r ỵ tt st ỵ ợ õ t số ợ õ r ❧➔ ✈➜♥ ✤➲ ✤➣ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉✳ ▲✉➟t sè ❧ỵ♥ ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ tr♦♥❣ t❤è♥❣ ❦➯✱ ❦✐♥❤ t➳✱ ② ❤å❝ ✈➔ ♠ët sè ♥❣➔♥❤ ❦❤♦❛ ❤å❝ t❤ü❝ ♥❣❤✐➺♠ ❦❤→❝✳ ❈❤➼♥❤ ✈➻ ✈➟②✱ ✈✐➺❝ ự ổ õ ỵ ỵ tt ỏ õ ỳ ỵ tỹ t t ợ r ữớ ỹ ỵ t❤✉②➳t ①→❝ s✉➜t ❞ü❛ ✈➔♦ ❤➺ t✐➯♥ ✤➲ ✈➔ ✤➣ t❤✐➳t ❧➟♣ ❧✉➟t sè ❧ỵ♥ ♥ê✐ t✐➳♥❣ ♠❛♥❣ t➯♥ ỉ♥❣✳ ▲✉➟t sè ❧ỵ♥ ✤è✐ ✈ỵ✐ ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ t✐➳♣ tư❝ ✤÷đ❝ r➜t ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ự ởt số t số ợ ữủ t t ợ t ữ ❏✳ ▼❛r❝✐♥❦✐❡✇✐❝③✱ ❆✳ ❩②❣♠✉♥❞✱ ❍✳ ❉✳ ❇r✉♥❦✱ ❨✳ ❱✳ Pr♦❦❤♦r♦✈✱ ❑✳ ▲✳ ❈❤✉♥❣✱ ❲✳ ❋❡❧❧❡r✱ ✳✳✳ ❈❤♦ tỵ✐ ♥❛②✱ ♥❣❤✐➯♥ ❝ù✉ ♣❤→t tr✐➸♥ ❦➳t q✉↔ ❝õ❛ ❆✳ ◆✳ ❑♦❧♠♦❣♦r♦✈ ✈➔ ♠ët sè ❞↕♥❣ ❧✉➟t sè ❧ỵ♥ ❦❤→❝ ✈➝♥ ❧➔ ♠ët õ t tớ sỹ ỵ tt s✉➜t✳ ❈→❝ ❧✉➟t sè ❧ỵ♥ ❝ê ✤✐➸♥ ❝❤õ ②➳✉ t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ 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 tt t r sỹ t ỳ ỵ tt st t ❑❤â❛ ❧✉➟♥ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ ▲✉➟t ②➳✉ sè ❧ỵ♥ ✤è✐ ✈ỵ✐ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❑❤â❛ ❧✉➟♥ ỗ ữỡ r ữỡ ú tổ tr ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ①→❝ s✉➜t tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❈→❝ ❦✐➳♥ t❤ù❝ ♥➔② ❧➔ ❝ì sð ✤➸ tr➻♥❤ ❜➔② ❝→❝ ✈➜♥ ✤➲ ❝õ❛ ❝❤÷ì♥❣ ✷✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ▲✉➟t ②➳✉ sè ❧ỵ♥ ✤è✐ ✈ỵ✐ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤æ♥❣ ❣✐❛♥ 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 ②➳✉ ❝❤♦ ❞➣② ✤ë❝ ❧➟♣ ♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❘❛❞❡♠❛❝❤❡r ❞↕♥❤ ♣ ✭1 ≤ p ≤ 2✮✳ ❈✉è✐ ❝ị♥❣✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ✈➲ ♠ët sè ❧✉➟t ②➳✉ ✤è✐ ✈ỵ✐ ❞➣② ♣❤ị ❤đ♣ ♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ♣✲ ❦❤↔ trì♥ ✭1 ≤ p ≤ 2✮✳ ❑❤â❛ ❧✉➟♥ ♥➔② ♥➔② ✤÷đ❝ ❤♦➔♥ t ợ sỹ ữợ P ◗✉↔♥❣✳ ◆❤➙♥ ❞à♣ ♥➔②✱ t→❝ ❣✐↔ ①✐♥ tä ❧á♥❣ ❜✐➳t ỡ sỹ ữợ t t ❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝ỉ ð ❇ë ♠ỉ♥ ❳→❝ s✉➜t t❤è♥❣ ❦➯ ✈➔ ❚♦→♥ ù♥❣ ❞ư♥❣✱ ❑❤♦❛ ❚♦→♥✱tr÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ q✉❛♥ t➙♠ ✤ë♥❣ ✈✐➯♥ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ▼➦❝ ❞ò t→❝ ❣✐↔ ✤➣ r➜t ❝è ❣➢♥❣✱ ♥❤÷♥❣ ❝❤➢❝ ❝❤➢♥ ❦❤â❛ ❧✉➟♥ ✈➝♥ ❝á♥ ♥❤✐➲✉ ❦❤✐➳♠ ❦❤✉②➳t✳ ▼♦♥❣ ✤÷đ❝ sü tr❛♦ ✤ê✐✱ õ ỵ t ổ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✷ ❚→❝ ❣✐↔ ✶ ❈❍×❒◆● ✶ ❈⑩❈ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✶✳✶✳ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ · ❑❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì E ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ : E → R t❤♦↔ ♠➣♥ ✭✐✮ x 0, ∀x ∈ E❀ ✭✐✐✮ x = ⇔ x = 0❀ ✭✐✐✐✮ kx = |k| x , ∀k ∈ R, ∀x ∈ E❀ ✭✐✈✮ x + y x + y , x, y E tỗ t →♥❤ ①↕ d(x, y) = x − y (x, y ∈ E) t❤➻ (E, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ❑❤✐ ✤â d ✤÷đ❝ ❣å✐ ❧➔ ♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ · ✳ ◆➳✉ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì t❤ü❝ t❤➻ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ E, · ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ◆➳✉ ✤➦t ✤à♥❤ ❝❤✉➞♥ t❤ü❝✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ E, · ✤÷đ❝ ❣å✐ ❧➔ d ❧➔ ♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ · ❇❛♥❛❝❤✱ t❤➻ E, · ✤÷đ❝ ❣å✐ ❧➔ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤➛② ✤õ✱ tr♦♥❣ ✤â E, · ✈➔ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ●✐↔ sû E ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♥➳✉ (E, d) ✳ ◆➳✉ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì t❤ü❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝✳ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝✳ ỵ E = {f : E R|f ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤✱ ❧✐➯♥ tư❝} E∗ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ ❝õ❛ E✳ ∗ ❱ỵ✐ f ∈ E ✱ ❝❤✉➞♥ ❝õ❛ f ✤÷đ❝ ①→❝ ✤à♥❤ ❚❛ ❣å✐ ❜ð✐ ❝ỉ♥❣ t❤ù❝ f = sup |f (x)|, x ♥➯♥ |f (x)| f x ✈ỵ✐ ♠å✐ x ∈ E✳ ✣à♥❤ ỵ sỷ E ổ t❤ü❝ ✈➔ F ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✤â♥❣ ❝õ❛ E✳ õ ợ ộ f F tỗ t f ∈ E∗ s❛♦ ❝❤♦ f | = f ✈➔ f = f ✳ F ❍➺ q✉↔ ✶✳✶✳✹✳ ●✐↔ sû E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ x E, x = 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♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ❦❤↔ ❧②✳ ❑❤✐ ✤â tỗ t {f , n s x = supn |fn (x)| n ✈ỵ✐ ♠å✐ x ∈ E✳ ✷ 1} ⊂ E∗ ❈❤ó♥❣ t❛ ❝❤✉②➸♥ s❛♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❝→❝ t➟♣ ❇♦r❡❧ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✼✳ ❚➟♣ A⊂E ●✐↔ sû E t➟♣ trư ✤÷đ❝ ❣å✐ ❧➔ ∗ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ✈➔ E ❧➔ ❦❤æ♥❣ ủ n tỗ t n ∈ N ; f1 , f2 , , fn ∈ E ; A ∈ B(R ) s❛♦ ❝❤♦ E✳ A = {x ∈ E : f1 (x), , fn (x) A} ỵ ❤✐➺✉ t➟♣ ❝→❝ t➟♣ trö ❧➔ F(E)✳ ❱➼ ❞ö ✶✳✶✳✽✳ ✶✳ ▲➜② ✷✳ ▲➜② f ∈ E∗ ✈➔ a ∈ R t❤➻ A = {x ∈ E : f (x) = a} ❧➔ t➟♣ trö✳ 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➟♣ ❇♦r❡❧ ❝õ❛ E✳ ✭✐✮ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ❦❤↔ ❧② t❤➻ σ F(E) = B(E)✱ ✈ỵ✐ B(E) ❧➔ σ✲✤↕✐ sè ❝→❝ ✶✳✷✳ P❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❚r♦♥❣ ♠ư❝ ♥➔② ❝❤ó♥❣ t❛ s➩ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❝ị♥❣ ❝→❝ t➼♥❤ ❝❤➜t ❝ì (Ω, F, P) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t ✤➛② ✤õ✱ E ❧➔ ❦❤æ♥❣ ❝♦♥ ❝õ❛ F ✈➔ B(E) ❧➔ σ ✲✤↕✐ sè ❝→❝ t➟♣ ❇♦r❡❧ ❝õ❛ E✳ ❜↔♥ ❝õ❛ ❝❤ó♥❣✳❈❤ó♥❣ t❛ ❧✉ỉ♥ ❣✐↔ sû ❇❛♥❛❝❤ t❤ü❝ ❦❤↔ ❧②✱ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ →♥❤ ①↕ G/B(E) G ❧➔ X sè X : Ω −→ E ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ −1 ✈ỵ✐ ♠å✐ B ∈ B(E) t❤➻ X (B) ∈ G ✮✳ ❚❛ ♥â✐ →♥❤ ①↕ ✤♦ ✤÷đ❝ ✭♥❣❤➽❛ ❧➔ P❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ♥❤✐➯♥✱ ♥➳✉ σ ✲✤↕✐ F ✲✤♦ ✤÷đ❝ s➩ ✤÷đ❝ ❣å✐ ♠ët ❝→❝❤ ✤ì♥ ❣✐↔♥ ❧➔ ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❞➔♥❣ t❤➜② r➡♥❣ ♥➳✉ X G ✲✤♦ ✤÷đ❝ t❤➻ X ❣✐❛♥ ✤÷đ❝✱ ♥➳✉ X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✳ ❍✐➸♥ ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✳ ▼➦t ❦❤→❝✱ ❞➵ ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ t❤➻ ❤å σ(X) = {X −1 (B) : B ∈ B(E)} ❧➟♣ t❤➔♥❤ ♠ët ♥ú❛✱ σ(X) ✈➔ ❝❤➾ ❦❤✐ σ ✲✤↕✐ σ ✲✤↕✐ sè σ(X) ⊂ G ✳ ❧➔ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✷✳ σ ✲✤↕✐ sè F ✳ σ ✲✤↕✐ sè ♥➔② ✤÷đ❝ ❣å✐ ❧➔ σ ✲✤↕✐ sè s✐♥❤ ❜ð✐ X ✳ ❍ì♥ ♠➔ X ✤♦ ✤÷đ❝✳ ❉♦ ✤â X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝ ❦❤✐ sè ❝♦♥ ❝õ❛ ❜➨ ♥❤➜t 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)✮✳ ❱➼ ❞ư ✶✳✷✳✸✳ P❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ●✐↔ sû A ∈ F ✱ a ∈ E✱ a = 0✳ ✣➦t X(ω) = ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ ♥➳✉ ♥➳✉ ω ∈ A; ω∈ / A B ∈ B(E)✱ ∅, A, X −1 (B) = A, Ω, ♥➯♥ a X −1 (B) ∈ F ✳ ❉♦ ✤â X ♥➳✉ ♥➳✉ ♥➳✉ ♥➳✉ 0∈ / B, a ∈ / B; 0∈ / B, a ∈ B; ∈ B, a ∈ / B; ∈ B, a ∈ B ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✳ ❍ì♥ ♥ú❛✱ ✈➻ ♥❣➝✉ ♥❤✐➯♥ ✤ì♥ ❣✐↔♥✳ ✸ |X(Ω)| ♥➯♥ X ❧➔ ♣❤➛♥ tû ❈❤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 ✤➦❝ tr÷♥❣ q✉❛♥ trå♥❣ ❝õ❛ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✳ ✣à♥❤ ỵ X : E ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈ỵ✐ ♠å✐ 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❝✳ f ∈ E∗ ❍➺ q✉↔ ✶✳✷✳✽✳ ◆➳✉ {X , n ❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝ ✈➔ Xn → X ❦❤✐ n → ∞ t❤➻ X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝✳ n 1} ❚✐➳♣ t❤❡♦ ❝❤ó♥❣ t❛ ♥❣❤✐➯♥ ❝ù✉ ❝➜✉ tró❝ ❝õ❛ t tỷ ỵ ①↕ X : Ω → E ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ X ❧➔ ❣✐ỵ✐ ❤↕♥ ✤➲✉ ❝õ❛ ♠ët ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ rớ r G ữủ tỗ t ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ rí✐ r↕❝ G ✲✤♦ ✤÷đ❝ {Xn, n 1}✱ s❛♦ ❝❤♦ lim sup Xn (ω) − X(ω) = n ỵ X : Ω → E ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ữủ X ợ ✭t❤❡♦ ❝❤✉➞♥ ✮ ❝õ❛ ♠ët ❞➣② ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ì♥ ❣✐↔♥ G ✲✤♦ ✤÷đ❝ {Xn, n 1}✱ s❛♦ ❝❤♦ Xn(ω) X(ω) ✈ỵ✐ ♠å✐ n ✈➔ ♠å✐ tỗ t tỷ ♥❣➝✉ ♥❤✐➯♥ ✤ì♥ ❣✐↔♥ G ✲✤♦ ✤÷đ❝ {Xn, n 1} t❤♦↔ ♠➣♥ n→∞ lim Xn (ω) − X(ω) = ✈➔ Xn (ω) X(ω) ✈ỵ✐ ♠å✐ n ✈➔ ♠å✐ ω ∈ Ω✳ ❈❤ó♥❣ t❛ ❦➳t t❤ó❝ ♠ư❝ ♥➔② ✈ỵ✐ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✶✳ {Xt , t ∈ ∆} ❧➔ ❤å ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❝ò♥❣ ①→❝ ✤à♥❤ (Ω, F, P)✱ ♥❤➟♥ ❣✐→ trà tr➯♥ (E, B(E))✳ ❑❤✐ ✤â✱ ❤å {Xt , t ∈ ∆} ✤÷đ❝ ❣å✐ ❧➔ ✤ë❝ ❧➟♣ ✤æ✐ ♠ët ❧➟♣ ✮ ♥➳✉ ❤å σ✲✤↕✐ sè {σ(Xt), t ∈ ∆} ✤ë❝ ❧➟♣ ✤æ✐ ♠ët ✭✤ë❝ ❧➟♣✮✳ ●✐↔ sû tr➯♥ ✤ë❝ ✭ ❚ø ✤à♥❤ ♥❣❤➽❛ tr➯♥✱ t❛ s✉② r ỵ s ỵ sû E1, E2 ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ 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 ♥❣➝✉ ♥❤✐➯♥ tr tr E2 ỵ ●✐↔ sû X , X , , X ❧➔ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❝ò♥❣ ①→❝ ✤à♥❤ tr➯♥ (Ω, F, P)✱ n ♥❤➟♥ ❣✐→ trà tr➯♥ (E, B(E))✳ ❑❤✐ ✤â✱ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ X1, X2, , Xn ✤ë❝ ❧➟♣ ❧➔ ✈ỵ✐ ♠å✐ f1 , f2 , , fn ∈ E∗ ✱ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ f (X1 ), f (X2 ), , f (Xn ) ✤ë❝ ❧➟♣✳ ✹ ✶✳✸✳ ❑ý ✈å♥❣ ❝õ❛ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳ X:Ω→E f ∈ E∗ t❛ ❝â ●✐↔ sû ✈å♥❣ ❝õ❛ X ♥➳✉ ✈ỵ✐ ♠å✐ ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✳ P❤➛♥ tû m∈E ✤÷đ❝ ❣å✐ ý f (m) = E(f (X)) ỵ m = EX ✳ ❱➼ ❞ö ✶✳✸✳✷✳ ●✐↔ sû a ∈ E✱ A ∈ F ✈➔ X = aIA ✱ X(ω) = ❑❤✐ ✤â✱ ✈➻ ✈ỵ✐ ♠å✐ a tù❝ ❧➔ ♥➳✉ ♥➳✉ ω ∈ A; ω∈ / A✳ f ∈ E∗ ✱ f (P(A)a) = P(A)f (a) ✈➔ E(f (X)) = E(f (a)IA ) = f (a)EIA = f (a)P(A), EX = P(A)a ỵ sỷ X ✱ Y ❧➔ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✱ ξ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝ò♥❣ ①→❝ ✤à♥❤ tr➯♥ (Ω, F, P), a R E õ 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) = t❤➻ EX = α❀ ∗ ✺✳ ◆➳✉ ξ ✈➔ f (X) ✤ë❝ ❧➟♣ ợ f E t tỗ t E(X) E(ξX) = EξEX ❀ ✻✳ ❱ỵ✐ ♠å✐ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ T : E → E ✭E ❧➔ ổ tỹ t tỗ t E(T (X)) E(T (X)) = T (E(X)) ỵ E X < t tỗ t EX E X EX ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✺✳ ●✐↔ sû ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✻✳ ❚❛ ♥â✐ →♥❤ ①↕ X ❦❤↔ t➼❝❤ ❜➟❝ p✳ ◆➳✉ X ❦❤↔ t➼❝❤ ❜➟❝ 1✱ t❤➻ ✤➸ ✤ì♥ ❣✐↔♥✱ ϕ:E→R ϕ(ax + (1 − a)y) ✈ỵ✐ ♠å✐ ❧➔ p < t t õ X ỗ aϕ(x) + (1 − a)ϕ(y) a ∈ [0, 1], x, y E ỵ t t p > 0✳ ◆➳✉ E X t❛ ♥â✐ X ❦❤↔ t➼❝❤✳ ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✈➔ ✭❇➜t ✤➥♥❣ t❤ù❝ ❏❡♥s❡♥✮ ◆➳✉ : E R ỗ tử X ✈➔ ϕ(X) ❦❤↔ 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 F ✳ ❑❤✐ ✤â ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❨✿ Ω → E ❣å✐ ❧➔ ❦ý ✈å♥❣ ❝â ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❳ ✤è✐ ✈ỵ✐ G ♥➳✉ ✭✐✮ Y ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝❀ ✭✐✐✮ E(Y IA ) = E(XIA )✱ ✈ỵ✐ ♠å✐ A ∈ G ✳ ❑➼ ❤✐➺✉ Y = E(X|G ✮✳ ❱➼ ❞ö G = {, } tỗ t EX ∈ E✱ t❤➻ E(X|G) = EX Y = EX ✱ ❦❤✐ B ∈ B(E) t❛ ❝â ❚❤➟t ✈➟②✱ ✤➦t ✭✐✮ ❱ỵ✐ ♠å✐ ✤â Y −1 (B) = ∅ Ω ♥➳✉ ♥➳✉ EX ∈ / B; EX ∈ B, Y −1 (B) ∈ G ✳ ❱➟② Y ❧➔ ♣❤➛♥ tû ♥❣➝✉ G ữủ ợ 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 ) ❱➼ ❞ư ✶✳✹✳✸✳ ✈ỵ✐ ♠å✐ A ∈ G✳ ❉♦ ✤â A ∈ F, a ∈ E, X = aIA ✳ ●✐↔ sû Y = E(X|G) ❤❛② E(X|G) = EX ✳ ❑❤✐ ✤â E(X|G) = aE(IA |G) ❚❤➟t ✈➟②✱ ✤➦t ✭✐✮ E(IA |G) Y = aE(IA |G)✳ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ G ✲✤♦ ✤÷đ❝ ♥➯♥ Y = aE(IA |G) ❧➔ ♣❤➛♥ tû G ữủ ợ BG (2.1) ✈➔ (2.2) s✉② ❱➟② Y = E(X|G)✳ ❚ø t❛ ❝â E(Y IB ) = E(aE(IA |G)IB ) = aE(E(IAB |G)) = aE(IAB ) = aP(AB), ✭✷✳✶✮ E(XIB ) = E(aIAB ) = aP(AB) ✭✷✳✷✮ r❛✱ ✈ỵ✐ ♠å✐ B ∈ G✱ t õ E(Y IB ) = E(XIB ) ỵ s❛✉ ✤➙② ❝❤♦ t❛ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❦❤→❝ ✤➸ ✤à♥❤ ♥❣❤➽❛ ❦ý ✈å♥❣ ❝â ✤✐➲✉ ❦✐➺♥✱ t÷ì♥❣ tü ♥❤÷ ✤à♥❤ ý ỵ sỷ X, Y ❧➔ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✳ ❑❤✐ ✤â Y f (Y ) = E(f (X)|G) ✈ỵ✐ ♠å✐ f ∈ E ✳ ∗ = E(X|G) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ⑩♣ ❞ö♥❣ ỵ tr t t ý õ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ỵ s ỵ sỷ X, Y ❧➔ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥✱ ξ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✱ α ∈ E✱ a ∈ R✱ f ∈ E∗ ✳ ❑❤✐ ✤â ✶✳ ◆➳✉ 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 σ ✲✤↕✐ sè ❝♦♥ ❝õ❛ σ ✲✤↕✐ sè F ✳ ❑❤✐ ✤â ❞➣② {Xn , Fn , n 1✱ Xn ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ Fn ✲✤♦ ✤÷đ❝✳ ❝→❝ n ♥❣➝✉ ♥❤✐➯♥✱ 1} ❣å✐ ❧➔ {Fn , n 1} ❧➔ ❞➣② t➠♥❣ ❞➣② ♣❤ị ❤đ♣ ♥➳✉ ✈ỵ✐ ♠å✐ {Xn , n 1} ❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❜➜t ❦ý ✈➔ Fn = σ(X1 , · · · , Xn ) X1 , · · · , Xn ✮ t❤➻ ❞➣② {Xn , Fn , n 1} ❧➔ ❞➣② ♣❤ị ❤đ♣✳ ❈❤➥♥❣ ❤↕♥✱ ♥➳✉ σ ✲✤↕✐ sè s✐♥❤ ❜ð✐ ✭❧➔ ✣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 = ✈ỵ✐ ♠å✐ n ✈➔ Fn = σ(X1 , · · · , Xn )✳ ❑❤✐ ✤â {Xn , Fn , n 1} ❧➔ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡ ✈➔ {Sn = n 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 = ✈➔ 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 ✶✳✹✳✾✳ ♠å✐ n ∈ N✱ ◆➳✉ {Xn , Fn , n 1} ❧➔ ❞➣② ♣❤ò {Xn , Fn , n 1} ❧➔ ♠❛rt✐♥❣❛❧❡✳ ✈ỵ✐ m n✱ t❤❡♦ t➼♥❤ ❝❤➜t ❤ót ❝õ❛ ❦ý ❤ñ♣✱ Xn ❦❤↔ t➼❝❤ ✈➔ E(Xn+1 |Fn ) = Xn t❤➻ ❚❤➟t ✈➟②✱ ✈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 ◆❤➟♥ ①➨t t÷ì♥❣ tü ❝ơ♥❣ ✤ó♥❣ ✤è✐ ✈ỵ✐ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡✳ ✼ 1} ❧➔ ♠❛rt✐♥❣❛❧❡✳ ✈ỵ✐ ❚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 1} ❧➔ ❞➣② ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ t❤ü❝ ❧➟♣ t❤➔♥❤ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡ ✈➔ {xn , n 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 = ✈ỵ✐ ♠å✐ m > 1✳ ✶✳ ✽ ì r sốt ữỡ ❝❤ó♥❣ tỉ✐ ❧✉ỉ♥ ❣✐↔ sû r➡♥❣ E B(E) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ❦❤↔ ❧②✱ ❧➔ σ (Ω, F, P) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t ✤➛② ✤õ✱ ✲ ✤↕✐ sè t r tr E C ỵ s ữủ ❞ị♥❣ ✤➸ ❝❤➾ ❤➡♥❣ sè ❞÷ì♥❣ ✈➔ ❣✐→ trà ❝õ❛ ♥â ❝â t❤➸ ❦❤→❝ ♥❤❛✉ ❣✐ú❛ ❝→❝ ❧➛♥ ①✉➜t ❤✐➺♥✳ ✷✳✶✳ ❈→❝ ❞↕♥❣ ❤ë✐ tö ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✳ ●✐↔ sû {X, Xn , n 1} ❧➔ ❤å ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❝ò♥❣ ①→❝ ✤à♥❤ tr➯♥ Ω ✈➔ E✳ ❚❛ ♥â✐✿ ❉➣② {Xn , n 1} ❤ë✐ tö ❤➛✉ ❝❤➢❝ ❝❤➢♥ X n tỗ t t N ∈ F s❛♦ ❝❤♦ P(N ) = ✈➔ Xn (ω) → X(ω) ✭t❤❡♦ ❝❤✉➞♥✱ ❦❤✐ n → ∞✮ ợ \N h.c.c ỵ Xn → X ❤✳❝✳❝✳✱ ❤♦➦❝ Xn − −−→ X ✭❦❤✐ n → ∞✮✳ ❉➣② {Xn , n 1} ❤ë✐ tö ✤➛② ✤õ ✤➳♥ X ✭❦❤✐ n → ∞✮✱ ♥➳✉ ✈ỵ✐ ♠å✐ ε > t❤➻ ♥❤➟♥ ❣✐→ trà tr♦♥❣ ∞ P( Xn X > ) < n=1 ỵ ❉➣② c Xn → − X ✭❦❤✐ n → ∞✮✳ {Xn , n 1} ❤ë✐ tö t❤❡♦ ①→❝ s✉➜t ✤➳♥ X ✭❦❤✐ n → ∞✮✱ ♥➳✉ ✈ỵ✐ ♠å✐ ε > t❤➻ lim P( Xn − X > ε) = n→∞ P Xn → − X ✭❦❤✐ n → ∞✮✳ ❉➣② {Xn , n 1} ❤ë✐ tö t❤❡♦ tr✉♥❣ ❜➻♥❤ p ❦❤↔ t➼❝❤ ❜➟❝ p ✈➔ lim E Xn X = ỵ p > ✤➳♥ X ✭❦❤✐ n → ∞✮✱ ♥➳✉ X, Xn (n 1) n ỵ Lp Xn X n → ∞✮✳ {Xn , n 1} ❤ë✐ tö ②➳✉ ✭t❤❡♦ ♣❤➙♥ ♣❤è✐✮ ✤➳♥ X ✭❦❤✐ n → ∞✮✱ ♥➳✉ PX n w − → PX ✱ tr♦♥❣ ✤â PX : B(E) → R B → P X −1 (B) ỵ w Xn X ✭❦❤✐ ❈❤♦ n → ∞✮✳ Xn → X ❤✳❝✳❝✳ ✈➔ Yn → Y ❤✳❝✳❝✳ ❑❤✐ ✤â Xn + Yn → X + Y n → ∞✮✳ ❚❤➟t ✈➟②✱ ✤➦t Ω1 = ω : lim Xn (ω) − X(ω) = , n→∞ Ω2 = ω : lim Yn (ω) − Y (ω) = n→∞ ✾ ❤✳❝✳❝✳ ✭❦❤✐ ❚❤❡♦ ❣✐↔ t❤✐➳t Xn → X ❤✳❝✳❝✳ ✈➔ Yn → Y ω ∈ Ω1 ∩ Ω2 t❤➻ ❤✳❝✳❝✳ ♥➯♥ P(Ω1 ) = P(Ω2 ) = 1✱ s✉② r❛ P(Ω1 ∩ Ω2 ) = 1✳ ❑❤✐ ✤â ✈ỵ✐ ♠å✐ lim Xn (ω) − X(ω) = 0, lim Yn (ω) − Y (ω) = n→∞ n→∞ ❤❛② · · Xn (ω) → X(ω), ❱➻ E Yn (ω) → Y (ω) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♥➯♥ · Xn (ω) + Yn (ω) → X(ω) + Y (ω) ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä r➡♥❣ ω ∈ ω : lim Xn + Yn − X − Y (ω) = n→∞ ❉♦ ✤â Ω1 ∩ Ω2 ⊂ {ω : lim Xn + Yn − X − Y (ω) = 0}, n→∞ ♥➯♥ P lim Xn + Yn − X − Y (ω) = = n→∞ ❱➟② h.c.c Xn + Yn −−−→ X + Y ❱➼ ❞ö ✷✳✶✳✸✳ ①→❝ s✉➜t ❚❤➟t ✭❦❤✐ n → ∞✮✳ a ∈ E ✈➔ Xn ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ì♥ ❣✐↔♥ ♥❤➟♥ ❝→❝ ❣✐→ trà ✈➔ a ợ L2 P tữỡ ự 1/n ✈➔ 1/n✳ ❑❤✐ ✤â Xn → ✈➔ Xn → ✭❦❤✐ n → ∞✮✳ ✈➟②✱ t❛ ❝â Xn : Ω → R+ ①→❝ ✤à♥❤ ❜ð✐ ❈❤♦ Xn (ω) = ❑❤✐ ✤â ✈ỵ✐ ♠å✐ ♥➳✉ a ♥➳✉ Xn (ω) = 0, Xn (ω) = a ε > 0✱ P Xn − > ε = P Xn > ε P Xn = a = P(Xn = a) = → ❦❤✐ n → ∞ n ▼➦t ❦❤→❝✱ ✈➻ ◆❤➟♥ ①➨t ✷✳✶✳✹✳ P Xn → ✭❦❤✐ n → ∞✮✳ E Xn − = a · 1/n → ✣✐➲✉ ♥➔② ✤↔♠ ❜↔♦ r➡♥❣ ❦❤✐ n→∞ L Xn →2 ❚ø ✤à♥❤ ♥❣❤➽❛ tr➯♥ s✉② r❛ r➡♥❣ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✭✤➛② ✤õ✱ t❤❡♦ ①→❝ s✉➜t✱ ✭❦❤✐ n → ∞✮✳ {Xn , n 1} ❤ë✐ tö ♥❤✐➯♥ X ✭❦❤✐ p✮ ✤➳♥ ♣❤➛♥ tû ♥❣➝✉ ❞➣② ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✭t❤ü❝✮ { Xn − X , n 1} ❤ë✐ tö ❤➛✉ ❝❤➢❝ ❝❤➢♥ t❤❡♦ tr✉♥❣ ❜➻♥❤ ❝➜♣ p✮ ✤➳♥ ✭❦❤✐ n → ∞✮✳ ❉♦ ✤â✱ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❤➛✉ ❝❤➢❝ ❝❤➢♥ ✭✤➛② ✤õ✱ t❤❡♦ ①→❝ s✉➜t✱ t❤❡♦ tr✉♥❣ ❜➻♥❤ ❝➜♣ n → ∞✮ ♥➯♥ ❄ ❝→❝ 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 = n→∞ m n ✶✵ c h.c.c ◆➳✉ Xn → − X t❤➻ Xn −−−→ X ✭❦❤✐ n → ∞✮✳ h.c.c c ✸✳ ◆➳✉ {Xn , n 1} ❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ✈➔ Xn −−−→ C ∈ E t❤➻ Xn → − C ✭❦❤✐ n → ∞✮✳ L P h.c.c − X ✭❦❤✐ n → ∞✮✳ ✹✳ ◆➳✉ Xn − −−→ X ❤♦➦❝ Xn −→ X t❤➻ Xn → ✺✳ ◆➳✉ ❞➣② {Xn , n 1} tử t st t tỗ t ❞➣② ❝♦♥ {Xn ; k 1) ⊂ (Xn , n ≥ 1} s❛♦ ❝❤♦ {Xn ; k 1} ❤ë✐ tö ❤✳❝✳❝✳ ✷✳ p k k ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✺✳ ❚❛ ♥â✐ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ {Xn , n 1} ❧➔ ❞➣② ❝ì ❜↔♥✿ ❤➛✉ ❝❤➢❝ ❝❤➢♥ ✭❤✳❝✳❝✳✮ ♥➳✉ P(m,n→∞ lim Xm − Xn = 0) = 1❀ t❤❡♦ ①→❝ s✉➜t ♥➳✉ m,n→∞ lim P( Xm − Xn > ε) = ✈ỵ✐ ♠å✐ ε > 0❀ t❤❡♦ tr✉♥❣ ❜➻♥❤ ❝➜♣ p > ♥➳✉ m,n→∞ lim E Xm − Xn p = ỵ {X , n ỵ {X , n n n s ✤÷đ❝ t❤♦↔ ♠➣♥✿ ✭✐✮ ✭✐✐✮ 1} ❝ì ❜↔♥ ❤✳❝✳❝✳ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❞➣② {Xn, n 1} ❧➔ ❞➣② ❝ì ❜↔♥ ❤✳❝✳❝✳ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♠ët tr♦♥❣ ❤❛✐ ✤✐➲✉ ❦✐➺♥ ❤ë✐ tư ❤✳❝✳❝✳ ✈ỵ✐ ♠å✐ ε > 0❀ = ✈ỵ✐ ♠å✐ ε > 0✳ lim P sup Xk − Xl > ε = n→∞ k,l n lim P sup Xk − Xn > ε n→∞ k n ỵ {X , n {Xn , n 1} 1} s❛♦ ❝❤♦ {Xn , k n k ỵ {X , n n st ỵ {X , n n ỡ t st t tỗ t {Xn , k 1} ❤ë✐ tö ❤✳❝✳❝✳ 1} 1} ✭✐✮ ●✐↔ sû G ❧➔ σ ✲✤↕✐ 1} sè ❝♦♥ ❝õ❛ ❤ë✐ tö t❤❡♦ tr✉♥❣ ❜➻♥❤ ❝➜♣ p (p σ ✲✤↕✐ sè F✳ 1) p 1✱ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â ❧➔ ❉➵ ❞➔♥❣ t r t t ỵ tr ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✈➝♥ ✤ó♥❣ ❝❤♦ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✭✐✐✮ ❱ỵ✐ ♠é✐ sè t❤ü❝ 1} ⊂ ❤ë✐ tö t❤❡♦ ①→❝ s✉➜t ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â ❧➔ ❞➣② ❝ì ❜↔♥ t❤❡♦ ①→❝ ❞➣② ❝ì ❜↔♥ t❤❡♦ tr✉♥❣ ❜➻♥❤ ❝➜♣ p✳ ◆❤➟♥ ①➨t ✷✳✶✳✶✶✳ k G ✲✤♦ ✤÷đ❝✳ ✤➦t Lp (G, E) = X : Ω → E : X ❧➔ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ E X p G ✲✤♦ ✤÷đ❝ bn ) → n → ∞, ✭✷✳✷✮ i=1 n b−1 n P E(Yni |Fi−1 ) → − ❦❤✐ n → ∞, i=1 n b−p n E Yni − E(Yni |Fi−1 ) p →0 ✭✷✳✸✮ ❦❤✐ n → ∞ ✭✷✳✹✮ i=1 ❈❤ù♥❣ ♠✐♥❤ ✳ n ✣➦t Sn = n Xi , Snn = i=1 Yni ✱ ❦❤✐ ✤â t❤❡♦ ✭✷✳✷✮✱ i=1 n Sn Snn = P bn bn P n (Yni = Xi ) i=1 P(Yni = Xi ) i=1 n P( Xi > bn ) → = ❦❤✐ n → ∞ i=1 ❉♦ ✤â✱ ✤➸ ❝❤ù♥❣ ♠✐♥❤ P b−1 − n Sn → ❦❤✐ n → ∞✱ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ▼➦t ❦❤→❝✱ ❞♦ ✭✷✳✸✮ ♥➯♥ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ n b−1 n P Yni − E(Yni |Fi−1 ) → − i=1 ✶✷ ❦❤✐ n → ∞ P b−1 − n Snn → ❦❤✐ n → ∞✳ ◆❤➟♥ t❤➜②✱ ✈ỵ✐ ♠é✐ trà tr➯♥ E✳ n ❉♦ ✤â✱ ✈ỵ✐ ♠å✐ 1✱ {Yni − E(Yni |Fi−1 ), Fi , ε > 0✱ i n} ❧➔ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡ ♥❤➟♥ ❣✐→ n b−1 n P E p ε bpn Yni − E(Yni |Fi−1 ) i=1 n >ε p Yni − E(Yni |Fi−1 ) i=1 n C εp bpn E Yni − E(Yni |Fi−1 ) p i=1 ✣✐➲✉ ♥➔② ❝ị♥❣ ✈ỵ✐ ✭✷✳✹✮ ✤↔♠ ❜↔♦ r➡♥❣ n b−1 n P Yni − E(Yni |Fi−1 ) → − ❦❤✐ n → ∞ i=1 ❚ø õ t ữủ ỵ ữủ ự ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ tỉ✐ s➩ t❤✐➳t ❧➟♣ ❧✉➟t ②➳✉ sè ❧ỵ♥ ❑♦❧♠♦❣♦r♦✈✲❋❡❧❧❡r ❝❤♦ ❞➣② ♣❤ị ❤đ♣✳ ✣➸ ❧➔♠ ✤✐➲✉ ✤â✱ t❛ ❝➛♥ ❜ê ✤➲ s❛✉✳ ❇ê ✤➲ ✷✳✷✳✸✳ ◆➳✉ p ❧➔ ♠ët sè t❤ü❝ (1 ✭✐✮ p r k p/r k ip/r−1 , i i = 1, 2, , k ✱ ✭✐✮ ❱➻ ✈➔ k ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣ t❤➻ ✈ỵ✐ ♠å✐ r ∈ (0, p), r k p/r−1 − (i0 − 1)p/r−1 , p−r p/r−2 i=i0 ❈❤ù♥❣ ♠✐♥❤ ✳ 2) i=1 k ✭✐✐✮ p r ∈ (0, p) ♥➯♥ ❤➔♠ y = xp/r−1 ✈ỵ✐ ♠å✐ i0 ∈ N, ỗ tr t r (p/2, p) (0, ) ❉♦ ✤â✱ ✈ỵ✐ ♠å✐ t❛ ❝â i i p/r−1 i p/r−1 = i xp/r−1 dx dx i−1 i−1 ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ k k i p/r−1 xp/r−1 dx i i=1 i−1 i=1 k xp/r−1 dx = = ✭✐✐✮ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ y = xp/r−2 r p/r k p ❧➔ ❤➔♠ ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ t➟♣ i = 1, 2, , k ✱ i i ip/r−2 = ip/r−2 dx i−1 xp/r−2 dx i−1 ❉♦ ✈➟② k k i i=i0 i p/r−2 xp/r−2 dx i=i0 i−1 r = k p/r−1 − (i0 − 1)p/r−1 p−r ❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✶✸ (0, ∞)✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ ❑❤→✐ ♥✐➺♠ s❛✉ ✤➙② tê♥❣ q✉→t ❦❤→✐ ♥✐➺♠ ❤å ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ❝ò♥❣ ♣❤➙♥ ♣❤è✐✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✹✳ tû ♥❣➝✉ ♥❤✐➯♥ X {Xi , i ∈ I} ❜à ❝❤♦✱ ✈ỵ✐ ♠å✐ t ❝❤➦♥ ♥❣➝✉ ♥❤✐➯♥ ❜ð✐ ♣❤➛♥ ❚❛ ♥â✐ ❤å tỷ C>0 tỗ t ởt ❤➡♥❣ sè P( Xi t) s❛♦ C P( X i I t) ỵ sû p✱ r ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ ♠➣♥ p ✈➔ r < p✱ {Xn , Fn , n 1} {Xn , n 1} ❜à ❝❤➦♥ ♥❣➝✉ ♥❤✐➯♥ ❧➔ ❞➣② ♣❤ị ❤đ♣ ♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ p✲❦❤↔ trì♥ E ✈➔ ❜ð✐ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ X ✳ ✣➦t Yni = Xi I( X n )✳ ❑❤✐ ✤â✱ ♥➳✉ 1/r i lim λ P( X > λ1/r ) = λ→∞ t❤➻ n P Xi − E(Yni |Fi−1 ) → n1/r ❈❤ù♥❣ ♠✐♥❤ ✳ {Xn , n 1} ❦❤✐ n → ∞ i=1 ❚❛ ❝➛♥ ❝❤➾ r❛ ❤❛✐ ✤✐➲✉ ❦✐➺♥ ✭✷✳✷✮ ✈➔ ✭✷✳✹✮ t❤ä❛ ♠➣♥ ✈ỵ✐ ❜à ❝❤➦♥ ♥❣➝✉ ♥❤✐➯♥ ❜ð✐ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ n X bn = n1/r ✳ ♥➯♥ t❛ ❝â n P( Xi > n1/r ) P( X > n1/r ) C i=1 i=1 1/r = C n P( X > n )→0 n → ∞, ❦❤✐ ❞♦ ✤â ✭✷✳✷✮ ✤ó♥❣✳ ❚✐➳♣ t❤❡♦✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ n −p/r p E Yni − E(Yni |Fi−1 ) n →0 ❦❤✐ n → ∞ i=1 ❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❏❡♥s❡♥ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ E Yni − E(Yni |Fi−1 ) p Cr ✱ t❛ ✤÷đ❝ E Yni + E(Yni |Fi−1 ) p E Yni + E( Yni |Fi−1 ) p 2p−1 (E Yni = 2p E Yni p p + E E( Yni p |Fi−1 )) ❉♦ ✤â n n−p/r E Yni − E(Yni |Fi−1 ) p i=1 n p −p/r n E Yni p i=1 n = 2p n−p/r E Xi i=1 n p I( Xi n1/r ) n = 2p n−p/r E Xi p I((k−1)1/r < Xi k1/r ) i=1 k=1 n n k p/r P (k − 1)1/r < Xi 2p n−p/r i=1 k=1 ✶✹ k 1/r ❚❤➟t ✈➟②✱ ✈➻ ❑➳t ❤ñ♣ ✤✐➲✉ ♥➔② ợ t ữủ n np/r E Yni − E(Yni |Fi−1 ) p i=1 n p p −p/r n r = C n−p/r n k lp/r−1 P (k − 1)1/r < Xi i=1 k=1 l=1 n n p/r−1 n P (k − 1)1/r < Xi l i=1 l=1 n n k 1/r k=l lp/r−1 P (l − 1)1/r < Xi = C n−p/r k 1/r n1/r i=1 l=1 n n lp/r−1 P Xi > (l − 1)1/r C n−p/r i=1 l=1 n n lp/r−1 P X > (l − 1)1/r −p/r Cn i=1 l=1 n −p/r p/r−1 =Cn n l P X > (l − 1)1/r l=1 n lp/r−2 l P( X > (l − 1)1/r ) = C n−p/r+1 l=1 ❚r♦♥❣ tr÷í♥❣ ❤đ♣ 0 (l − 1)1/r = l→∞ ♥➯♥ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ỵ t t ữủ n p/r E Yni − E(Yni |Fi−1 ) n p i=1 C n n l P( X > (l − 1)1/r ) → ❦❤✐ n → ∞ l=1 ❇➙② ❣✐í ❝❤ó♥❣ t❛ ①➨t t✐➳♣ tr÷í♥❣ ❤đ♣ p/2 < r < p✳ ❱➻ lim l P X > (l − 1)1/r = l→∞ ợ > tỗ t l0 N t❤ä❛ ♠➣♥ l P X > (l − 1)1/r < ε ✶✺ ✈ỵ✐ ♠å✐ l l0 ❉♦ ❉♦ ✤â n n lp/r−2 l P( X > (l − 1)1/r ) −p/r+1 l=1 l0 −1 lp/r−2 l P( X > (l − 1)1/r ) = n−p/r+1 l=1 n lp/r−2 l P( X > (l − 1)1/r ) + l=l0 n −p/r+1 Cn +n −p/r+1 lp/r−2 ε l=l0 ❇➡♥❣ ❝→❝❤ sû ❞ư♥❣ ❇ê ✤➲ ✷✳✷✳✸✭✐✐✮✱ t❛ ✤÷đ❝ n n−p/r+1 n−p/r+1 lp/r−2 ε l=l0 r np/r−1 − (l0 − 1)p/r−1 ε p−r r = pr ữ ợ n > l0 n lp/r−2 l P(|X > (l − 1)1/r ) → −p/r+1 n l=1 ❦❤✐ n → ∞✳ ✣à♥❤ ỵ ữủ ự t số ợ ố ✈ỵ✐ ❞➣② tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❘❛❞❡♠❛❝❤❡r ❞↕♥❣ p ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ s➩ tr➻♥❤ ❜➔② ✈➲ t✐➯✉ ❝❤✉➞♥ ❤ë✐ tư s✉② ❜✐➳♥ ✈➔ ❧✉➟t ②➳✉ sè ❧ỵ♥ ❑♦❧♠♦❣♦r♦✈✲❋❡❧❧❡r ✤è✐ ✈ỵ✐ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà tr➯♥ ổ p trỡ rữợ t t ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✶✳ p 2✮ ♥➳✉ ♥❣➝✉ ♥❤✐➯♥ E ✤÷đ❝ ổ r p tỗ t ởt số ữỡ C = Cp s ợ ♠å✐ ❞➣② {Xj , j 1} ❝→❝ ♣❤➛♥ tû ✤ë❝ ❧➟♣✱ ❝â ❝→❝ ❦ý ✈å♥❣ ❜➡♥❣ ✵✱ ❝â ❝→❝ ♠♦♠❡♥t ❜➟❝ p ❤ú✉ ❤↕♥✱ t❤➻ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ i i p Xj E E Xj p , C j=1 i j=1 ❚ø ✤à♥❤ ♥❣❤➽❛ tr➯♥ s✉② r❛ r➡♥❣ ♥➳✉ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❧➔ ❦❤æ♥❣ ❣✐❛♥ t❤➻ ♥â ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❘❛❞❡♠❛❝❤❡r ❞↕♥❣ p✲❦❤↔ trì♥ (1 p 2) p✳ tt tữỡ tỹ ữ trữợ ú t õ t ự ữủ ỵ s ỵ sỷ E ổ r p (1 ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ♥❤➟♥ ❣✐→ trà tr➯♥ E✱ {bn, n Xi I( X b ) ✳ ❑❤✐ ✤â i n n b−1 n P Xi → − i=1 ✶✻ ❦❤✐ n→∞ 1} p 2)✱ {Xn , n ❧➔ ❞➣② sè ❞÷ì♥❣✳ ✣➦t 1} ❧➔ Yni = ♥➳✉ n ❦❤✐ P( Xi > bn ) → n → ∞, i=1 n b−1 n ❦❤✐ EYni → n → ∞, i=1 n b−p n E Yni − EYni p →0 ❦❤✐ n → ∞ i=1 ❈❤ù♥❣ ♠✐♥❤ ✳ n ✣➦t n Sn = Xi , Snn = Yni ✱ i=1 ❦❤✐ ✤â✱ i=1 n Snn Sn P = bn bn n (Yni = Xi ) P P(Yni = Xi ) i=1 i=1 n P( Xi > bn ) → = ❦❤✐ n → ∞ i=1 ❉♦ ✤â✱ ✤➸ ❝❤ù♥❣ ♠✐♥❤ P b−1 − n Sn → ❦❤✐ ▼➦t ❦❤→❝✱ ❞♦ n → ∞✱ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ n b−1 n ❦❤✐ EYni → P b−1 − n Snn → ❦❤✐ n → ∞✳ n→∞ i=1 ♥➯♥ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ n b−1 n P Yni − EYni → − ❦❤✐ n → ∞ i=1 ◆❤➟♥ t❤➜②✱ ✈ỵ✐ ♠é✐ trà tr➯♥ E✳ n ❉♦ ✤â✱ ✈ỵ✐ ♠å✐ 1✱ {Yni − EYni , ε > 0✱ i n} ❧➔ ❞➣② ✤ë❝ ❧➟♣✱ ❦ý ✈å♥❣ ❦❤æ♥❣✱ ♥❤➟♥ ❣✐→ n b−1 n P Yni − EYni E εp bpn C εp bpn ❙✉② r❛ >ε i=1 n p Yni − EYni i=1 n E Yni − EYni p →0 ❦❤✐ n → ∞ i=1 n b−1 n P Yni − EYni → − ❦❤✐ n → ∞ i=1 ỵ ữủ ự ỵ sû E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❘❛❞❡♠❛❝❤❡r ❞↕♥❣ p (1 (r < p)✱ {Xn , n p 2)✱ r ❧➔ sè t❤ü❝ ❞÷ì♥❣ tr➯♥ E ✈➔ {Xn, n 1} ❜à ❧➔ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ♥❤➟♥ ❣✐→ trà ❝❤➦♥ ♥❣➝✉ ♥❤✐➯♥ ❜ð✐ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ X ✳ ✣➦t Yni = Xi I( X n )✳ ❑❤✐ ✤â✱ ♥➳✉ 1} i lim λ P X > λ1/r = λ→∞ ✶✼ 1/r t❤➻ n ❈❤ù♥❣ ♠✐♥❤ ✳ ❱➻ {Xn , n 1} ❦❤✐ P (Xi − EYni ) → n1/r n → ∞ i=1 ❜à ❝❤➦♥ ♥❣➝✉ ♥❤✐➯♥ ❜ð✐ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ n X n 1/r P( Xi > n ) P( X > n1/r ) C i=1 i=1 1/r = C n P( X > n )→0 n → ∞, ❦❤✐ ❚✐➳♣ t❤❡♦✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ n −p/r p E Yni − EYni n →0 ❦❤✐ n → ∞ i=1 ❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❏❡♥s❡♥ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ E Yni − EYni p Cr ✱ t❛ ✤÷đ❝ p E Yni + EYni p E Yni + E Yni 2p−1 (E Yni = 2p E Yni p p + E E( Yni p ❉♦ ✤â n n −p/r p E Yni − EYni i=1 n 2p n−p/r p E Yni i=1 n = 2p n−p/r E Xi i=1 n p =2 n p I( Xi n1/r ) n −p/r E Xi p I((k−1)1/r < Xi k1/r ) i=1 k=1 n n k p/r P (k − 1)1/r < Xi 2p n−p/r k 1/r i=1 k=1 ❑➳t ❤ñ♣ ✤✐➲✉ ♥➔② ợ t ữủ n n p/r E Yni − EYni p i=1 2p p −p/r n r −p/r =Cn n n k lp/r−1 P (k − 1)1/r < Xi i=1 k=1 l=1 n n p/r−1 n P (k − 1)1/r < Xi l i=1 l=1 n n k=l lp/r−1 P (l − 1)1/r < Xi = C n−p/r i=1 l=1 n n lp/r−1 P Xi > (l − 1)1/r C n−p/r k 1/r i=1 l=1 ✶✽ n1/r k 1/r ♥➯♥ t❛ ❝â n n lp/r−1 P X > (l − 1)1/r C n−p/r i=1 l=1 n −p/r p/r−1 =Cn n l P X > (l − 1)1/r l=1 n lp/r−2 l P( X > (l − 1)1/r ) = C n−p/r+1 l=1 ❚r♦♥❣ tr÷í♥❣ ❤đ♣ 0 (l − 1)1/r = l→∞ ♥➯♥ ❜➡♥❣ ❝→❝❤ sû ỵ t t ữủ n np/r p E Yni − EYni i=1 n C n l P( X > (l − 1)1/r ) → ❦❤✐ n → ∞ l=1 ❇➙② ❣✐í ❝❤ó♥❣ t❛ ①➨t t✐➳♣ tr÷í♥❣ ❤ñ♣ p/2 < r < p✳ ❱➻ lim l P X > (l − 1)1/r = l→∞ ♥➯♥ ✈ỵ✐ > N tỗ t l0 tọ l P X > (l − 1)1/r < ε ✈ỵ✐ ♠å✐ l l0 ❉♦ ✤â n n lp/r−2 l P( X > (l − 1)1/r ) −p/r+1 l=1 l0 −1 lp/r−2 l P( X > (l − 1)1/r ) = n−p/r+1 l=1 n lp/r−2 l P( X > (l − 1)1/r ) + l=l0 n −p/r+1 Cn +n −p/r+1 lp/r−2 ε l=l0 ❇➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❇ê ✤➲ ✷✳✷✳✸✭✐✐✮✱ t❛ ✤÷đ❝ n −p/r+1 n−p/r+1 lp/r−2 ε n l=l0 r np/r−1 − (l0 − 1)p/r−1 ε p−r r ε=ε p−r ◆❤÷ ✈➟② lp/r−2 l P(|X > (l − 1)1/r ) → n l=1 n → ∞✳ n > l0 n p/r+1 ợ ỵ ữủ ự ♠✐♥❤✳ ✶✾ ❉♦ ❑➌❚ ▲❯❾◆ ❑❤â❛ ❧✉➟♥ ✤➣ t❤✉ ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ s❛✉✿ ✲❚r➻♥❤ ❜➔② ✤÷đ❝ ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ①→❝ s✉➜t tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✲❚r➻♥❤ ❜➔② ✤÷đ❝ ✈➲ ❝→❝ ❞↕♥❣ ❤ë✐ tư ❝õ❛ ❞➣② ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✲❚r➻♥❤ ❜➔② ✤÷đ❝ ✈➲ ♠ët sè ❧✉➟t ②➳✉ ❝❤♦ ❞➣② ✤ë❝ ❧➟♣ ♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❘❛❞❡♠❛❝❤❡r ❞↕♥❤ ♣ ✭1 ≤ p ≤ 2✮✳ ✲❚r➻♥❤ ❜➔② ✤÷đ❝ ✈➲ ♠ët sè ❧✉➟t ②➳✉ ✤è✐ ✈ỵ✐ ❞➣② ♣❤ị ❤đ♣ ♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ♣✲ ❦❤↔ trì♥ ✭1 p ỳ ữợ ❝ù✉ t✐➳♣ t❤❡♦ ❚r♦♥❣ t❤í✐ ❣✐❛♥ tỵ✐ ❝❤ó♥❣ tỉ✐ ♠♦♥❣ ♠✉è♥ t✐➳♣ tö❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✈➜♥ ✤➲ s❛✉✿ ✲ ▲✉➟t ②➳✉ sè ❧ỵ♥ ✤è✐ ✈ỵ✐ ♠↔♥❣ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❘❛❞❡♠❛❝❤❡r ❞↕♥❣ ♣ ✭1 ≤ p ≤ 2✮✳ ✲ ▲✉➟t ②➳✉ sè ❧ỵ♥ ✤è✐ ✈ỵ✐ ♠↔♥❣ ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♣✲❦❤↔ trì♥ ✭1 ≤ p ≤ 2✮✳ ✷✵ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ◆❣✉②➵♥ ❱➠♥ ◗✉↔♥❣✱ ❳→❝ s✉➜t tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✱ ✷✵✶✷ Pr♦❜❛❜✐❧✐t② ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✳ ■s♦♣❡r✐♠❡tr② ❛♥❞ ♣r♦✲ ❝❡ss❡s✳ ❊r❣❡❜♥✐ss❡ ❞❡r ▼❛t❤❡♠❛t✐❦ ✉♥❞ ✐❤r❡r ●r❡♥③❣❡❜✐❡t❡ ✭✸✮✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✳ ❬✷❪ ▲❡❞♦✉① ▼✳ ❛♥❞ ❚❛❧❛❣r❛♥❞ ▼✳ ✭✶✾✾✶✮✱ ❬✸❪ ◗✉❛♥❣ ◆✳ ❱✳ ❛♥❞ ❍✉❛♥ ◆✳ ❱✳ ✭✷✵✵✾✮✱ ❖♥ t❤❡ str♦♥❣ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❛♥❞ ❢♦r ❞♦✉❜❧❡ ❛rr❛②s ♦❢ r❛♥❞♦♠ ❡❧❡♠❡♥ts ✐♥ ▲❡tt✳✱ ✼✾✭✶✽✮✱ ✶✽✾✶✲✶✽✾✾✳ Lp ✲❝♦♥✈❡r❣❡♥❝❡ p✲✉♥✐❢♦r♠❧② s♠♦♦t❤ ❇❛♥❛❝❤ s♣❛❝❡s✱ ❙t❛t✐st✳ Pr♦❜❛❜✳ ❬✹❪ ◗✉❛♥❣ ◆✳ ❱✳ ❛♥❞ ❍✉② ◆✳ ◆✳ ✭✷✵✵✽✮✱ ✏❲❡❛❦ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❢♦r ❛❞❛♣t❡❞ ❞♦✉❜❧❡ ❛rr❛②s ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✑✱ ❏✳ ❑♦r❡❛♥ ▼❛t❤✳ ❙♦❝✳✱ ✹✺✭✸✮✱ ✼✾✺✲✽✵✺✳ ❬✺❪ ◗✉❛♥❣ ◆✳ ❱✳ ❛♥❞ ❙♦♥ ▲✳ ❍✳ ✭✷✵✵✻✮✱ ✏❖♥ t❤❡ ✇❡❛❦ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❢♦r s❡q✉❡♥❝❡s ♦❢ ❇❛♥❛❝❤ s♣❛❝❡ ✈❛❧✉❡❞ r❛♥❞♦♠ ❡❧❡♠❡♥ts✑✱ ❇✉❧❧✳ ❑♦r❡❛♥ ▼❛t❤✳ ❙♦❝✳✱ ✹✸✭✸✮✱ ✺✺✶✲✺✺✽✳ ❬✻❪ ❘♦s❛❧s❦② ❆✳ ❛♥❞ ❚❤❛♥❤ ▲✳ ❱✳ ✭✷✵✵✻✮✱ ❙tr♦♥❣ ❛♥❞ ✇❡❛❦ ❧❛✇s ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❢♦r ❞♦✉❜❧❡ s✉♠s ♦❢ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ❡❧❡♠❡♥ts ✐♥ ❘❛❞❡♠❛❝❤❡r t②♣❡ ❆♣♣❧✳✱ ✷✹✭✻✮✱ ✶✵✾✼✲✶✶✶✼✳ p ❇❛♥❛❝❤ s♣❛❝❡s✱ ❬✼❪ ❱❛❦❤❛♥✐❛ ◆✳ ◆✳✱ ❚❛r✐❡❧❛❞③❡ ❱✳ ■✳ ❛♥❞ ❈❤♦❜❛♥②❛♥ ❙✳ ❆✳ ✭✶✾✽✼✮✱ t✐♦♥s ♦♥ ❇❛♥❛❝❤ s♣❛❝❡s✱ ❙t♦❝❤✳ ❆♥❛❧✳ Pr♦❜❛❜✐❧✐t② ❞✐str✐❜✉✲ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✱ ❉✳ ❘❡✐❞❡❧ P✉❜❧✐s❤✐♥❣ ❈♦✳✱ ❉♦r❞r❡❝❤t✳ ✷✶ ... ♥❣➝✉ ♥❤✐➯♥ ❜ð✐ ♣❤➛♥ ❚❛ ♥â✐ ❤å ❝→❝ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ C>0 ♥➳✉ tỗ t ởt số P( Xi t) s C P( X ✈➔ ♠å✐ i ∈ I✱ t) ✣à♥❤ ỵ sỷ p r số tỹ ❞÷ì♥❣ t❤ä❛ ♠➣♥ p ✈➔ r < p✱ {Xn , Fn , n 1} {Xn , n 1} ❜à ❝❤➦♥... ♥❤✐➯♥ ♥❤➟♥ ❣✐→ trà tr ổ p trỡ rữợ t t ♥❣❤➽❛ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✶✳ ❜➟❝ p E ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ tỗ t số ữỡ C = Cp ữủ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ p✲❦❤↔ trì♥ ✭1 s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ ❤✐➺✉ ♠❛rt✐♥❣❛❧❡ {Xj , Fj , j 1} p 2✮... ✤➲ ✷✳✷✳✸✳ ◆➳✉ p ❧➔ ♠ët sè t❤ü❝ (1 ✭✐✮ p r k p/r k ip/r−1 , i i = 1, 2, , k ✱ ✭✐✮ ❱➻ ✈➔ k ❧➔ ♠ët số ữỡ t ợ r (0, p), r k p/r−1 − (i0 − 1)p/r−1 , p−r p/r−2 i=i0 ❈❤ù♥❣ ♠✐♥❤ ✳ 2) i=1 k ✭✐✐✮ p r