▲✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ Ù♥❣ ❉ö♥❣ ▼❐❚ ❙➮ ❚➑◆❍ ❈❍❻❚ ❈❒ ❇❷◆ ❈Õ❆ ◗❯⑩ ❚❘➐◆❍ ▼❆❘❑❖❱ ❱⑨ ệ ữợ ữỡ ❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥✿ ✣é ❚❤➔♥❤ ❚➔✐ ▲ỵ♣✿ ❚♦→♥ Ù♥❣ ❉ö♥❣ ❑✸✷ ◆❣➔② ✷✹ t❤→♥❣ ✼ ♥➠♠ ✷✵✶✵ ✷ ▲❮■ ❈❷▼ ❒◆ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝→♠ ì♥ ❇❛♥ ❣✐→♠ ❤✐➺✉ tr÷í♥❣ ✣↕✐ ❤å❝ ❈➛♥ ❚❤ì✱ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❑❤♦❛ ❑❤♦❛ ❍å❝ ❚ü ◆❤✐➯♥✱ ❝→❝ t❤➛② ❝æ ❜ë ♠æ♥ t♦→♥ ỹ ú ù ữợ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❡♠ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ✣➦❝ ❜✐➺t✱ ❡♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝→♠ ì♥ ❚❤➛② ▲➙♠ ❍♦➔♥❣ ❈❤÷ì♥❣✱ ❜ë ♠ỉ♥ t♦→♥ ❑❤♦❛ ❑❤♦❛ ❍å❝ ❚ü ◆❤✐➯♥ ✲ ỡ ữớ trỹ t ữợ ❧✉➟♥ ✈➠♥✳ ❚❤➛② ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï✱ ✤ë♥❣ ✈✐➯♥✱ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ✤➸ ❡♠ ❤♦➔♥ t❤➔♥❤ ❜➔✐ ❧✉➟♥ ✈➠♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝→♠ ì♥ ❝è ✈➜♥ ❤å❝ t➟♣ ❈ỉ ❉÷ì♥❣ ❚❤à ❚✉②➲♥ ✤➣ ❞↕② ❞é✱ r ữợ sốt ố t ✤➦❝ ❜✐➺t tr♦♥❣ t❤í✐ ❣✐❛♥ ❡♠ ❧➔♠ ❜➔✐ ❧✉➟♥ ✈➠♥✳ ❊♠ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝→♠ ì♥ ✤➳♥ ❝→❝ ❜↕♥ ❧ỵ♣ ❚♦→♥ Ù♥❣ ❉ư♥❣ ❑✸✷ ✲ ❑❤♦❛ ❑❤♦❛ ❍å❝ ❚ü ◆❤✐➯♥ ✲ ✣↕✐ ❤å❝ ❈➛♥ ❚❤ì ✤➣ tr❛♦ ✤ê✐✱ ❣â♣ þ ❝❤♦ ❜➔✐ ❧✉➟♥ ✈➠♥✳ ❈✉è✐ ❝ò♥❣ ❡♠ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ✈➟t ❝❤➜t✱ t✐♥❤ t❤➛♥ tr♦♥❣ t❤í✐ ❣✐❛♥ ❡♠ ❤♦➔♥ t❤➔♥❤ ❜➔✐ ❧✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤✳ ❈➛♥ ❚❤ì✱ ♥❣➔② ✸✵ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✵ ộ P é ỵ ❝❤å♥ ✤➲ t➔✐ ✈➔ ♠ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ◗✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥ ❧➔ ♠æ ❤➻♥❤ t♦→♥ ❤å❝ ❝õ❛ r➜t ♥❤✐➲✉ ❜➔✐ t♦→♥ t❤ü❝ t✐➵♥ ①✉➜t ❤✐➺♥ tr♦♥❣ ❦❤♦❛ ❤å❝ ✈➔ ❝ỉ♥❣ ♥❣❤➺✳ ◆â ♠ỉ t↔ sü t✐➳♥ ❤â❛ t❤❡♦ t❤í✐ ❣✐❛♥ ❝õ❛ ♠ët ❤➺ t❤è♥❣ ❝❤à✉ sü t→❝ ✤ë♥❣ ❝õ❛ ❝→❝ ♥❤➙♥ tè ♥❣➝✉ ♥❤✐➯♥✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ❧ỵ♣ q✉→ tr q trồ ỵ tt ụ ữ ù♥❣ ❞ö♥❣ ✤â ❧➔ ✧◗✉→ tr➻♥❤ ▼❛r❦♦✈✧✳ ◗✉→ tr➻♥❤ ♥➔② t t ỵ t ữớ ◆❣❛ ❆✳❆✳▼❛r❦♦✈ ✭✶✹✴✻✴✶✽✺✻ ✲ ✷✵✴✼✴✶✾✷✷✮ ✤÷❛ r❛ ✈➔♦ ✤➛✉ t❤➳ ❦➾ ❳❳ ✤➸ ♠æ t↔ ❝❤✉②➸♥ ✤ë♥❣ ❝õ❛ ❝→❝ ♣❤➙♥ tû ❝❤➜t ❧ä♥❣ tr♦♥❣ ♠ët ❜➻♥❤ ❦➼♥✳ ❱➲ s❛✉ ♠æ ❤➻♥❤ ✤÷đ❝ ♣❤→t tr✐➸♥ ✈➔ sû ❞ư♥❣ ♥❤✐➲✉ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❝ì ❤å❝✱ ② ❤å❝✱ s✐♥❤ ❤å❝✱ ❦✐♥❤ t➳ ✳ ✳ ✳ ✣➸ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ♠ët q✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥ ♥â✐ ❝❤✉♥❣ ❤❛② ♠ët q✉→ tr➻♥❤ ▼❛r❦♦✈ ♥â✐ r✐➯♥❣✱ ♥❣♦➔✐ ✈✐➺❝ ❦❤↔♦ s→t ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ú t ởt ọ ợ ổ ữủ t r❛ ✤â ❧➔✿ ▲✐➺✉ q✉→ tr➻♥❤ ♥➔② ❝â ❤ë✐ tö ②➳✉ ✭t❤❡♦ ♣❤➙♥ ♣❤è✐✮ ✈➲ ♠ët ✤↕✐ ❧÷đ♥❣ ♥❣➝✉ ♥❤✐➯♥ ♥➔♦ ✤â ❤❛② ❦❤æ♥❣❄ ◆➳✉ ❦❤æ♥❣ t❤➻ ❝➛♥ ♣❤↔✐ t❤❛② ✤ê✐ ❤❛② t❤➯♠ ❜ỵt ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ ❣➻❄ ✣â ❝ơ♥❣ ❝❤➼♥❤ ❧➔ ✈➜♥ ✤➲ ♠➔ ❧✉➟♥ ✈➠♥ ♥➔② ✤÷❛ r❛ ✈➔ t➻♠ ❝→❝❤ ❣✐↔✐ q✉②➳t✳ ■■✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ự rữợ t ữ r ♥❣❤➽❛ ✧◗✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥✧✱ ❞ü❛ tr➯♥ ❝ì sð ✤â ✤÷❛ r❛ ✤à♥❤ ♥❣❤➽❛ ✧◗✉→ tr➻♥❤ ▼❛r❦♦✈✧✱ ❝ư t❤➸✿ ✰ ❳➼❝❤ ▼❛r❦♦✈ t÷ì♥❣ ù♥❣ q✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥ rí✐ r↕❝ t❤❡♦ t❤í✐ ❣✐❛♥✳ ✰ ◗✉→ tr➻♥❤ ▼❛r❦♦✈ t÷ì♥❣ ù♥❣ q✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥ ❧✐➯♥ tư❝ t❤❡♦ t❤í✐ ❣✐❛♥✳ ❚✐➳♣ t❤❡♦✱ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥✱ ❝➛♥ tt tổ q ỵ ✳ ✮ ✤➸ ù♥❣ ❞ö♥❣ ✈➔♦ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❤❛✐ ỵ r s ỵ r P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ✣å❝ ❝→❝ t➔✐ ❧✐➺✉ ❧✐➯♥ q✉❛♥ ✈➔ ❝→❝ s→❝❤ t❤❛♠ ❦❤↔♦ ❝❤✉②➯♥ ♥❣❤➔♥❤✱ ❦➳t ❤đ♣ ✈ỵ✐ tổ t tứ trt ữợ sỹ ữợ ❚❤➛② ▲➙♠ ❍♦➔♥❣ ❈❤÷ì♥❣✱ ❡♠ ✤➣ ❧ü❛ ❝❤å♥ ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ q✉❛♥ trå♥❣✱ ♥❤✐➲✉ ù♥❣ ❞ư♥❣ ✤➸ tr➻♥❤ tr ỗ tớ ữ r ự ợ tự ữ r❛✳ ✷ ■❱✳ ❈➜✉ tró❝ ❝õ❛ ❧✉➟♥ ✈➠♥ ▲✉➟♥ ✈➠♥ ỗ ữỡ ữỡ tr ✈➔ ①➼❝❤ ▼❛r❦♦✈ ❚r➻♥❤ ❜➔② ❝→❝ ✤à♥❤ ♥❣❤➽❛ ❝ì ❜↔♥ ✈➲ q✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥✱ ①➼❝❤ ▼❛r❦♦✈ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝õ❛ ❝❤ó♥❣✳ ❈❤÷ì♥❣ ♥➔② ❝á♥ ♠ët số ỵ t ❝❤÷ì♥❣ s❛✉✿ ♣❤÷ì♥❣ tr➻♥❤ ❈❤❛♣♠❛♥ ✲ ❑♦❧♠♦❣♦r♦✈✱ ♣❤➙♥ ❜è ❜❛♥ ✤➛✉✱ ♣❤➙♥ ❜è ❞ø♥❣✱ ♣❤➙♥ ❜è ❣✐ỵ✐ ❤↕♥✱ ♣❤➙♥ ❧♦↕✐ tr t r ữỡ ◗✉→ tr➻♥❤ ▼❛r❦♦✈ ❚r➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ q✉→ tr➻♥❤ ▼❛r❦♦✈✳ ỵ t t q tr➻♥❤ tr♦♥❣ ❜❛ tr÷í♥❣ ❤đ♣✿ ❦❤ỉ♥❣ ❣✐❛♥ tr↕♥❣ t❤→✐ ✤➳♠ ✤÷đ❝✱ ❦❤ỉ♥❣ ❣✐❛♥ tr↕♥❣ t❤→✐ ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝ ✈➔ trữớ ủ tờ qt ữỡ ởt số ự ❞ö♥❣ ❝❤♦ q✉→ tr➻♥❤ ▼❛r❦♦✈ ❚r➻♥❤ ❜➔② ❤❛✐ ù♥❣ ❞ö♥❣ ❝õ❛ q✉→ tr➻♥❤ ▼❛r❦♦✈✳ ❈ö t❤➸ ❝❤ù♥❣ ♠✐♥❤ ♠ët ♣❤➛♥ ỵ r ỵ ợ tr t➙♠ ✭●♦r❞✐♥✮✳ ✣é ❚❤➔♥❤ ❚➔✐ ✷ ▼ư❝ ❧ư❝ ❈❤÷ì♥❣ ✶✳ ◗✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥ ✈➔ ❳➼❝❤ ▼❛r❦♦✈ ✶✳✶✳ ✶✳✷✳ ✶✳✸✳ ✶✳✹✳ ✶✳✺✳ ✶✳✻✳ ◗✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❳➼❝❤ ▼❛r❦♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ P❤➙♥ ❜è ❜❛♥ ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ P❤÷ì♥❣ tr➻♥❤ ❈❤❛♣♠❛♥ ✲ ❑♦❧♠♦❣♦r♦✈ ✳ P❤➙♥ ❜è ❞ø♥❣ ✈➔ ♣❤➙♥ ❜è ❣✐ỵ✐ ❤↕♥ ✳ ✳ P❤➙♥ ❧♦↕✐ tr↕♥❣ t❤→✐ ①➼❝❤ ▼❛r❦♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤÷ì♥❣ ✷✳ ◗✉→ tr➻♥❤ ▼❛r❦♦✈ ✷✳✶✳ ✷✳✷✳ ✷✳✸✳ ✷✳✹✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚r÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ ❣✐❛♥ tr↕♥❣ t❤→✐ ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ❚r÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ ❣✐❛♥ tr↕♥❣ t❤→✐ ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝ ❚r÷í♥❣ ❤đ♣ tê♥❣ q✉→t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝❤♦ q✉→ tr➻♥❤ ▼❛r❦♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✳ ✺ ✳ ✺ ✳ ✻ ✳ ✼ ✳ ✶✶ ✳ ✶✼ ✳ ✳ ✳ ỵ ❣✐ỵ✐ ❤↕♥ tr✉♥❣ t➙♠ ❝❤♦ ①➼❝❤ ▼❛r❦♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ❍♦♣❢ ❊r❣♦❞✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸ ✸ ✹ ▼ư❝ ❧ư❝ ✹✾ ✷✳✹✳ ❚r÷í♥❣ ❤đ♣ tê♥❣ q✉→t ✂ −1 (u − x)2 P (t, x, du) = b(x) lim t t→0+ B(x, ) ❑❤✐ ✤â ❤➔♠ φ(t, x) ①→❝ ✤à♥❤ ❜ð✐ ✷✳✶✾ t❤ä❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ s❛✉✿ φ φ b(x) φ = a(x) + t x x2 ✭✷✳✷✵✮ lim φ(t, x) = ξ(x) ✭✷✳✷✶✮ ợ t0+ ữủ trữợ ❤❛✐ ❤➔♠ a(x)✱ b(x)✳ ❑❤✐ ✤â ✈ỵ✐ ♠ët sè ❣✐↔ t❤✐➳t ✈➲ ❤❛✐ ❤➔♠ a(x)✱ b(x) ✈ỵ✐ ♠é✐ ❤➔♠ ❧✐➯♥ tử ữỡ tr ợ ❜❛♥ ✤➛✉ ✷✳✷✶ s➩ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t φ(t, x) ✈➔ ❞♦ ✤â ①→❝ ✤à♥❤ ❝❤♦ t❛ ✤ë ✤♦ P (t, x, )✳ ❍å ✤ë ✤♦ ♥➔② ❧➟♣ t❤➔♥❤ ♠ët ❤å ①→❝ s✉➜t ❝❤✉②➸♥✳ ❱➔ q✉→ tr➻♥❤ ▼❛r❦♦✈ t÷ì♥❣ ù♥❣ ✈ỵ✐ ❤å ①→❝ s✉➜t ❝❤✉②➸♥ ♥➔② ❝â q✉ÿ ✤↕♦ ❧✐➯♥ tư❝✳ ✺✵ ❈❤÷ì♥❣ ✷✳ ◗✉→ tr➻♥❤ ▼❛r❦♦✈ ❈❤÷ì♥❣ ✸ ▼ët số ự q tr r ỵ ❣✐ỵ✐ ❤↕♥ tr✉♥❣ t➙♠ ❝❤♦ ①➼❝❤ ▼❛r❦♦✈ ●✐↔ sû (Xn)n≥1 ❧➔ ♠ët ❞➣② ❝→❝ ✤↕✐ ❧÷đ♥❣ ♥❣➝✉ ♥❤✐➯♥ ✈➔ (Fn)n≥1 ❧➔ ♠ët ❜ë ❧å❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ①→❝ s✉➜t (Ω, F, µ)✳ ●✐↔ sû t❤➯♠ r➡♥❣ tê♥❣ r✐➯♥❣ ❝õ❛ Xn ①→❝ ✤à♥❤ ♠ët q✉→ tr➻♥❤ ▼❛rt✐♥❣❛❧❡✿ Xn ❧➔ ❤➔♠ ✤♦ ữủ tữỡ ự ợ Fn E(Xn /Fn1 ) = ✈ỵ✐ ♠å✐ n ≥ 1✳ ✣➦t k E(Xk2 ) s2n = k=1 ỵ r sỷ ❣✐ỵ✐ ❤↕♥ s❛✉ ❤ë✐ tư t❤❡♦ ①→❝ s✉➜t n E(Xk2 /Fk −1 ) = ✭✸✳✶✮ E(Xk2 ✶|Xk |>δsn /Fk −1 ) = 0, ∀δ > ✭✸✳✷✮ n→∞ s2 n lim lim n→∞ sn ❑❤✐ ✤â✱ sn n Xk k=1 k=1 n k=1 s➩ ❤ë✐ tö ②➳✉ ✭t❤❡♦ ♣❤➙♥ ♣❤è✐✮ ✤➳♥ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ t➢❝ N (0, 1) ú t sỷ ỵ ự ỵ t t (Xn)n ❧➔ ♠ët ①➼❝❤ ▼❛r❦♦✈✳ ●✐↔ sû (Xn)n ❝â ♣❤➙♥ ♣❤è✐ ❞ø♥❣ ✈➔ ❣å✐ µ ❧➔ ✤ë ✤♦ ①→❝ s✉➜t tr X0 ợ f L1(à) t♦→♥ tû ♥❤÷ s❛✉✿ P f (Xn−1 ) = E(f (Xn )|Xn1 ), n à.p.p ỵ r s ợ f L (à) t g = P f − f ✱ t❛ ❝❤➾ r❛ r➡♥❣✿ √ n tr♦♥❣ ✤â σg2 = ✁ ✁ ∞ g(Xk ) → N (0, σg2 ) k=1 f dµ − (P f ) dµ ✺✶ ✺✷ ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝❤♦ q✉→ tr➻♥❤ ▼❛r❦♦✈ ự rữợ t t t r g(Xk ) = P f (Xk ) − f (Xk ) = P f (Xk ) − f (Xk+1 ) + f (Xk+1 ) − f (Xk ) n ⇒ ✭✸✳✸✮ n (P f (Xk ) − f (Xk+1 )) + f (Xn+1 ) − f (X1 ) g(Xk ) = k=1 ⇒√ n k=1 n g(Xk ) = √ n k=1 n Mk+1 + √ (f (Xn+1 ) − f (X1 )) n k=1 ✈ỵ✐ Mk+1 = P f (Xk ) − f (Xk+1) ✣➦t Fn = σ(X0, X1, , Xn) t❛ s➩ ❝❤➾ r❛ r➡♥❣ ♠ët ♣❤➛♥ ❝õ❛ tê♥❣ Mn+1 ❧➔ ♠ët q✉→ tr➻♥❤ ▼❛rt✐♥❣❛❧❡ tọ ỵ r ❝á♥ ❧↕✐ √1n (f (Xn+1) − f (X1)) t❤➻ tr✐➺t t✐➯✉✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② ❝á♥ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ▼❛rt✐♥❣❛❧❡✳ n−1 ✣➦t Sn+1 = Mk+1✳ ❚❛ ❝â✿ k=0 Sn+1 = Sn + P f (Xn ) − f (Xn+1 ) ⇒ E(Sn+1 |Fn ) = Sn + P f (Xn ) − E(f (Xn+1 )|Fn ) = Sn ✭✸✳✹✮ ❱➔✿ E(Mk+1 |Fk ) = P f (Xk ) − E(f (Xk+1 )|Fk ) = (k ❚✐➳♣ t❤❡♦✱ t❛ ❝❤ù♥❣ ♠✐♥❤✿ lim n→∞ s2 n lim n→∞ s2 n tr♦♥❣ ✤â s2n = E(Mk+1 ) k=0 ❚❛ ❝â✿ n−1 n−1 E(Mk+1 |Fk ) = k=0 n−1 E(Mk+1 k=0 ✶ |Mk+1 |>δsn σg |Fk ) =0 1) ỵ ợ tr t➙♠ ❝❤♦ ①➼❝❤ ▼❛r❦♦✈ ✺✸ • n−1 s2n = ) E(Mk+1 k=0 n−1 E(Mk+1 |Fk )) = E( k=0 n−1 E(P (f (Xk )) − (P f )2 (Xk )) = k=0 n−1 ✂ (P f − (P f )2 )dµ = k=0 ✂ = n dà n1 s2n = E(Mk+1 ) k=0 n−1 E(Mk+1 |Fk )) = E( k=0 n−1 E(P (f (Xk )) − (P f )2 (Xk )) = k=0 n−1 ✂ (P f − (P f )2 )dà = k=0 = n dà E(Mk+1 |Fk ) = = = = = = = E((P f (Xk ) − f (Xk+1 ))2 |Fk ) E((P f (Xk ) − f (Xk+1 ))2 |Xk ) E[((P f )2 (Xk ) − 2P f (Xk )f (Xk+1 ) + f (Xk+1 ))|Xk ] (P f )2 (Xk ) − 2P f (Xk )E(f (Xk+1 )|Xk ) + E(f (Xk+1 )|Xk ) (P f )2 (Xk ) − 2(P f )2 (Xk )E(f (Xk+1 )|Xk ) + P (f )(Xk ) P (f )2 (Xk ) − (P f )2 (Xk ) ϕ(Xk ) tr♦♥❣ ✤â ϕ = P (f 2) − (P f )2 ❚❤❡♦ ❧✉➟t sè ❧ỵ♥ (Yn = ϕ(Xn))n t❛ ❝â✿ ✭✸✳✼✮ ✺✹ ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝❤♦ q✉→ tr➻♥❤ ▼❛r❦♦✈ L − lim n→∞ n n−1 n−1 E(Mk+1 k=0 |Fk ) = lim ϕ(Xk ) n→∞ n k=0 = E(ϕ(X )) ✂ = ✭✸✳✽✮ ϕdµ ◆➯♥✿ n−1 ϕ(Xk ) n k=0 E(Mk+1 |Fk ) = lim =1 lim n→∞ n→∞ s2 n k=0 s n n n−1 • ❈è ✤à♥❤ M > 0✱ ✤➦t ϕM (x) = Ex((P (f 2) − (P f )2)2✶|P (f )−(P f ) |>δM σ ) ∀δ > 0✳ ❚❛ ❝â✿ lim n→∞ n n−1 E(Mk+1 k=0 ✶ g n−1 ϕM (Xk ) lim |Mk+1 >δM |Fk ) = n→∞ n k=0 = E(ϕ ✂ M (X0 )) = (P (f ) − (P f )2 )2 ✶ ✭✸✳✾✮ |P (f )−(P f )2 |>δM σg dµ ❱➻ n→∞ lim Sn = +∞, ∃N > s❛♦ ❝❤♦ ∀n > N t❤➻ sn > M ❱➔ ❞♦ ✤â✿ s2n n−1 E(Mk+1 ✶ |Mk+1 |>δsn σg |Fk ) ≤ k=0 s2n n−1 E(Mk+1 ✶ |Mk+1 |>δM σg |Fk ), ∀n > N k=0 ❈❤♦ n → ∞ t❛ ✤÷đ❝✿ L − lim n→∞ sn n−1 E(Mk+1 k=0 ✶ limn→∞ |Mk+1 |>δsn σg |Fk ) n−1 E(Mk+1 n k=0 ✁ ϕM dµ = ✁ →0 limn→∞ ϕdµ ❚✐➳♣ t❤❡♦✱ t❛ t❤➜②✿ √ (f (Xn+1 ) − f (X1 )) → n ❦❤✐ n→∞ ✶ |Mk+1 |>δM σg |Fk ) s2n n M → +∞ ỵ r ố ũ t ✤÷đ❝✿ √ n tr♦♥❣ ✤â σ2(g) = f2 L2 (µ) n−1 g(Xk ) → N (0; σ (g)) k=0 (P f )2 L2 (à) ỵ ❍♦♣❢ ❊r❣♦❞✐❝ ❈❤♦ (Xn)n ❧➔ ♠ët ①➼❝❤ ▼❛r❦♦✈✳ ●✐↔ sû (Xn)n ❝â ♣❤➙♥ ♣❤è✐ ❞ø♥❣ ✈➔ ❣å✐ µ ❧➔ ✤ë ✤♦ ①→❝ s✉➜t tr➯♥ X0✳ ❱ỵ✐ ♠å✐ ❤➔♠ f L1(à) t tỷ ữ s P f (Xn−1 ) = E(f (Xn )|Xn−1 ) ✣➦t µ.p.p k P if Sk f = i=0 Sn∗ f = max0 k n Sk f ỵ ợ f ∈ L (µ)✱ t❛ ❝â✿ ✂ f dµ S ∗ f >0 ✈ỵ✐ S ∗f = supk Sk f ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ a ∈ R✱ ✤➦t a+ = max {a, 0} ❱ỵ✐ ♠å✐ a, b ∈ R ⇒ max {a, a + b} = a + max {b, 0} = a + b+ ❚❛ ❝â✿ ∗ Sn+1 f = max0 k n+1 Sk f n+1 P if = max f, f + P f, , f + i=1 = f + max {0, max0 k n P (Sk f )} = f + (max0 k n P (Sk f ))+ ❈❤♦ ❤❛✐ ❤➔♠ g✱ h ∈ L1(µ) P (max {g, h}) = P (g + (h − g)+ ) = P (g) + P ((h − g)+ ) ✭✸✳✶✶✮ ✺✻ ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝❤♦ q✉→ tr➻♥❤ ▼❛r❦♦✈ ✭✸✳✶✷✮ P (g) ❚÷ì♥❣ tü P (max {g, h}) ❙✉② r❛ P (max {g, h}) P (h) max {P (g), P (h)} ❱➔✿ P (g + ) = P (max(0, g)) ❱➟② P (g+) ❚❛ ❝ô♥❣ ❝â✿ max {P (0), P (g)} = max {0, P (g)} = (P (g))+ ✭✸✳✶✸✮ (P (g))+ ∗ Sn+1 f = f + (max0 k n P (Sk f ))+ = f + max {P (0), maxk n P (Sk f )} f + P (max {0, maxk n Sk f }) = f + P max {0, Sn∗ f } = f + P ((Sn∗ f )+ ) ✭✸✳✶✹✮ ∗ ❱➟② Sn+1 f f + P ((Sn∗ f )+ ) ✣➦t✿ En = (Sn∗ f > 0)✳ ❚❛ ❝â✿ Sn∗ f ∗ f + P ((Sn−1 f )+ ) f + P ((Sn∗ f )+ ) ❙✉② r❛✿ ✂ ✂ Sn∗ f dµ ✂ ✂En En ✂En ✂En = f dµ + f dµ + f dµ + ✂En ✂R ✂R P ((Sn∗ f )+ )dµ (Sn∗ f )+ dµ Sn∗ f dµ f dµ + En P ((Sn∗ )+ )dµ En ✂ ⇔ ❈❤♦ n → ∞ f dµ ∀n En ✂ f dµ S ∗ f >0 ✈ỵ✐ S ∗ f = supk Sk f ỵ r ❍➺ q✉↔ ✸✳✶✳ ✁ (M ∗ f >α) f dµ ợ {M f > } ự ỵ Sk f | k+1 f dµ − αµ {(M ∗ (f − α) (f − α)dµ = (M ∗ (f −α)>0) ❙✉② r❛✿ M ∗ f = supk | 0} (M ∗ (f −α)>0) ✂ αµ {M ∗ f > α} f dµ (M ∗ f >α) ✣à♥❤ ♥❣❤➽❛ ✸✳✶✳ ❳➼❝❤ ▼❛r❦♦✈ ✤÷đ❝ ❣å✐ ❧➔ ❊r❣♦❞✐❝ ♥➳✉ P h = h✱ ❦❤✐ ✤â h ỵ r (X ) ❧➔ ♠ët ❊r❣♦❞✐❝ t❤➻ ✈ỵ✐ ♠å✐ g ∈ L (µ) ∞ n n 1 lim Sk g = k→∞ k + ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t✿ Mk g = ✂ gdµ µ.a.s Sk g k+1 h = lim infk Mk g rữợ t t t tr g ❚❤❡♦ ❜ê ✤➲ ❋❛t♦✉ ✈ỵ✐ ♠å✐ l > 0✿ P (infl k Mk g) infl k Mk P g ❈❤♦ l → ∞✱ t❛ ❝â✿ Ph lim infl 0} = ✈ỵ✐ M f = limk→∞ Mk f àa.s àa.s ữỡ ởt số ù♥❣ ❞ö♥❣ ❝❤♦ q✉→ tr➻♥❤ ▼❛r❦♦✈ ❈❤ù♥❣ ♠✐♥❤✳ ∀ >0 tỗ t g L(à) s gf t ❝â✿ M f = M g + M (f − g) ∗ ⇒ |M f | |M ✂ g| + M (f − g) | gdµ| + M ∗ (f − g) + M ∗ (f − g) ∀a > t ỵ {|M f | > a} µ {M ∗ |f − g| > a − } g−f a− a− ❈❤♦ → s✉② r❛ µ {|M f | > a} = ❈❤♦ a → s✉② r❛ µ {|M f | > 0} = ✭✸✳✷✹✮ ❑➌❚ ▲❯❾◆ ❱⑨ ✣➋ ◆●❍➚ ◗✉→ tr➻♥❤ ▼❛r❦♦✈ ❧➔ ♠ët ❞↕♥❣ q✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥ r➜t q✉❛♥ trå♥❣✱ ♥â ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ♥❤✐➲✉ ❤✐➺♥ t÷đ♥❣ t ỵ tr số ú t ữợ qt ữủ ỳ t t ỡ ❜↔♥ ❝õ❛ ♠ët q✉→ tr➻♥❤ ✭①➼❝❤✮ ▼❛r❦♦✈✱ tø ✤â ❣✐ó♣ ❝❤♦ ♥❣÷í✐ ✤å❝ ❝â ❝→✐ ♥❤➻♥ t♦➔♥ ❞✐➺♥ ❤ì♥ ✈➲ ♠ët q✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥✱ ✤➦❝ ❜✐➺t ❧➔ q✉→ tr➻♥❤ ▼❛r❦♦✈✳ ❱➲ ♣❤➛♥ ù♥❣ ❞ư♥❣✱ ❧✉➟♥ ✈➠♥ ✤÷❛ r❛ ❤❛✐ ự rt q trồ õ ỵ r s ỵ r ỵ ●♦r❞✐♥ ▲✐❢s✐❝ ♥â✐ ✈➲ sü ❤ë✐ tö ②➳✉ ❝õ❛ ♠ët ❞➣② ❤➔♠ ❝õ❛ ♠ët q✉→ tr➻♥❤ ▼❛r❦♦✈ ✈➲ ✤↕✐ ❧÷đ♥❣ ♥❣➝✉ ♥❤✐➯♥ ❝â ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥✳ ✣➙② ✤÷đ❝ ①❡♠ ❧➔ ỵ ỡ t õ ữợ rt ỵ ❣✐ỵ✐ ❤↕♥ tr✉♥❣ t➙♠ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ❞➣② ❤➔♠ ởt q tr r õ ỵ ❊r❣♦❞✐❝ ✤÷đ❝ sû ❞ư♥❣ ❝❤♦ ❝❤ù♥❣ ♠✐♥❤ ❞➣② ❤➔♠ ❝õ❛ ♠ët q✉→ tr➻♥❤ ▼❛r❦♦✈ ①✉➜t ♣❤→t tø ♠ët ✤✐➸♠ (X0 = x)✳ ✣➙② ❝ô♥❣ ❝❤➼♥❤ ❧➔ ✈➜♥ ✤➲ ❝➛♥ ♠ð rë♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥➔②✳ ✷ ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ö♥❣ ❝❤♦ q✉→ tr➻♥❤ ▼❛r❦♦✈ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ t tr ỵ tt q tr ♥❣➝✉ ♥❤✐➯♥✱ ◆❣÷í✐ ❞à❝❤ ✿ ◆❣✉②➵♥ ❱✐➳t P❤ó ✲ ◆❣✉②➵♥ ❉✉② ❚✐➳♥✱ ◆❳❇ ✣↕✐ ❤å❝ ✈➔ ❚r✉♥❣ ❤å❝ ❈❤✉②➯♥ ♥❣❤✐➺♣ ❍➔ ◆ë✐ ♥➠♠ ✶✾✽✼✳ ❬✷❪ ❉÷ì♥❣ ❚❤à ❚✉②➲♥✱ ❇➔✐ ❣✐↔♥❣ ❈❤✉②➯♥ ✤➲ ①→❝ s✉➜t t❤è♥❣ ❦➯✱ ✣↕✐ ❤å❝ ❈➛♥ ❚❤ì✱ ✷✵✵✾✳ ❬✸❪ ❉÷ì♥❣ ❚❤à ❚✉②➲♥✱ ❇➔✐ ❣✐↔♥❣ ❳→❝ s✉➜t ♥➙♥❣ ❝❛♦✱ ✣↕✐ ❤å❝ ❈➛♥ ❚❤ì✱ ✷✵✵✾✳ ❬✹❪ ✣➦♥❣ ❍ò♥❣ ❚❤➢♥❣✱ ◗✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥ ✈➔ t➼♥❤ t♦→♥ ♥❣➝✉ ♥❤✐➯♥✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐ ♥➠♠ ✷✵✵✻✳ ❬✺❪ ▲➙♠ ❍♦➔♥❣ ❈❤÷ì♥❣✱ ❈❤✉②➯♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ s✐♥❤✱ ✷✵✵✾✳ ❬✻❪ ▲➙♠ ❍♦➔♥❣ ❈❤÷ì♥❣✱ ❇➔✐ ❣✐↔♥❣ ◗✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥✱ ✣↕✐ ❤å❝ ❈➛♥ ❚❤ì✱ ✷✵✵✾✳ ❬✼❪ ◆❣✉②➵♥ ❉✉② ❚✐➳♥✱ ❈→❝ ♠ỉ ❤➻♥❤ ①→❝ s✉➜t ✈➔ ù♥❣ ❞ö♥❣✱ P❤➛♥ ✶✿ ❳➼❝❤ ▼❛r❦♦✈ ✈➔ ù♥❣ ❞ö♥❣✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐ ♥➠♠ ụ t ỵ t❤✉②➳t ①→❝ s✉➜t✱ ◆❳❇ ●✐→♦ ❞ö❝ ♥➠♠ ✷✵✵✵✳ ❬✾❪ ❈❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠ ❢♦r st❛t✐♦♥❛r② ▼❛r❦♦✈ ♣r♦❝❡ss❡s✱ ●♦r❞✐♥✱ ▲✐❢s✐❝ ❇❆✱ ❉♦❦❧ ❆❦❛❞ ◆❛✉❦ ❙❙❙❘ ✷✸✾ ✭✹✮✿ ✼✻✻✲✼✻✼✱ ✶✾✼✽✳ ❬✶✵❪ ▼❛rt✐♥❣❛❧❡ ❈❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✱ ❇✳▼✳ ❇r♦✇♥✱ ❚❤❡ ❛♥♥❛❧s ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❙t❛t✐st✐❝s✱ ❱♦❧ ✹✷✱ ◆♦ ✶✱ ✺✾✲✻✻✱ ✶✾✼✶✳ ✸ ... ♥❤✐➺♠ ❑❤♦❛ ❑❤♦❛ ❍å❝ ❚ü ◆❤✐➯♥✱ ❝→❝ t❤➛② ❝æ ❜ë ♠ỉ♥ t♦→♥ ❑❤♦❛ ❑❤♦❛ ❍å❝ ❚ü ◆❤✐➯♥ ✤➣ ❣✐ó♣ ✤ï✱ ữợ tr sốt tớ t t tr÷í♥❣✳ ✣➦❝ ❜✐➺t✱ ❡♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝→♠ ì♥ ❚❤➛② ▲➙♠ ❍♦➔♥❣ ❈❤÷ì♥❣✱ ❜ë ♠ỉ♥ t♦→♥ ❑❤♦❛... ❤♦➔♥ t❤➔♥❤ ❜➔✐ ❧✉➟♥ ✈➠♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝→♠ ì♥ ❝è ✈➜♥ ❤å❝ t➟♣ ❈ỉ ❉÷ì♥❣ ❚❤à ❚✉②➲♥ ✤➣ ộ r ữợ sốt ố t➟♣ ✈➔ ✤➦❝ ❜✐➺t tr♦♥❣ t❤í✐ ❣✐❛♥ ❡♠ ❧➔♠ ❜➔✐ ❧✉➟♥ ✈➠♥✳ ❊♠ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝→♠ ì♥ ✤➳♥ ❝→❝ ❜↕♥... ✈➲ q✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥✱ ①➼❝❤ ▼❛r❦♦✈ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝õ❛ ❝❤ó♥❣✳ ❈❤÷ì♥❣ ♥➔② ỏ ởt số ỵ t ❝❤♦ ❝→❝ ❝❤÷ì♥❣ s❛✉✿ ♣❤÷ì♥❣ tr➻♥❤ ❈❤❛♣♠❛♥ ✲ ❑♦❧♠♦❣♦r♦✈✱ ♣❤➙♥ ❜è ❜❛♥ ✤➛✉✱ ♣❤➙♥ ❜è ❞ø♥❣✱