→∞

t❤❡♦ st

ỵ {X , n
ỵ ❉➣② {X , n

1} ❝ì ❜↔♥ ❤✳❝✳❝✳ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❞➣② {Xn , n

n

1} ❤ë✐ tö ❤✳❝✳❝✳

1} ❧➔ ❞➣② ❝ì ❜↔♥ ❤✳❝✳❝✳ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♠ët tr♦♥❣ ❤❛✐ ✤✐➲✉ ❦✐➺♥
n
s❛✉ ✤÷đ❝ t❤♦↔ ♠➣♥✿
✭✐✮ lim P sup Xk − Xl > ε = 0 ✈ỵ✐ ♠å✐ ε > 0❀
n→∞

✭✐✐✮ n→∞
lim P

k,l n

sup Xk − Xn > ε = 0 ợ ồ > 0
k n

ỵ {X , n

{Xn , n

1} s❛♦ ❝❤♦ {Xnk , k

n

✣à♥❤ ỵ {X , n
st

n

1} ỡ t st t tỗ t {Xnk , k
1} ở tö ❤✳❝✳❝✳

1} ⊂

1} ❤ë✐ tö t❤❡♦ ①→❝ s✉➜t ❦❤✐ ✈➔ õ ỡ t

ỵ ✷✳✶✳✶✵✳ ❉➣② {X , n

1} ❤ë✐ tö t❤❡♦ tr✉♥❣ ❜➻♥❤ ❝➜♣ p (p
❞➣② ❝ì ❜↔♥ t❤❡♦ tr✉♥❣ ❜➻♥❤ ❝➜♣ p✳
n

1) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â ❧➔

◆❤➟♥ ①➨t ✷✳✶✳✶✶✳

✭✐✮ ●✐↔ sû G ❧➔ σ✲✤↕✐ sè ❝♦♥ ❝õ❛ σ✲✤↕✐ sè F ✳ t r t t ỵ
tr ❝õ❛ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✈➝♥ ✤ó♥❣ ❝❤♦ ❞➣② tỷ G ữủ
ợ ộ số t❤ü❝ p 1✱ ✤➦t
Lp (G, E) = X : Ω → E : X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ữủ
E X

p

< .

õ ỗ t ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝✱ ❜➡♥❣ ♥❤❛✉ ❤➛✉
tứ ờ ỵ t s✉② r❛ r➡♥❣ Lp(G, E) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥
X p = (E X p )1/p ✳
✭✐✐✐✮ ⑩♥❤ ①↕
EG : L1 (F, E) −→ L1 (G, E)
X −→ E(X|G)

❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤✱ ❧✐➯♥ tö❝ ✈➔ EG = 1. ❚❤➟t ✈➟②✱ t➼♥❤ t✉②➳♥ t➼♥❤ ❝õ❛ EG ✤÷đ❝ s✉② tø
✣à♥❤ ỵ ự EG tử EG = 1✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠å✐ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥
X ∈ L1 (F, E)✱ t❛ ❝â
E E(X|G)

E E( X |G) = E X .

✶✶


✣✐➲✉ ♥➔② ♥❣❤➽❛ ❧➔
❱➟② EG ❧✐➯♥ tö❝ ✈➔ EG

X 1
= 1✳ ❉♦ ✤â
EG
X

1✳

E(X|G)

X 1.

1

▼➦t ❦❤→❝✱ ✈ỵ✐ ♠å✐ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ X

∈ L1 (G, E)✱

t❛ ❝â

EG = 1.

1

✷✳✷✳ ❙ü ❤ë✐ tö ❝õ❛ ❝❤✉é✐ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥
✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✶✳ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ p✲❦❤↔ trỡ 1
tỗ t số ữỡ C = Cp s ❝❤♦✱ ✈ỵ✐ ♠å✐ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡ {Xj , Fj , j
❜➟❝ p ❤ú✉ ❤↕♥ t❤➻
i

2✮

i

p

Xj

E

♥➳✉
1} ❝â ❝→❝ ♠♦♠❡♥t
p

E Xj p ,

C

j=1

i

1.

j=1

❚ø ✤à♥❤ ♥❣❤➽❛ tr➯♥ s✉② r❛ r➡♥❣ ✤÷í♥❣ t❤➥♥❣ t❤ü❝ R ổ 2 trỡ

ỵ sỷ E ổ p trỡ (1

p 2) õ tỗ t↕✐ ❤➡♥❣ sè C = Cp

1} ♥❤➟♥ ❣✐→ trà tr➯♥ E ✈➔ ❦❤↔ t➼❝❤ ❜➟❝ p ✈➔ ♠å✐

s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡ {Xn , Fn , n
ε > 0✱ t❛ ❝â

C
εp

P max Sn > ε
1 n N

N

E Xn p ,
n=1

tr♦♥❣ ✤â Sn = X1 + X2 + · · · + Xn .
❈❤ù♥❣ ♠✐♥❤ ✳ ❱➻ {Xn , Fn , n
✈ỵ✐ ♠å✐ ε > 0✱

1}

❧➔ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡ ♥➯♥ {Sn, Fn, n
1
E SN
εp

P max Sn > ε
1 n N

p

1}

❧➔ ♠❛rt✐♥❣❛❧❡✳ ❉♦ ✤â✱

.

▼➦t ❦❤→❝✱ ❞♦ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ p✲❦❤↔ trì♥ ♥➯♥
N

E SN

p

E Xn p .

C
n=1

❚ø ✤â s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

❍➺ q✉↔ ✷✳✷✳✸✳ ●✐↔ sû E ❧➔ ❦❤æ♥❣ ❣✐❛♥ p trỡ (1

p 2) õ tỗ t số C = Cp
1} ♥❤➟♥ ❣✐→ trà tr➯♥ E✱ ❦❤↔ t➼❝❤ ❜➟❝ p✱ ✈➔ ♠å✐

s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡ {Xn , Fn , n
ε > 0✱ t❛ ❝â
k

C
✭✐✮ P nmax
Sm − Sn > ε
E Xm p ❀
p
m k
ε m=n+1

✭✐✐✮ P

sup Sm − Sn > ε
m n

❈❤ù♥❣ ♠✐♥❤ ✳

❱ỵ✐ k

n✱

C
εp

∞

E Xm p .
m=n+1

✤➦t

Y1 = 0, . . . , Yn = 0, Yn+1 = Xn+1 , . . . , Yk = Xk .

✶✷


❑❤✐ ✤â {Yn, Fn, n

1}

t❤♦↔ ♠➣♥ ❣✐↔ t❤✐➳t ❝õ❛ ✣à♥❤ ỵ
k

k

k

Ym = Sk Sn ;
m=1

p

E Ym

E Xm p .

=

m=1

m=n+1

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

max Sm − Sn > ε ,

n m k

k

n.

❑❤✐ ✤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 = lim P(Bk )
k→∞

k=n+1
k

C
lim
E Xm
εp k→∞ m=n+1
C
= p


p



E Xm p .
m=n+1

ỵ sỷ E ❦❤ỉ♥❣ ❣✐❛♥ p✲❦❤↔ trì♥ (1

s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡ {Xn , Fn , n

p 2) õ tỗ t ❤➡♥❣ sè C > 0
1} ♥❤➟♥ ❣✐→ trà tr➯♥ E ✈➔ ❦❤↔ t➼❝❤ ❜➟❝ p✱ t❛ ❝â
N

E( max Sn )p

E Xn p .

C

1 n N

n=1

❈❤ù♥❣ ♠✐♥❤ ✳ ❉♦ {Xn , Fn , n
1} ❧➔ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡✱ ♥➯♥ {Sn , Fn , n
1} ❧➔ ♠❛rt✐♥❣❛❧❡✳ ❉♦
✤â✱ { Sn , Fn, n 1} rt ữợ ổ r trữớ ủ 1 < p 2✱ t❤❡♦ ❜➜t
✤➥♥❣ t❤ù❝ ❉♦♦❜ ✭①❡♠ ❬ ✱ tr✳ ✶✺✹❪✮✱ ✈ỵ✐ ♠å✐ n 1 t❤➻

❄

✶✸


p
p−1

p

E max Sn
1 n N

▼➦t ❦❤→❝✱ ❞♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ p✲❦❤↔ trì♥✱ ♥➯♥

p

p

E SN

.

N

E SN

p

E Xn p .

C
n=1

❚ø ✤â s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳
❚r♦♥❣ tr÷í♥❣ ❤đ♣ p = 1✱ ✈ỵ✐ n 1✱ t❛ ❝â
n

E max Sn

E

1 n N

Xi

max

1 n N

i=1

N

N

=E

Xi

=

i=1

E Xi .
i=1

ỵ sỷ E ❦❤ỉ♥❣ ❣✐❛♥ p✲❦❤↔ trì♥ (1

2)✱ {Xn , Fn , n

p

♠❛rt✐♥❣❛❧❡ ♥❤➟♥ ❣✐→ trà tr➯♥ E✳ ❑❤✐ ✤â ♥➳✉

1} ❧➔ ❤✐➺✉

∞

E Xn

p

<∞

n=1

t❤➻ ❝❤✉é✐

∞
n=1

Xn ❤ë✐ tö ❤✳❝✳❝✳

❈❤ù♥❣ ♠✐♥❤ ✳

✣➦t Sn = X1 + X2 + · · · + Xn✳ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ {Sn, n 1} ❤ë✐ tö ❤✳❝✳❝✳ ✣➸
❧➔♠ ✤✐➲✉ ✤â✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ {Sn, n 1} ❧➔ ❞➣② ❝ì ❜↔♥ ❤✳❝✳❝✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠å✐ ε > 0✱
C
lim p
n→∞ ε

lim P(sup Sm − Sn > ε)

n→∞

m n

=

∞

E Xm

p

m=n+1
∞

C
E Xm
lim
εp n→∞ m=n+1

p

= 0.

❱➟② {Sn, n 1} ❧➔ ❞➣② ❝ì ỵ ữủ ự
sỷ 0 < c < ∞ ✈➔ X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❜➜t ❦ý✱ ✤➦t
X ♥➳✉ X
c❀
X c = XI( X c) =
0 X > c
ỵ ỵ ❝❤✉é✐✮ ●✐↔ sû {Xn, Fn, n 1} ❧➔ ❞➣② ♣❤ò ủ õ tỗ t
số c > 0 s❛♦ ❝❤♦

∞

E Xnc − E(Xnc |Fn−1 )

p

❤ë✐ tö,

n=1
∞

E(Xnc |Fn−1 ) ❤ë✐ tö ❤✳❝✳❝✳,
n=1
∞

P( Xn > c) ❤ë✐ tö,
n=1

t❤➻ ❝❤✉é✐

∞
n=1

Xn ❤ë✐ tö ❤✳❝✳❝✳

✶✹


❈❤ù♥❣ ♠✐♥❤ ✳ ❚ø ❣✐↔ t❤✐➳t {Xn , Fn , n 1} ❧➔ ❞➣② ♣❤ị ❤đ♣✱ s✉② r❛ r➡♥❣ ❞➣② {Xnc −E(Xnc |Fn−1 ), Fn , n
1} ❧➔ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡✳ ❉♦ ✤â✱ tø ❣✐↔ t❤✐➳t
∞

E Xnc − E(Xnc |Fn−1 )

p

❤ë✐ tö,

n=1

s✉② r❛ ❝❤✉é✐

∞
c
n=1 (Xn

− E(Xnc |Fn−1 ))
∞

❤ë✐ tö ❤✳❝✳❝✳ ❑➳t ❤đ♣ ✤✐➲✉ ♥➔② ✈ỵ✐ ❣✐↔ t❤✐➳t

E(Xnc |Fn−1 ) ❤ë✐

tư ❤✳❝✳❝✳,

n=1

s✉② r❛ ❝❤✉é✐

∞
n=1

Xnc

❤ë✐ tư ❤✳❝✳❝✳ ❈✉è✐ ❝ị♥❣✱ ❣✐↔ t❤✐➳t
∞

P( Xn > c) ❤ë✐

tư

n=1

❝ị♥❣ ✈ỵ✐ ❧✉➟t∞ 0 − 1 ❇♦r❡❧✲❈❛♥t❡❧❧✐ ✤↔♠ ❜↔♦ r➡♥❣✱ ✈ỵ✐ ①→❝ s✉➜t ✶ t❤➻ Xn = Xnc ❦❤✐ n ❦❤→ ❧ỵ♥✳ ❉♦
✤â ❝❤✉é✐ n=1 Xn ❝ơ♥❣ ❤ë✐ tư ❤✳❝✳❝✳
✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✼✳ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ❣å✐ ❧➔ ổ r p 1
p 2 tỗ t ởt số ữỡ C = Cp s ợ ♠å✐ ❞➣② {Xj , j 1} ❝→❝ ♣❤➛♥ tû
♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ❝â ❝→❝ ❦ý ✈å♥❣ ❜➡♥❣ ✵✱ ❝â ❝→❝ ♠♦♠❡♥t ❜➟❝ p ❤ú✉ ❤↕♥✱ t❤➻
i

i

p

Xj

E

E Xj p ,

C

j=1

i

1.

j=1

❚ø ✤à♥❤ ♥❣❤➽❛ tr➯♥ s✉② r❛ r➡♥❣ ♥➳✉ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ p✲❦❤↔ trì♥ (1
t❤➻ ♥â ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ r p

p

2)

ỵ sỷ E ổ r p (1

p 2) õ tỗ t
số ữỡ C = Cp s ợ ồ ❞➣② {Xn , n 1} ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ♥❤➟♥ ❣✐→
trà tr➯♥ E✱ ❝â ❦ý ✈å♥❣ ❜➡♥❣ 0✱ ❦❤↔ t➼❝❤ ❜➟❝ p ✈➔ ♠å✐ ε > 0✱ t❤➻
C
εp

P( max Sn > ε)
1 n N

N

E Xn p ,
n=1

tr♦♥❣ ✤â Sn = X1 + X2 + · · · + Xn .
❈❤ù♥❣ ♠✐♥❤ ✳ ✣➦t Fn = σ(Xk , 1
{Sn , Fn , n 1} ❧➔ ♠❛rt✐♥❣❛❧❡✳ ❉♦

❑❤✐ ✤â {Xn, Fn, n
✤â✱ ✈ỵ✐ ♠å✐ ε > 0✱
k

n).

P( max Sn > ε)
1 n N

1
E SN
εp

p

1}

❧➔ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡✱ ♥➯♥

.

▼➦t ❦❤→❝✱ ❞♦ E ổ r p tỗ t ❤➡♥❣ sè ❞÷ì♥❣ C = Cp✱ s❛♦ ❝❤♦
N

E SN

p

E Xn p .

C
n=1

❚ø ✤â s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳
❍➺ q✉↔ s❛✉ ✤➙② ✤÷đ❝ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü ♥❤÷ ❤➺ q✉↔ t÷ì♥❣ ù♥❣
❝❤♦ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡ ♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ p✲❦❤↔ trì♥✳
✶✺


❍➺ q✉↔ ✷✳✷✳✾✳ ●✐↔ sû E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❘❛❞❡♠❛❝❤❡r p (1

p 2) õ tỗ t
số ữỡ C = Cp ✱ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ ❞➣② {Xn , n 1} ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ♥❤➟♥ ❣✐→
trà tr➯♥ E✱ ❝â ❦ý ✈å♥❣ ❜➡♥❣ 0✱ ❦❤↔ t➼❝❤ ❜➟❝ p ✈➔ ♠å✐ ε > 0✱ t❤➻
k
C
E Xm p ❀
✭✐✮ P nmax
Sm − Sn > ε
m k
εp m=n+1

✭✐✐✮ P

sup Sm − Sn > ε
m n

∞

C
εp

E Xm p .
m=n+1

✣à♥❤ ỵ sỷ E ổ r p (1

p

♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ♥❤➟♥ ❣✐→ trà tr➯♥ E✳ ❑❤✐ ✤â ♥➳✉

2)✱ {Xn , n

1} ❧➔ ❞➣②

∞

E Xn − EXn

p

<∞

n=1

t❤➻ ❝❤✉é✐

∞
n=1

❈❤ù♥❣ ♠✐♥❤ ✳

Xn − EXn ❤ë✐ tö ❤✳❝✳❝✳

❱ỵ✐ ♠é✐ n

1✱

✤➦t

Sn = (X1 − EX1 ) + (X2 − EX2 ) + · · · + (Xn − EXn )
= Y1 + Y2 + · · · + Yn .

❇➡♥❣ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü ♥❤÷ ❧➟♣ ❧✉➟♥ tr♦♥❣ ự ừ ỵ t ữủ {Sn, n
❞➣② ❝ì ❜↔♥ ❤✳❝✳❝✳✱ ❞♦ ✤â ❤ë✐ ❤✳❝✳❝✳ ✣â ❧➔ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳

1}

❈ơ♥❣ ❜➡♥❣ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü ♥❤÷ tr ự ừ ỵ t ữủ
ỵ ỵ ộ sỷ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❘❛❞❡♠❛❝❤❡r ❞↕♥❣ p (1 p 2)✱

{Xn , n 1} ❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ♥❤➟♥ ❣✐→ trà tr➯♥ E✳ ❑❤✐ ✤â ♥➳✉ ✈ỵ✐ c ♥➔♦ ✤â
✭0
<
c
< ∞✮✱ ❜❛ ❝❤✉é✐

∞

∞

E Xnc − EXnc p ;
n=1

❤ë✐ tö✱ t❤➻ ❝❤✉é✐

∞
n=1

∞

EXnc ;
n=1

P( Xn > c)
n=1

Xn ❤ë✐ tö ❤✳❝✳❝✳

✷✳✸✳ ▲✉➟t ♠↕♥❤ sè ❧ỵ♥ ✤è✐ ✈ỵ✐ ❞➣② ❦❤ỉ♥❣ ❝ị♥❣ ♣❤➙♥ ố
rữợ t ú t ự t số ❧ỵ♥ ❝❤♦ ❞➣② ❦❤ỉ♥❣ ❝ị♥❣ ♣❤➙♥ ♣❤è✐ tr➯♥
❦❤ỉ♥❣ ❣✐❛♥ ❘❛❞❡♠❛❝❤❡r ❞↕♥❣ p ✈➔ ❦❤ỉ♥❣ ❣✐❛♥ p✲❦❤↔ trì♥✳ ❚❛ ❝➛♥ ❜ê ✤➲ s❛✉✳

❇ê ✤➲ ✷✳✸✳✶✳ ✭❑r♦♥❡❝❦❡r✮ ●✐↔ sû 0 < b

n

❤ë✐ tö✳ ❑❤✐ ✤â

1
bn

↑ ∞ ❦❤✐ n → ∞✱ {xn , n

1} ⊂ E ✈➔ ❝❤✉é✐

∞

xk /bk
k=1

n

xk → 0 khi n → ∞.
k=1

✶✻


❈❤ù♥❣ ♠✐♥❤ ✳ ✣➦t rn = ∞
k=n+1 xk /bk ✳ ❘ã r➔♥❣ rn → 0 ❦❤✐ n → ∞
❑❤✐ ✤â✱ xn/bn = rn−1 − rn (n 1) ♥➯♥ xn = bn(rn−1 − rn)✱ s✉② r❛
n

✈➔ r = supn

rn < ∞✳

n

bk (rk−1 − rk )

xk =
k=1

k=1

= (b1 r0 − b1 r1 ) + (b2 r1 − b2 r2 ) + · · · + (bn rn−1 − bn rn )
= r1 (b2 − b1 ) + · · · + rn−1 (bn − bn−1 ) + b1 r0 − bn rn .

❉♦ ✤â
n

1
lim
n
bn
lim
n

= lim
n

xk
k=1

rk (bk+1 − bk ) + lim

n

k=1
n−1

1
bn

▼➦t ❦❤→❝✱ ✈ỵ✐ ♠å✐ ε > 0✱ ❞♦
✤â✱ ✈ỵ✐ ♠å✐ n > n0✱

xk

k=1
n−1

1
bn

n

1
= lim
n bn

b1
r0 + lim rn
n
bn

rk (bk+1 − bk ).
k=1

rn 0

1
bn
1
bn

tỗ t n0 s ợ ồ n

n0

t

rn < ε✳

❑❤✐

n−1

rk (bk+1 − bk )
k=1
n0 −1

n−1

r(bk+1 − bk ) +

ε(bk+1 − bk )
k=n0

k=1

1
=
(bn0 − b1 )r + ε(bn − bn0 ) .
bn

❉♦ ✤â
s✉② r❛

1
lim
n
bn

n

xk

lim
n

k=1

lim
n

1
bn

❱➟②
1
lim
n
bn

bn
r(bn0 − b1 )
+ε−ε 0
bn
bn

= ε,

n

xk = 0.
k=1
n

xk = 0.
k=1

❚ø ✤â s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣
sỷ ử ờ rr ỵ ỵ t õ t
số ợ s

ỵ

sỷ E ổ p✲❦❤↔ trì♥ (1
p
2)✱ {Xn , Fn , n
1} ❧➔ ❤✐➺✉
♠❛rt✐♥❣❛❧❡ ♥❤➟♥ ❣✐→ trà tr➯♥ E✱ {bn , n 1} ❧➔ ❞➣② sè ❞÷ì♥❣✱ bn ↑ ∞ ❦❤✐ n → ∞✳ ❑❤✐ ✤â✱ ♥➳✉
∞

n=1

E Xn
bpn

✶✼

p

<∞


t❤➻

1
bn
❈❤ù♥❣ ♠✐♥❤ ✳ ❱ỵ✐ ♠é✐ n
♥➯♥ {Yn, Fn, n 1} ❝ô♥❣

n

Xk → 0 h.c.c. ❦❤✐ n → ∞.
k=1

1✱ ✤➦t Yn = Xn /bn ✳ ❑❤✐ ✤â✱ ❞♦ {Xn , Fn , n
❧➔ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡✳ ▼➦t ❦❤→❝✱
∞

∞

E Yn

p

=

n=1

n=1

❙✉② r❛

∞

n=1

Xn
=
bn

E Xn

bpn

1}

❧➔ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡

p

< ∞.

∞

Yn
n=1

❤ë✐ tư ❤✳❝✳❝✳ ✣✐➲✉ ♥➔② ❝ị♥❣ ✈ỵ✐ ❜ê ✤➲ rr t ự

ỵ sû E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❘❛❞❡♠❛❝❤❡r ❞↕♥❣ p (1

p 2)✱ {Xn , n 1} ❧➔ ❞➣②
♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ❦ý ✈å♥❣ ✵ ✈➔ ♥❤➟♥ ❣✐→ trà tr➯♥ E✱ {bn , n
1} ❧➔ ❞➣② sè ❞÷ì♥❣✱
bn ↑ ∞ ❦❤✐ n → ∞✳ ❑❤✐ ✤â✱ ♥➳✉
∞

n=1

E Xn
bpn

p

<∞

t❤➻

1
bn

n

Xk → 0 h.c.c. ❦❤✐ n → ∞.
k=1

❈❤ù♥❣ ♠✐♥❤ ✳

❱ỵ✐ ♠é✐ n 1✱ ✤➦t Yn = Xn/bn✳ ❑❤✐ ✤â✱ ❞♦ {Xn, n 1} ❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉
♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ❦ý ✈å♥❣ ✵ ♥➯♥ {Yn, n 1} ❝ô♥❣ ❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ❦ý ✈å♥❣ ✵✳
▼➦t ❦❤→❝✱
∞
∞
E Yn

❙✉② r❛

p

=
n=1

n=1
∞

n=1

Xn
=
bn

E Xn
bpn

p

< ∞.

∞

Yn
n=1

❤ë✐ tư ❤✳❝✳❝✳ ✣✐➲✉ ♥➔② ❝ị♥❣ ✈ỵ✐ ❜ê ✤➲ ❑r♦♥❡❝❦❡r ❝❤♦ t❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

✷✳✹✳ ▲✉➟t ♠↕♥❤ sè ❧ỵ♥ ✤è✐ ợ ũ ố
rữợ t ú t tt ❧✉➟t ♠↕♥❤ sè ❧ỵ♥ ❝❤♦ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ❝ị♥❣
♣❤➙♥ ♣❤è✐✱ ♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ tỹ t ý ú ỵ r E ❧➔
❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝✱ ❦❤↔ ❧② t❤➻ ✈ỵ✐ ♠å✐ n 1✱ ①→❝ ✤à♥❤ ✤÷đ❝ →♥❤ ①↕ ✤♦ ✤÷đ❝ ϕn : E → E
s❛♦ ❝❤♦✱ ✈ỵ✐ ♠é✐ ♣❤➛♥ tû ♥❣➝✉ t X : E tỗ t ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ì♥
❣✐↔♥ {Xn = ϕn(X), n 1} ✤➸ Xn → X ❦❤✐ n → ∞ ✈➔ E Xn − X → 0 ❦❤✐ n → ∞✳
❉ü❛ ú ỵ tr t số ợ t ỵ t t ữủ ỵ

s



ỵ {X, X

1} ồ tû ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤æ♥❣
n : n
❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ❦❤↔ ❧② E✳ ●✐↔ sû {Xn : n
1} ✤ë❝ ❧➟♣ ✤ỉ✐ ♠ët✱ ❝ị♥❣ ♣❤➙♥ ♣❤è✐ ✈ỵ✐ X ✈➔
E X < ∞✳ ❑❤✐ ✤â
1
n

n

Xi → EX h.c.c. khi n → ∞.
i=1

❈❤ù♥❣ ♠✐♥❤ ✳

✣➛✉ t✐➯♥✱ ❣✐↔ sû X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ì♥ ❣✐↔♥ ♥❤➟♥ ❝→❝ ❣✐→ trà x1, x2, ..., xk
❧➛♥ ❧÷đt tr➯♥ ❝→❝ t➟♣ A1, A2, ..., Ak ✈ỵ✐ P(Ai) > 0✱ i = 1, 2, ..., k✳ ❱➻ {X, Xn : n 1} ❝ò♥❣ ♣❤➙♥
♣❤è✐ ♥➯♥ Xn ❝ơ♥❣ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ì♥ ❣✐↔♥ ♥❤➟♥ ❣✐→ trà x1, x2, ..., xk ✈ỵ✐ P(Xn = xt) = P(At)✳
❱ỵ✐ ♠é✐ t = 1, 2, .., k✱ ✤➦t
n

Znt

=

I(Xi =xt ) = ❝❛r❞{i ∈ {1, 2, ..., n} : Xi = xt }.

i=1

❉♦ {I(X =x ) : i 1} ❧➔ ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ✤ỉ✐ ♠ët✱ ❝ị♥❣ ♣❤➙♥ ♣❤è✐ ✈➔ E I(X =x ) =
P(At ) ♥➯♥ t❤❡♦ ❧✉➟t ♠↕♥❤ sè ❧ỵ♥ ❊t❡♠❛❞✐ t❛ ❝â
i

t

i

Znt
→ P(At ) > 0
n

❉♦ ✤â
1
n

n

i=1

1
Xi =
n

k

k

Znt xt

=

t=1

t=1

❤✳❝✳❝✳ ❦❤✐ n → ∞.
Znt
xt →
n

k

P(At )xt = EX h.c.c.
t=1

❦❤✐ n → ∞.
❳➨t tr÷í♥❣ ❤đ♣ X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❦❤↔ t➼❝❤ ❜➜t ❦ý✳ ❱ỵ✐ ♠å✐ ε > 0✱ tø E
0 ❦❤✐ n → ∞✱ tỗ t m s E n (X) X < ε ✈ỵ✐ ♠å✐ n m✳ ❚❛ ❝â
1
n

n

1

n

(Xi − EX)
i=1

+
+

t

1
n
1
n

ϕn (X)−X →

n

(Xi − ϕm (Xi ))
i=1
n

(ϕm (Xi ) − Eϕm (X))
i=1
n

(Eϕm (X) − EX)
i=1

:= ✭■✮✰✭■■✮✰✭■■■✮.

❉♦ {Xn : n

❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ✤ỉ✐ ♠ët✱ ❝ị♥❣ ♣❤➙♥ ♣❤è✐ ✈ỵ✐ X ✱ ♥➯♥
1} ❧➔ ❞➣② ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ❝ị♥❣ ♣❤➙♥ ♣❤è✐ ✈ỵ✐ X − ϕm (X) ✳
❚❤❡♦ ❧✉➟t ♠↕♥❤ sè ❧ỵ♥ ❊t❡♠❛❞✐ t❛ ❝â
1}
{ Xn − ϕm (Xn ) : n

✭■✮

1
n

n

Xi − ϕm (Xi ) → E X − ϕm (X) < ε

❤✳❝✳❝✳ ❦❤✐ n → ∞.

i=1

❚❤❡♦ ❝❤ù♥❣ ♠✐♥❤ tr➯♥ t❤➻ ✭■■✮ → 0 ❤✳❝✳❝✳ ❦❤✐ n → ∞✳
❱ỵ✐ ✭■■■✮✱ t❛ ❝â
✭■■■✮ E ϕm(X) − X < ε.
❑➳t ❤đ♣ ❝→❝ ❧➟♣ ❧✉➟♥ tr➯♥ t❛ ✤÷đ❝ ✤✐➲✉ ♣❤↔✐ ự
ỵ s t sè ❧ỵ♥ ❞↕♥❣ ❧✉➟t ♠↕♥❤ sè ❧ỵ♥ ▼❛r❝✐♥❦✐❡✇✐❝③✲❩②❣♠✉♥❞
❝❤♦ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❘❛❞❡♠❛❝❤❡r ❞↕♥❣ p ✈➔ ❦❤ỉ♥❣ ❣✐❛♥
p✲❦❤↔ trì♥✳

✶✾


ỵ

1 < r < p 2 E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❘❛❞❡♠❛❝❤❡r ❞↕♥❣ p✳ ●✐↔ sû {Xn , n 1}
❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ♥❤➟♥ ❣✐→ trà tr➯♥ E ✈➔ ❜à ❝❤➦♥ ♥❣➝✉ ♥❤✐➯♥ ❜ð✐ ♣❤➛♥ tû ♥❣➝✉
♥❤✐➯♥ X ✳ ◆➳✉ E X r < ∞ t❤➻
n

1
n1/r
❈❤ù♥❣ ♠✐♥❤ ✳

❱ỵ✐ ♠é✐ k

(Xk − EXk ) → 0 h.c.c

✭✷✳✶✮

khi n → ∞.

k=1

1✱

✤➦t

Xk = Xk I(

k1/r ) ,

Xk

Xk = Xk I(

Xk >k1/r ) .

❑❤✐ ✤â
✭✷✳✷✮

Xk − EXk = (Xk − EXk ) + (Xk − EXk ).

✣➛✉ t✐➯♥ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤
n

1

❚❤➟t ✈➟②✱ t❛ ❝â E

p

2p E Xk

∞

k=1
∞

k p/r

∞

p

E Xk

=

I( Xk
k p/r

k=1

1

k

1

✈➔

pxp−1 P( Xk > x)dx
0

i1/r

pxp−1 P( X > x)dx
i=1

(i−1)1/r
∞

i1/r

pxp−1 dx

1/r

P X > (i − 1)

C

(i−1)1/r

i=1
∞

C

r

> i − 1)

dy
k=i

∞

P( X

r

> i − 1)ip/r−1

i=1
∞

C

y

p/r−1

(i−1)

i=1
∞

k=i

P( X

r

> i − 1)

k=i
∞

i

P( X

C

k1/r )

k1/r

k p/r

k=1

p

p

E Xk
k p/r

k=1
∞

C

✭✷✳✸✮

n → ∞.

k=1

Xk − EXk

∞

❤✳❝✳❝ ❦❤✐

(Xk − EXk ) → 0

n1/r

1
k p/r

1
k p/r

(✈ỵ✐ y = xr )

1
k p/r

CE X

r

< .

i=1

ử ỵ ợ bn = n1/r t❛ t❤✉ ✤÷đ❝ ✭✷✳✸✮✳
❚✐➳♣ t❤❡♦ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤
1
n1/r

n

(Xk − EXk ) → 0
k=1

✷✵

❤✳❝✳❝ ❦❤✐

n → ∞.

✭✷✳✹✮


❚❛ ❝â E

Xk − EXk

✈➔

2E Xk
∞

k=1

∞

∞

E Xk
k 1/r

=

=
k=1

1

k=1
∞

Xk I(

P
0

Xk >k1/r )

∞

P( X > k 1/r )dx +

k 1/r

∞

P( X > k

1/r

)+C

k=1

k=1
∞

1
k 1/r

(i+1)1/r

P( X > x)dx
i=k

i1/r
i

y 1/r−1 dy
(i−1)

i=1
∞

C +C

∞

i

P X > i1/r

C +C

P( X > x)dx
k1/r

0

∞

=C

> x dx

k1/r

1

k=1

>k1/r )

∞

k 1/r

C

E Xk I( Xk
k 1/r

P( X

r

k=1

1
k 1/r

> i) < ∞.

i=1

⑩♣ ❞ö♥❣ ✣à♥❤ ỵ ợ bn = n1/r t t ữủ
t ❧✉➟♥ ✭✷✳✶✮ ✤÷đ❝ s✉② r❛ tø ✭✷✳✷✮✱ ✭✷✳✸✮ ✈➔ ✭✷✳✹✮✳
❚÷ì♥❣ tỹ t õ

ỵ 1 < r < p

trì♥ E✳ ●✐↔ sû {Xn , n
t❤➻

1
n1/r

2 ✈➔ {Xn , Fn , n 1} ❧➔ ❞➣② ♣❤ị ❤đ♣ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ p✲❦❤↔
1} ❜à ❝❤➦♥ ♥❣➝✉ ♥❤✐➯♥ ❜ð✐ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ X ✳ ◆➳✉ E X r < ∞
n

Xk − E(Xk |Fk−1 ) → 0 h.c.c. khi n → ∞.
k=1

✷✶


❑➌❚ ▲❯❾◆
❑❤â❛ ❧✉➟♥ ✤➣ t❤✉ ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ s❛✉✿

✲❚r➻♥❤ ❜➔② ✤÷đ❝ ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ①→❝ s✉➜t tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
✲❚r➻♥❤ ❜➔② ✤÷đ❝ ✈➲ ❝→❝ ❞↕♥❣ ❤ë✐ tö ❝õ❛ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤ỉ♥❣
❣✐❛♥ ❇❛♥❛❝❤✳
✲❚r➻♥❤ ❜➔② ✤÷đ❝ ✈➲ ❧✉➟t ♠↕♥❤ sè ❧ỵ♥ ✤è✐ ✈ỵ✐ ❞➣② ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ tr♦♥❣ ❦❤æ♥❣
❣✐❛♥ ❇❛♥❛❝❤ ❜➜t ❦ý✱ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❘❛❞❡♠❛❝❤❡r ❞↕♥❣ ♣ ✈➔ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♣✲❦❤↔ trì♥
✭1 ≤ p ≤ 2✮✳

❑✐➳♥ ỳ ữợ ự t t

r tớ tỵ✐ ❝❤ó♥❣ tỉ✐ ♠♦♥❣ ♠✉è♥ t✐➳♣ tư❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✈➜♥ ✤➲ s❛✉✿
✲ ▲✉➟t ♠↕♥❤ sè ❧ỵ♥ ✤è✐ ✈ỵ✐ ♠↔♥❣ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❜➜t ❦ý✳
✲ ▲✉➟t ♠↕♥❤ sè ❧ỵ♥ ✤è✐ ✈ỵ✐ ♠↔♥❣ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❘❛❞❡♠❛❝❤❡r
❞↕♥❣ ♣ ✭1 ≤ p ≤ 2✮✳
✲ ▲✉➟t ♠↕♥❤ sè ❧ỵ♥ ✤è✐ ✈ỵ✐ ♠↔♥❣ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♣✲❦❤↔ trì♥

✭1 ≤ p ≤ 2✮✳

✷✷


❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖
❬✶❪ ◆❣✉②➵♥ ❱➠♥ ◗✉↔♥❣✱ ❳→❝ s✉➜t tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✱
✷✵✶✷
❬✷❪ ❊t❡♠❛❞✐ ◆✳ ✭✶✾✽✶✮✱ ❆♥ ❡❧❡♠❡♥t❛r② ♣r♦♦❢ ♦❢ t❤❡ str♦♥❣ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❩✳ ❲❛❤rs❝❤✳
❱❡r✇✳ ●❡❜✐❡t❡✱ ✺✺✭✶✮✱ ✶✶✾✲✶✷✷✳
❬✸❪ ▲❡❞♦✉① ▼✳ ❛♥❞ ❚❛❧❛❣r❛♥❞ ▼✳ ✭✶✾✾✶✮✱ Pr♦❜❛❜✐❧✐t② ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✳ ■s♦♣❡r✐♠❡tr② ❛♥❞ ♣r♦✲
❝❡ss❡s✳ ❊r❣❡❜♥✐ss❡ ❞❡r ▼❛t❤❡♠❛t✐❦ ✉♥❞ ✐❤r❡r ●r❡♥③❣❡❜✐❡t❡ ✭✸✮✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✳
❬✹❪ ◗✉❛♥❣ ◆✳ ❱✳ ❛♥❞ ❍✉❛♥ ◆✳ ❱✳ ✭✷✵✵✾✮✱ ❖♥ t❤❡ str♦♥❣ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❛♥❞ Lp✲❝♦♥✈❡r❣❡♥❝❡
❢♦r ❞♦✉❜❧❡ ❛rr❛②s ♦❢ r❛♥❞♦♠ ❡❧❡♠❡♥ts ✐♥ p✲✉♥✐❢♦r♠❧② s♠♦♦t❤ ❇❛♥❛❝❤ s♣❛❝❡s✱ ❙t❛t✐st✳ Pr♦❜❛❜✳
▲❡tt✳✱ ✼✾✭✶✽✮✱ ✶✽✾✶✲✶✽✾✾✳
❬✺❪ ◗✉❛♥❣ ◆✳ ❱✳ ❛♥❞ ❚❤✉❛♥ ◆✳ ❚✳ ✭✷✵✶✶✮✱ ❖♥ t❤❡ str♦♥❣ ❧❛✇s ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❢♦r ❞♦✉✲
❜❧❡ ❛rr❛②s ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✐♥ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥ s♣❛❝❡s✱ ❆❝t❛ ▼❛t❤✳ ❍✉♥❣❛r✳ ❉❖■✿
✶✵✳✶✵✵✼✴s✶✵✹✼✹✲✵✶✶✲✵✶✻✽✲✶✳
❬✻❪ ❘♦s❛❧s❦② ❆✳ ❛♥❞ ❚❤❛♥❤ ▲✳ ❱✳ ✭✷✵✵✻✮✱ ❙tr♦♥❣ ❛♥❞ ✇❡❛❦ ❧❛✇s ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❢♦r ❞♦✉❜❧❡
s✉♠s ♦❢ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ❡❧❡♠❡♥ts ✐♥ ❘❛❞❡♠❛❝❤❡r t②♣❡ p ❇❛♥❛❝❤ s♣❛❝❡s✱ ❙t♦❝❤✳ ❆♥❛❧✳
❆♣♣❧✳✱ ✷✹✭✻✮✱ ✶✵✾✼✲✶✶✶✼✳
❬✼❪ ❚❡r→♥ P✳ ❛♥❞ ▼♦❧❝❤❛♥♦✈ ■✳ ✭✷✵✵✻✮✱ ❚❤❡ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs
✐♥ ❛ ♠❡tr✐❝ s♣❛❝❡ ✇✐t❤ ❛ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥ ♦♣❡r❛t✐♦♥✱
❏✳ ❚❤❡♦r✳ Pr♦❜❛❜✳✱ ✶✾✱ ✽✼✺✲✽✾✽✳
❬✽❪ ❱❛❦❤❛♥✐❛ ◆✳ ◆✳✱ ❚❛r✐❡❧❛❞③❡ ❱✳ ■✳ ❛♥❞ ❈❤♦❜❛♥②❛♥ ❙✳ ❆✳ ✭✶✾✽✼✮✱ Pr♦❜❛❜✐❧✐t② ❞✐str✐❜✉✲
t✐♦♥s ♦♥ ❇❛♥❛❝❤ s♣❛❝❡s✱ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✱ ❉✳ ❘❡✐❞❡❧ P✉❜❧✐s❤✐♥❣ ❈♦✳✱
❉♦r❞r❡❝❤t✳

✷✸