Xấp xỉ chuẩn bằng phương pháp stein cho cặp hoán đổi được và một số ứng dụng

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ◆●❯❨➍◆ ❈❍➓ ❉Ơ◆● ❳❻P ❳➓ ❈❍❯❽◆ ❇➀◆● P❍×❒◆● P❍⑩P ❙❚❊■◆ ❈❍❖ ❈➄P ❍❖⑩◆ ✣✃■ ✣×Đ❈ ❱⑨ ▼❐❚ ❙➮ Ù◆● ❉Ư◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆❣❤➺ ❆♥ ✲ ✷✵✶✾ ❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ◆●❯❨➍◆ ❈❍➓ ❉Ơ◆● ❳❻P ❳➓ ❈❍❯❽◆ ❇➀◆● P❍×❒◆● P❍⑩P ❙❚❊■◆ ❈❍❖ ❈➄P ❍❖⑩◆ ✣✃■ ✣×Đ❈ ❱⑨ ▼❐❚ ❙➮ Ù◆● ❉Ư◆● ❈❤✉②➯♥ ♥❣➔♥❤ ✿ ▲Þ ❚❍❯❨➌❚ ❳⑩❈ ❙❯❻❚ ❱⑨ ❚❍➮◆● ❑➊ ❚❖⑩◆ ❍➴❈ ▼➣ sè✿ ✽✹✻✵✶✵✻ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ữợ P ❚❤➔♥❤ ◆❣❤➺ ❆♥ ✲ ✷✵✶✾ ✶ ▼❐❚ ❙➮ ❑Þ ❍■➏❯ ❚❍×❮◆● ❉Ị◆● ❚❘❖◆● ▲❯❾◆ ❱❿◆ R t➟♣ ❤đ♣ ❝→❝ sè t❤ü❝ R+ t➟♣ ❤đ♣ ❝→❝ sè t❤ü❝ ❦❤ỉ♥❣ ➙♠ B(X ) σ ✲ ✤↕✐ sè ❇♦r❡❧ ❝õ❛ X ❧♦❣x ❧♦❣❛r✐t ❝ì sè tü ♥❤✐➯♥ ❝õ❛ sè t❤ü❝ ❞÷ì♥❣ x exp(x) ❤➔♠ sè ♠ơ ✈ỵ✐ ❝ì sè e✱ sè ♠ơ ❧➔ x EX ❦➻ ✈å♥❣ ❝õ❛ ♣❤➛♥ tû ♥❣➝✉ ♥❤✐➯♥ X ❝❤✉➞♥ ❝õ❛ ❤➔♠ sè V ar(X) ♣❤÷ì♥❣ s❛✐ ❝õ❛ X I(A) ❤➔♠ ❝❤➾ t✐➯✉ ❝õ❛ t➟♣ ❤ñ♣ A ✷ t tú ự C ỵ ởt sè ❞÷ì♥❣ ✈➔ ❝â t❤➸ ❦❤ỉ♥❣ ❣✐è♥❣ ♥❤❛✉ ð ♠é✐ t d X=Y ỵ ♥❤✐➯♥ X, Y ❝ị♥❣ ♣❤➙♥ ♣❤è✐ ✷ ▼ư❝ ❧ư❝ ❈❤÷ì♥❣ Pữỡ t ố ợ ữủ ✺ ✶✳✶ ✣➦❝ tr÷♥❣ ❙t❡✐♥ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❙t❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷ P❤÷ì♥❣ ♣❤→♣ ❙t❡✐♥ ❝❤♦ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ❈❤÷ì♥❣ ✷ ❙❛✐ sè tr♦♥❣ ①➜♣ ố ợ ổ N tr ợ trữớ tr ổ N tr ợ trữớ tr✉♥❣ ❜➻♥❤ ✈➔ ♠ët sè ❦➳t q✉↔ ❧✐➯♥ q✉❛♥ ✳ ✷✵ ✷✳✷ ❙❛✐ sè tr♦♥❣ ①➜♣ ①➾ ❝❤✉➞♥ ✤è✐ ✈ỵ✐ ổ N tr ợ trữớ tr ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ õ ỵ t❤✉②➳t ①→❝ s✉➜t ♥û❛ ✤➛✉ t❤➳ ❦✛ ✷✵ ✤➣ ❝â ♥❤ú♥❣ t❤➔♥❤ tü✉ ✈÷đt ❜➟❝ tr♦♥❣ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ỵ ợ ữ t số ợ t rt ỵ ợ tr t ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✳✳✳ P❤÷ì♥❣ ♣❤→♣ ❝ê ✤✐➸♥ ự ỵ ợ tr t ỹ ✈➔♦ ❤➔♠ ✤➦❝ tr÷♥❣✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ ✤ë❝ ❧➟♣ t❤➻ ♣❤÷ì♥❣ ♣❤→♣ ❤➔♠ ✤➦❝ tr÷♥❣ r➜t ❦❤â →♣ ❞ư♥❣ ✈➔ t❤÷í♥❣ ❦❤ỉ♥❣ t➻♠ ✤÷đ❝ tè❝ ✤ë ❤ë✐ tư✳ ◆➠♠ rs t ợ t ởt ữỡ ợ ♠➔ ♥❣➔② ♥❛② ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❙t❡✐♥ ✤➸ ❝❤ù♥❣ ỵ ợ tr t ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ m✲♣❤ö t❤✉ë❝✳ ▼ö❝ ✤➼❝❤ ❜❛♥ ✤➛✉ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❙t❡✐♥ ❧➔ ①➜♣ ①➾ ♠ët t❤è♥❣ ❦➯ ♥➔♦ ✤â ♠➔ t❛ ✤❛♥❣ q✉❛♥ t➙♠ ✈ỵ✐ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ ✈➔ ✤→♥❤ ❣✐→ s❛✐ sè✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② ♠❛♥❣ ❧↕✐ ÷ỵ❝ ❧÷đ♥❣ t÷í♥❣ ♠✐♥❤ ❝õ❛ s❛✐ sè ①➜♣ ①➾ ❦❤✐ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ t❤ä❛ ♠➣♥ ♥❤✐➲✉ ❝➜✉ tró❝ ♣❤ư t❤✉ë❝ ❦❤→❝ ♥❤❛✉✳ ◆❤í ♥❤ú♥❣ ÷✉ ✤✐➸♠ ♥➔② ♠➔ ♥â ♥❣➔② ❝➔♥❣ ✤÷đ❝ ù♥❣ ❞ư♥❣ ♥❤✐➲✉ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉✳ ❚ø ✈✐➺❝ t➻♠ ❤✐➸✉ ❝❤õ ✤➲ ♥➔② ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➔✐ ❧✐➺✉ ❧✐➯♥ q✉❛♥✱ ❝❤ó♥❣ tỉ✐ ❧ü❛ ❝❤å♥ ✤➲ t➔✐ ❝❤♦ ❧✉➟♥ ✈➠♥ ❝❛♦ ❤å❝ ❝õ❛ ♠➻♥❤ ❧➔✿ ✏❳➜♣ ①➾ ❝❤✉➞♥ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❙t❡✐♥ ❝❤♦ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝ ✈➔ ♠ët sè ù♥❣ ❞ư♥❣✳✑ ▲✉➟♥ ✈➠♥ ữủ tỹ t trữớ ữợ sỹ ữợ t ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ❝õ❛ ♠➻♥❤ t ữớ trỹ t t t ữợ ❞➝♥ ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ r➜t ♥❤✐➲✉ tr♦♥❣ q✉→ tr t ự ỗ tớ t ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ð ❱✐➺♥ s÷ ♣❤↕♠ tü ♥❤✐➯♥ ✈➔ ♣❤á♥❣ s❛✉ ✤↕✐ ❤å❝ tr÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❚→❝ ❣✐↔ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧✱ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ ❜↕♥ tr ợ ỵ tt st t❤è♥❣ ❦➯ t♦→♥ ❤å❝ ✤➣ ❣✐ó♣ ✤ï✱ ✤ë♥❣ ✈✐➯♥ ✈➔ ỗ ũ t tr sốt q tr t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ❉♦ tr➻♥❤ ✤ë ✈➔ t❤í✐ ❣✐❛♥ ❤↕♥ ❝❤➳✱ ♠➦❝ ❞ị ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣✱ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚→❝ rt ữủ ỳ ỵ õ õ ❝õ❛ ❝→❝ t❤➛②✱ ❝æ ❣✐→♦ ✈➔ ❜↕♥ ❜➧ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ỡ ữỡ Pữỡ t ố ợ ữủ ỵ ợ tr t ởt tr ỳ ỵ q trồ ỵ t❤✉②➳t ①→❝ s✉➜t✳ ❈❤♦ ♠ët ❞➣② ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ {Xn }n1 ỵ ợ tr t r➡♥❣ ✈ỵ✐ ♠ët sè ✤✐➲✉ ❦✐➺♥ ♥➔♦ ✤â✱ t❛ ❝â lim P n→∞ tr♦♥❣ ✤â Sn = n Sn − ESn √ ≤ x = Φ(x) ✈ỵ✐ ♠å✐ x ∈ R, V arSn Xk , n ≥ 1✳ k=1 ✣➸ tố ở tử ỵ ợ ❤↕♥ tr✉♥❣ t➙♠ ♥❣÷í✐ t❛ ❞ị♥❣ ♠ët sè ❦❤♦↔♥❣ ❝→❝❤ ữ r ssrst ợ X Y trữợ õ dW = sup |Eh(X) − Eh(Y )| h ≤1 ✈➔ dK = sup|P (X ≤ x) − P (Y ≤ x)| x∈R ❧➛♥ ❧÷đt ✤÷đ❝ ❣å✐ ❧➔ ❦❤♦↔♥❣ ❝→❝❤ ❲❛ss❡rst❡✐♥ ✈➔ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❧♠♦❣♦r♦✈ ❣✐ú❛ ❤❛✐ ✻ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X, Y ✳ ✶✳✶ ✣➦❝ tr÷♥❣ ❙t❡✐♥ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❙t❡✐♥ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ s➩ tr➻♥❤ ❜➔② ✈➲ ✤➦❝ tr÷♥❣ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❙t❡✐♥ ❝ơ♥❣ ♥❤÷ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❙t❡✐♥✳ ❇ê ✤➲ s❛✉ ✤➙② ♥➯✉ ❧➯♥ ✤➦❝ tr÷♥❣ ❝ì ❜↔♥ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝â ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ t➢❝✳ ✶✳✶✳✶ ❇ê ✤➲ ✭✣➦❝ tr÷♥❣ ❙t❡✐♥✮✳ ◆➳✉ Z ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝â ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ t➢❝ N (0, 1) t❤➻ ✭✶✳✶✮ Ef (Z) = EZf (Z), ✈ỵ✐ ♠å✐ ❤➔♠ ❧✐➯♥ tư❝ t✉②➺t ✤è✐ ữủ ợ E|f (Z)| < ự ✭✶✳✶✮ t❤➻ f :R→R t❤ä❛ ♠➣♥ E|f (Z)| < ∞ ✤ó♥❣ ✈ỵ✐ ♠å✐ ❤➔♠ ❧✐➯♥ tư❝✱ ❜à ❝❤➦♥✱ ❦❤↔ ✈✐ tr➯♥ tø♥❣ ❦❤♦↔♥❣ Z ∼ N (0, 1) ❈❤✐➲✉ t❤✉➟♥✿ ●✐↔ sû Z ∼ N (0, 1) ❱ỵ✐ ❤➔♠ f : R −→ R ❧✐➯♥ tö❝ t✉②➺t ✤è✐ t❤ä❛ ♠➣♥ E|f (Z)| < ∞✱ t❛ ❝â ∞ Ef (Z) = −∞ − x2 f (x) √ e dx 2π  ∞  x2 x2 = √  f (x).e− dx + f (x).e− dx 2π −∞   x ∞ ∞ u2 u2 = √  f (x) −ue− dudx + f (x) ue− dudx 2π −∞ −∞ x ✼   ∞   u   −u2 −u2 = √   −f (x)ue dx du +  f (x)ue dx du 2π −∞ u 0   ∞ u2 u2 = √  (f (u) − f (0))ue− du + (f (u) − f (0))ue− du 2π −∞ ∞ =√ 2π =√ 2π u2 u2 f (u)ue− − f (0)ue− du −∞ ∞ u2 f (u)ue− du = EZf (Z) −∞ ❈❤✐➲✉ ♥❣❤à❝❤✳ ●✐↔ sû Ef (Z) = EZf (Z) ✈ỵ✐ ♠å✐ ❤➔♠ ❧✐➯♥ tư❝✱ ❜à ❝❤➦♥✱ ❦❤↔ ✈✐ tr➯♥ tø♥❣ ❦❤♦↔♥❣ t❤ä❛ ♠➣♥ E|f (Z)| < ∞ ❱ỵ✐ x ∈ R ❜➜t ❦ý✱ ①➨t ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✭✶✳✷✮ f (ω) − ωf (ω) = I(ω ≤ x) − Φ(x), tr♦♥❣ ✤â Φ(x) ❧➔ ❤➔♠ ♣❤➙♥ ♣❤è✐ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝â ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ t➢❝✳ ω2 ◆❤➙♥ ữỡ tr ợ e t❛ ✤÷đ❝ ω2 ω2 e− (f (ω) − ωf (ω)) = e− (I(ω ≤ x) − Φ(x)), ❤❛② ω2 e− f (ω) ω2 = e− I(ω ≤ x) − Φ(x) , ❚ø ✤➙② s✉② r❛ e − ω2 f (ω) =  ∞  − u2  − e (I(u ≤ x) − Φ(x))du  ω    ♥➳✉ ω > x, (3) ♥➳✉ ω ≤ x (4) ω −∞ u2 e− (I(u ≤ x) − Φ(x))du ✽ ✣✐➲✉ ♥➔② t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐ ∞  − u2  e Φ(x)du  − ω2 e f (ω) = ω ω u2    e− (1 − Φ(x))du −∞ √  2π.Φ(x)(1 − Φ(ω)) = √  2π.Φ(ω)(1 − Φ(x)) ♥➳✉ ω > x, ♥➳✉ ω ≤ x ♥➳✉ ω > x, ♥➳✉ ω ≤ x ❉♦ ✤â √  2π.e ω2 Φ(x)(1 − Φ(ω)) f (ω) = √  2π.e ω22 Φ(ω)(1 − Φ(x)) ♥➳✉ ω > x, ♥➳✉ ω ≤ x ✭✶✳✸✮ ❍➔♠ f ♥❤÷ tr➯♥ ❧➔ ❧✐➯♥ tö❝✱ ❜à ❝❤➦♥ ✈➔ ❝â ✤↕♦ ❤➔♠ t↕✐ ♠å✐ ✤✐➸♠ ❝❤➾ trø t↕✐ ✤✐➸♠ ω = x✳ ◆❤÷ ✈➟② ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ ❝â ♥❣❤✐➺♠ ❧✐➯♥ tư❝✱ ❜à ❝❤➦♥ ❞✉② ♥❤➜t✳ Ð ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ t❛ t❤❛② ω ❜ð✐ Z ý ú ỵ t❤✐➳t Ef (Z) = EZf (Z) t❛ ✤÷đ❝ = Ef (Z) − EZf (Z) = E(I(Z ≤ x)) − Φ(x), ❤❛② P (Z ≤ x) = Φ(x) ✈ỵ✐ ♠å✐ x ∈ R ◆❤÷ ✈➟② Z ∼ N (0, 1) ❚✐➳♣ t❤❡♦ ❝❤ó♥❣ t❛ s➩ tr➻♥❤ ❜➔② ❤❛✐ ❜ê ✤➲ ✈➲ t➼♥❤ ❝❤➜t ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❙t❡✐♥✳ ❈→❝ ❜ê ✤➲ ♥➔② ✤÷đ❝ sû ❞ư♥❣ ✤➸ ❝❤➦♥ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❧♠♦❣♦r♦✈ ✈➔ ❦❤♦↔♥❣ ❝→❝❤ ❲❛ss❡rst❡✐♥ ❝❤♦ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝✳ ✶✳✶✳✷ ❇ê ✤➲✳ ●✐↔ sû x ∈ R ✈➔ f ❧➔ ♥❣❤✐➺♠ ❜à ❝❤➦♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ f (ω) − ωf (ω) = I(ω ≤ x) − Φ(x)✳ ❑❤✐ ✤â ωf (ω) ❧➔ ❤➔♠ t➠♥❣ ❝õ❛ ω✳ ❍ì♥ ♥ú❛✱ ✈ỵ✐ ✷✺ ❑➳t ❤đ♣ ✭✷✳✽✮✲✭✷✳✶✵✮✱ ❝❤ó♥❣ t❛ ❝â < f (x) < f (x) < , x N −1 ✭✷✳✶✶✮ ✈➔ ✭✷✳✶✷✮ f (x) < ❉♦ ✤â |(xf (x)) | = |2f (x) + xf (x)| N −1 N −1 + 2f (x) + f (x) x x2 (N − 1)f (x) ≤ + (N − 1)f (x) + 2xf (x)f (x) + x ≤ + 2f (x) (t❤❡♦ ✭✷✳✶✶✮) = 2f (x) + x −f (x) (t❤❡♦ ✭✷✳✶✷✮) ≤6 ◆❤÷ ✈➟② ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ (i) ✈➔ (ii) ✤÷đ❝ ❤♦➔♥ t❤➔♥❤✳ ❇➙② ❣✐í ❝❤ó♥❣ t❛ s➩ t✐➳♣ tö❝ ❝❤ù♥❣ ♠✐♥❤ (iii)✳ ❚❛ ❝â f (x) x ❑➳t ❤đ♣ ✭✷✳✶✶✮ ✈➔✭✷✳✶✸✮✱ t❛ ✤÷đ❝ f (x) x = N f (x) 1− x x = f (x) f (x) − x x f (x) x − ✭✷✳✶✸✮ < 0✳ ❚ø ✭✷✳✼✮ ✈➔ ✭✷✳✶✶✮✲✭✷✳✶✸✮✱ t❛ s✉② r❛ f (x) N f (x) > 1− x x x − N −1 ✭✷✳✶✹✮ ỵ 2(a) s ú t ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ N f (x) 1− x x ❑➳t ❤ñ♣ ✭✷✳✶✹✮ ✈➔ ✭✷✳✶✺✮✱ t❛ ❝â f (x) x ❚ø ✤â (iii) ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ > > −4 N −1 −5 N −1 ✭✷✳✶✺✮ ✷✻ ✣➦t σ (1,2) β|σ (1,2) | = n = Sn − σ1 − σ2 ✈➔ b12 ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ ✤à♥❤ ♥❣❤➽❛ ♠ët ❤➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ♥❤÷ s❛✉ p12 (x, y) = β exp x + y, σ (1,2) Z12 n , x, y ∈ SN −1 , ✭✷✳✶✻✮ tr♦♥❣ ✤â Z12 ❧➔ ❤➡♥❣ sè ❝❤✉➞♥ ❤â❛✱ β ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✮✳ ●✐↔ sû (ξ1 , ξ2 ) ∼ p12 (x, y) ❝❤♦ ✈ỵ✐ (σj )j>2 ✱ ✈ỵ✐ i = 1, ∼ Vi = E ξi , σ (1,2) |(σj )j>2 ợ (1,2) b12 ữủ ♥❣❤➽❛ ♥❤÷ tr➯♥✱ t❛ ❝â (N − 1)f (b12 ) , i = 1, b12 ∼ Vi = |σ (1,2) |2 − ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû AN = 2π N/2 /Γ(N/2) ❧➔ ✤ë ✤♦ ▲❡❜❡s❣✉❡ ❝õ❛ SN −1 ❚ø ✭✷✳✶✻✮ t❛ s✉② r❛ Z12 = SN −1 SN −1 β x, σ (1,2) exp n SN −1 = 2π π  = AN = π  = β x + y, σ (1,2) n exp AN −1 AN AN −1 AN π dµ(x) (1,2) exp dµ(x)dµ(y) β|σ n | 2 cos ϕN −2 sinN −2 ϕN −2 sin2 ϕ2 sin ϕ1 dϕn−2 dϕ1 dθ 2 eb12 cos ϕN −2 sinN −2 ϕN −2 dϕN −2  √ πΓ(N/2 − 1/2) IN/2−1 (b12 ) (b12 /2)N/2−1 ❈❤♦ i = 1, 2✱ t❛ ❝â ✷✼ ∼ Vi = Z12 = Z12 θ, σ (1,2) exp SN −1 |σ (1,2) |2 SN −1 β θ, σ (1,2) n σ (1,2) θ, (1,2) |σ | dµ(θ) β|σ (1,2) | n exp θ, σ (1,2) |σ (1,2) | dµ(θ) π AN −1 = |σ (1,2) | AN Z12 cos2 ϕN −2 sinN −2 ϕN −2 eb12 cos ϕN −2 dϕN −2 π AN −1 = |σ (1,2) | AN Z12  AN −1 = 1 − AN Z12 π eb12 cos ϕN −2 sinN −2 ϕN −2 dϕN −2 − eb12 cos ϕN −2 sinN ϕN2 dϕN −2 π  eb12 cos ϕN −2 sinN ϕN −2 dϕN −2  |σ (1,2) | √ AN −1 πΓ(N/2 + 1/2) IN/2 (b12 ) |σ (1,2) | N/2−1 AN Z12 (b12 /2) (N − 1)f (b12 ) |σ (1,2) | = 1− b12 = 1− t ú tổ ợ t ởt ữủ sỷ ự ỵ ữỡ ♥➔②✳ ✷✳✷✳✹ ▼➺♥❤ ✤➲✳ ●✐↔ sû Wn ✈➔ Wn ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr♦♥❣ ✭✷✳✹✮ ✈➔ ✭✷✳✻✮✳ ❑❤✐ ✤â t❛ ❝â ✭✐✮ ✭✐✐✮ |Wn − Wn | ≤ Cn−1/2 EWn2 ≤ C E(Wn − Wn |Wn ) = λ(Wn + R)✱ ♥❤✐➯♥ t❤ä❛ ♠➣♥ ✭✐✐✐✮ ✈➔ tr♦♥❣ ✤â λ = E|R| ≤ Cn−1/2 E((Wn − Wn )2 |Wn ) − B ≤ Cn−1/2 ✱ 2λ ✭✷✳✺✮✳ E − βf (b) n tr♦♥❣ ✤â B2 ✈➔ R ❧➔ ❜✐➳♥ ♥❣➝✉ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr♦♥❣ ✷✽ ❈❤ù♥❣ ♠✐♥❤✳ (i) ❚❛ ❝â β2 |Sn |2 − |Sn |2 3/2 bn β2 = 3/2 | Sn + Sn , Sn − Sn | bn 2β n|Sn − Sn | ≤ b2 n3/2 2β |σI − σI | 4β = ≤ 1/2 b2 n1/2 bn ❈❤ù♥❣ ♠✐♥❤ ♣❤➛♥ ✤➛✉ ❝õ❛ (i) ❤♦➔♥ t❤➔♥❤✳ ❇➙② ❣✐í →♣ ❞ư♥❣ ❇ê ✤➲ ✷✳✷✳✶✱ t❛ ❝â |Wn − Wn | = EWn2 β|Sn | +1 = nE nb β|Sn | −1 ≤ CnE nb β|Sn | −1 nb 2 ≤ C (ii) ❑✐r❦♣❛tr✐❝❦ ✈➔ ◆❛✇❛③ ❬✻✱ ❡q✉❛t✐♦♥ ✭✾✮❪ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ E(Wn − Wn |Wn ) = 2 2β Wn + √ − 1/2 n n n b β|Sn | f n β|Sn | n + R1 , ✭✷✳✶✽✮ tr♦♥❣ ✤â R1 ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ t❤ä❛ ♠➣♥ E|R1 | ≤ Cn−3/2 ✳ ✣➦t g(x) = xf (x), x > 0✳ ❚❤❡♦ ❝æ♥❣ t❤ù❝ ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r t❛ ❝â g β|Sn | n β|Sn | g (ξ) = g(b) + g (b) −b + n ✣➦t V = β|Sn | + 1, nb β|Sn | −b n ✭✷✳✶✾✮ ✷✾ ❦❤✐ ✤â ≤ V ≤ C ✈➔ bWn β|Sn | β|Sn | −b=b −1 = √ n nb nV bWn bWn 1 = √ − √ − n n V bWn bWn β|Sn | = √ − √ −1 n nV nb bWn2 bWn = √ − n 2nV ✭✷✳✷✵✮ ❑➳t ❤ñ♣ ✭✷✳✶✽✮✲✭✷✳✷✵✮ ú ỵ r b = f (b) t õ E(Wn − Wn |Wn ) = 2 Wn + √ + R1 n n 2β g (ξ) β|Sn | bWn bWn2 − 1/2 g(b) + g (b) √ − + −b 2 n 2nV n n b 2 = Wn + √ + R1 n n 2β b bWn b2 bWn2 g (ξ)b2 Wn2 √ − − 1/2 + + bf (b) + β β n 2nV 2nV n b − βf (b) − βf (b) (Wn + R) = λ(Wn + R), ✈ỵ✐ λ = , = n n tr♦♥❣ ✤â βWn2 n R1 + 3/2 R= − βf (b) n V + f (b) − g (ξ) β ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✷✱ t❛ ❝â |g (ξ)| < ❚ø V ≥ 1, EWn2 ≤ C ✈➔ E|R1 | ≤ Cn−3/2 , t❛ ❦➳t ❧✉➟♥ r➡♥❣ E|R| ≤ Cn−1/2 ✳ ❈❤ù♥❣ ♠✐♥❤ ❝õ❛ (ii) ữủ t (iii) ỵ Id tr➟♥ ✤ì♥ ✈à ❝➜♣ n × n ✈➔ ✤➦t σ (i) = Sn − σi , bi = σ (i) β|σ (i) | , ri = (i) n |σ | ✸✵ ❚ø ❑✐r❦♣❛tr✐❝❦ ✈➔ ◆❛✇❛③ ❬✻✱ ❊q✉❛t✐♦♥ ✭✶✶✮ ✈➔ ✭✶✷✮❪✱ t❛ ❝â n 4β E((Wn − Wn ) |Wn ) = 2λB + 4 nb i=1 n 8β − nb (i) |σ | σi , σ i=1 n 4β + 4 nb σi , σ (i) i=1 n + 4β n4 b4 (n − 1)2 b2 |σ | − β2 N −1 1− β (i) (i) n2 b3 − β N −1 − 1− β (n − 1)2 b2 β2 σjT Ri σk , i=1 j,k=i tr♦♥❣ ✤â Ri = f (bi ) − bi β Id − N f (bi ) N − bi β Pi − f (bi ) − b β (ri σiT + σi riT ), ✈➔ Pi ❧➔ ❤➻♥❤ ❝❤✐➳✉ trü❝ ❣✐❛♦ ✈➔♦ ri ✳ ❉♦ ✤â 2β 2b 2 E((Wn − Wn ) |Wn ) − B = R2 − R3 + R4 + R5 , ✭✷✳✷✶✮ 2λ n b (1 − βf (b)) β tr♦♥❣ ✤â n R2 = i=1 n (n − 1)2 b2 |σ | − β2 N −1 1− β (i) |σ (i) | σi , σ (i) − R3 = i=1 n2 b3 β3 n σi , σ (i) R4 = − 1− i=1 n , , N −1 β (n − 1)2 b2 β2 , σjT Ri σk R5 = i=1 j,k=i ❱ỵ✐ R2 ✱ ú ỵ r | (i) Sn | ❦❤✐ ✤â t❤❡♦ ❇ê ✤➲ ✷✳✷✳✶ t❛ ❝â β|σ (i) | E −b n 2 β|σ (i) | β|Sn | C C ≤E −b ≤E −b + ≤ n n n n ✭✷✳✷✷✮ ✸✶ ❉♦ ✤â✱ n E |σ (i) |2 − E|R2 | ≤ C i=1 n ≤ Cn i=1 n ≤ Cn i=1 5/2 ≤ Cn (n − 1)2 b2 β2 (2n − 1)b2 β |σ (i) |2 −b + E n2 n2 ✭✷✳✷✸✮ β|σ (i) | E −b +C n ❱ỵ✐ R3 ✱ t❛ ❝â n E|R3 | = E |Sn | σi , Sn i=1 n2 b3 − + |σ (i) | σi , σ (i) − |Sn | σi , Sn β n3 b3 ≤ E |Sn | − + E β n ≤ Cn2 E |Sn | − ≤ Cn3 E ≤ Cn3 E nb +E β β|Sn | −b +E n |σ (i) | σi , σ (i) − |Sn | σi , Sn i=1 n |σ (i) | − |Sn | σi , σ (i) − |Sn | σi , σi i=1 n | σi , σ (i) | + |Sn | i=1 β|Sn | − b + Cn2 n ≤ Cn5/2 ✭✷✳✷✹✮ ✣➸ ❝❤➦♥ R5 ✱ ❝❤ó♥❣ t❛ ú ỵ r n n jT Ri k = i=1 j,k=i i=1 j,k=i n − i=1 j,k=i f (bi ) − bi β N f (bi ) N − bi β σj , σk − f (bi ) − σjT Pi σk b β σjT (ri σiT + σi riT )σk ✸✷ n f (bi ) − bi β = i=1 n N f (bi ) N − bi β − i=1 n f (bi ) − bi β = i=1 n N f (bi ) N − bi β − i=1 n |σ (i) |2 − f (bi ) − b β |σ (i) | σ (i) , σi ❚r❛❝❡(σk σjT ri riT ) j,k=i |σ (i) |2 − f (bi ) − σ (i) , ri f (bi ) = (1 − N ) − b β i i=1 b β |σ (i) | σ (i) , σi n (i) |σ | − f (bi ) − i=1 b β |σ (i) | σ (i) , σi := R51 − 2R52 f (b) β|σ (i) | ✈➔ bi = ✱ t❛ ❝â ❚ø = β b n n E|R51 | = E (1 − N ) i=1 n f (bi ) f (b) − |σ (i) |2 bi b E|bi − b| ✭t❤❡♦ ❇ê ✤➲ ✷✳✷✳✷ ✭✐✐✐✮ ✈➔ |σ (i) | ≤ n ✮ ≤ Cn i=1 n ≤ Cn2 E i=1 ✭✷✳✷✺✮ β β|Sn | − b + |σ (i) | − |Sn | n n ≤ Cn5/2 ✭t❤❡♦ ✭✷✳✷✷✮ ✈➔ ||σ (i) | − |Sn || ≤ ✮ ❚÷ì♥❣ tü t❛ ❝â n (f (bi ) − f (b))|σ (i) | σ (i) , σi E|R52 | = E i=1 n ≤ Cn i=1 ≤ Cn5/2 E|bi − b| ✭t❤❡♦ ❇ê ✤➲ ✷✳✷✳✷ ✭✐✮ ✈➔ |σ (i) | ≤ n ✮ ✭✷✳✷✻✮ ✸✸ ❑➳t ❤ñ♣ ✭✷✳✷✺✮ ✈➔ ✭✷✳✷✻✮✱ t❛ ❝â ✭✷✳✷✼✮ E|R5 | ≤ Cn5/2 ❱ỵ✐ R4 ✱ ✤➦t (n − 1)2 b2 , β2 N −1 a= 1− β ✈➔ V1 = σ1 , σ (1,2) , V2 = σ2 , σ (1,2) , t❛ ❝â σ1 , σ (1) − V1 ≤ Cn, σ1 , σ (2) − V2 ≤ Cn tr♦♥❣ ✤â σ (1,2) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✷✳✷✳✸✳ ❚ø ✤➙② t❛ s✉② r❛ ER42 = nE σ1 , σ (1) −a ≤ Cn5 + n(n − 1) E σ1 , σ (1) + n(n − 1)E σ1 , σ (1) 2 σ2 , σ (2) −a σ2 , σ (2) − V1 + V1 − a 2 −a − V2 + V2 − a ≤ Cn5 + n(n − 1)|E(V1 − a)(V2 − a)| ≤ Cn5 + n(n − 1)|E(V1 − E(V1 |(σj )j>2 ))(V2 − E(V2 |(σj )j>2 ))| + n(n − 1)|E(E(V1 |(σj )j>2 ) − a)(E(V2 |(σj )j>2 ) − a)| := Cn5 + n(n − 1)(|R41 | + |R42 |) ✭✷✳✷✽✮ ∼ ❱ỵ✐ ξi , Vi , (i = 1, 2) ❧➛♥ ❧÷đt ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✷✳✷✳✸✳ ❚÷ì♥❣ tü ♥❤÷ ❙❤❛♦ ✈➔ ❩❤❛♥❣ ❬✶✵✱ tr❛♥❣ ✸✾✱ ✹✵❪✱ t❛ ❝â t❤➸ ❝❤➾ r❛ r➡♥❣ R41 = E ✈➔ ξi , σ (1,2) ∼ ξi , σ (1,2) − V1 ∼ ∼ ∼ − V2 + H1 , R42 = E(V1 − a)(V2 − a) + H2 , ✭✷✳✷✾✮ ✭✷✳✸✵✮ tr♦♥❣ ✤â |H1 | ≤ Cn3 ✈➔ |H2 | ≤ Cn3 ✣➦t b12 = β|σ (1,2) | n ✭✷✳✸✶✮ ✸✹ ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✸ ✈➔ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ a✱ t❛ ❝â ∼ V1 − a = = N − (n − 1)2 b2 (N − 1)f (b12 ) |σ (1,2) |2 − − b12 β β2 (n − 1)2 b2 f (b12 ) N −1 (1,2) |σ | − + (N − 1) − |σ (1,2) |2 1− β β β b12 1− ≤ Cn2 β |σ (1,2) |2 (n − 1)2 b2 f (b) f (b12 ) − − + n2 n2 b b12 β |Sn |2 − b2 + |b12 − b| + Cn n β|Sn | β|σ (1,2) | −b + −b n n ≤ Cn ≤ Cn ≤ Cn2 + Cn β|Sn | − b + Cn n ∼ ❙û ❞ö♥❣ ✤→♥❤ ❣✐→ t÷ì♥❣ tü ❝❤♦ V2 − a ✱ ❦❤✐ ✤â t❛ ❝â E ∼ V1 −a ∼ V2 −a β|Sn | β|Sn | − b + n3 E − b + n2 nE n n ≤C ✭✷✳✸✷✮ Cn3 t ú ỵ r ợ (σj )j>2 ✱ ξ1 ✈➔ ξ2 ❧➔ ✤ë❝ ❧➟♣ ❝â ✤✐➲✉ ❦✐➺♥✳ ❚ø ✤â s✉② r❛ E ξi , σ (1,2) ∼ − V1 ξi , σ (1,2) ∼ − V2 = ✭✷✳✸✸✮ ❑➳t ❤ñ♣ ✭✷✳✷✽✮✲✭✷✳✸✸✮✱ t❛ ❝â ER42 ≤ Cn5 ✱ ✈➔ ✈➻ ✈➟② E|R4 | ≤ Cn5/2 ✭✷✳✸✹✮ ❑➳t ❤ñ♣ ✭✷✳✷✶✮✱✭✷✳✷✸✮✱✭✷✳✷✹✮✱✭✷✳✷✼✮ ✈➔ ✭✷✳✸✹✮ t❛ ❝â E E E((Wn − Wn )2 |Wn ) − B ≤ Cn−1/2 2λ ❈✉è✐ ❝ị♥❣✱ ❝❤ó♥❣ tổ tr ởt ỵ tr ❑♦❧♠♦❣♦r♦✈ ✸✺ ✈➔ ❦❤♦↔♥❣ ❝→❝❤ ❲❛ss❡rst❡✐♥ ❣✐ú❛ ❤❛✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ Wn /B ✈➔ Z ✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ✤à♥❤ ỵ ữủ t ữ s sû Wn ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr♦♥❣ B ữ tr ợ > N✳ ❑❤✐ ✤â✱ t❛ ❝â sup |Eh(Wn /B) − Eh(Z)| ≤ Cn−1/2 , h ≤1 ✈➔ sup |P (Wn /B ≤ z) − Φ(z)| ≤ Cn−1/2 , z∈R tr♦♥❣ ✤â C ❧➔ ❤➡♥❣ sè ❞÷ì♥❣ ❝❤➾ ♣❤ư t❤✉ë❝ ✈➔♦ ❝❤✉➞♥ t➢❝ ✈➔ Φ(z) β, Z ❧➔ ❤➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ❝õ❛ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝â ♣❤➙♥ ♣❤è✐ Z: z Φ(z) = √ 2π ❈❤ù♥❣ ♠✐♥❤✳ e−u /2 du −∞ ⑩♣ ❞ö♥❣ ▼➺♥❤ ✤➲ ✶✳✷✳✼✱ ▼➺♥❤ ✤➲ q t ỵ tr trỹ t✐➳♣ ✤÷đ❝ s✉② r❛✳ ✸✻ ❑➌❚ ▲❯❾◆ ❈❍❯◆● ❱⑨ ❑■➌◆ ◆●❍➚ ✶✳ ❑➳t ❧✉➟♥ ❝❤✉♥❣ ▲✉➟♥ ✈➠♥ ✤➣ t❤✉ ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ s❛✉ ✤➙②✿ ✲ ◆➯✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❜ê ✤➲✱ t➼♥❤ ❝❤➜t ✈➲ ✤➦❝ tr÷♥❣ ❙t❡✐♥✱ ❝→❝ t➼♥❤ ❝❤➜t ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❙t❡✐♥✳ ✲ ◆➯✉ ✤à♥❤ ♥❣❤➽❛✱ t➼♥❤ ❝❤➜t ✈➔ ♠ët sè ✈➼ ❞ư ✈➲ ❝➦♣ ❤♦→♥ ✤ê✐ ✤÷đ❝✱ ❝❤➦♥ tr➯♥ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❧♠♦❣♦r♦✈ ✈➔ ❦❤♦↔♥❣ ❝→❝❤ ❲❛ss❡rst❡✐♥ ✤è✐ ✈ỵ✐ ❝➦♣ ❤♦→♥ ữủ ợ t ổ s❛✐ sè tr♦♥❣ ①➜♣ ①➾ ❝❤✉➞♥ ✤è✐ ✈ỵ✐ ♠ỉ ❤➻♥❤ N tr ợ trữớ tr r t❤í✐ ❣✐❛♥ tỵ✐✱ ❝❤ó♥❣ tỉ✐ ❞ü ✤à♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✈➜♥ ✤➲ s❛✉ ✤➙②✿ ✲ ✣→♥❤ ❣✐→ s❛✐ sè tr♦♥❣ ①➜♣ ①➾ ❝❤✉➞♥ ✤è✐ ✈ỵ✐ ♠ỉ ❤➻♥❤ N ✲✈❡❝t♦r ✈ỵ✐ tr÷í♥❣ tr✉♥❣ ❜➻♥❤ ❝â trå♥❣ sè✳ ✸✼ ❉❆◆❍ ▼Ư❈ ❈⑩❈ ❈➷◆● ❚❘➐◆❍ ▲■➊◆ ◗❯❆◆ ✣➌◆ ▲❯❾◆ ❱❿◆ ✶✳ ◆❣✉②➵♥ ◆❣å❝ ❚ù✱ ◆❣✉②➵♥ ❈❤➾ ❉ô♥❣✱ ▲➯ ❱➠♥ ❚❤➔♥❤ ✈➔ ✣➦♥❣ ❚❤à P❤÷ì♥❣ ❨➳♥ ✭✷✵✶✾✮✱ ▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ ♠❡❛♥✲❢✐❡❧❞ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ✇✐t❤ ❡①✲ t❡r♥❛❧ ♠❛❣♥❡t✐❝ ❢✐❡❧❞✳ ❚↕♣ ❝❤➼ ❦❤♦❛ ❤å❝ tr÷í♥❣ ✤↕✐ ❤å❝ ❱✐♥❤ ✭✤➣ ♥❤➟♥ ✤➠♥❣✮✳ ✸✽ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❆♠♦s✱ ❉✳❊ ✭✶✾✼✹✮✱ ❈♦♠♣✉t❛t✐♦♥ ♦❢ ♠♦❞✐❢✐❡❞ ❜❡ss❡❧ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r r❛t✐♦s✳ ▼❛t❤✳ ❈♦♠♣✉t✳ ✷✽ ✱ ♥♦✳ ✶✷✺✱ ✷✸✾✕✷✺✶✳ ❬✷❪ ❆r❢❦❡♥✱ ●✳ ❇✳ ❛♥❞ ❲❡❜❡r✱ ❍✳ ❏ ✭✶✾✼✹✮✱ ▼❛t❤❡♠❛t✐❝❛❧ ♠❡t❤♦❞s ❢♦r P❤②s✐❝✐sts✳ ❊❧s❡✈✐❡r ❆❝❛❞❡♠✐❝ Pr❡ss✱ ✻t❤ ❡❞✐t✐♦♥✱ ①✳ ❬✸❪ ❈❤❡♥✱ ▲✳ ❍✳ ❨✳✱ ●♦❧❞st❡✐♥✱ ▲✳ ❛♥❞ ❙❤❛♦✱ ◗✳ ▼ ✭✷✵✶✶✮✱ ◆♦r♠❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❜② ❙t❡✐♥✬s ♠❡t❤♦❞✳ Pr♦❜❛❜✐❧✐t② ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s ✭◆❡✇ ❨♦r❦✮✳ ❙♣r✐♥❣❡r✱ ❍❡✐❞❡❧✲ ❜❡r❣✳ ❬✹❪ ❊✐❝❤❡❧s❜❛❝❤❡r✱ P✳ ❛♥❞ ▲☎♦✇❡✱ ▼ ✭✷✵✶✵✮✱ ❙t❡✐♥✬s ♠❡t❤♦❞ ❢♦r ❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ♦❝❝✉r✐♥❣ ✐♥ st❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s✳ ❊❧❡❝tr♦♥✳ ❏✳ Pr♦❜❛❜✳ ✶✺✱ ♥♦✳ ✸✵✱ ✾✻✷✕ ✾✽✽✳ ❬✺❪ ❑✐r❦♣❛tr✐❝❦✱ ❑✳ ❛♥❞ ▼❡❝❦❡s✱ ❊ ✭✷✵✶✸✮✱ ❆s②♠♣t♦t✐❝s ♦❢ t❤❡ ▼❡❛♥✲❋✐❡❧❞ ❍❡✐s❡♥✲ ❜❡r❣ ▼♦❞❡❧✳ ❏♦✉r♥❛❧ ♦❢ ❙t❛t✐st✐❝❛❧ P❤②s✐❝s✱ ✶✺✷✱ ✺✹✕✾✷✳ ❬✻❪ ❑✐r❦♣❛tr✐❝❦✱ ❑✳ ❛♥❞ ◆❛✇❛③✱ ❚ ✭✷✵✶✻✮✱ ❆s②♠♣t♦t✐❝s ♦❢ t❤❡ ▼❡❛♥✲❋✐❡❧❞ ▼♦❞❡❧✳ ❏♦✉r♥❛❧ ♦❢ ❙t❛t✐st✐❝❛❧ P❤②s✐❝s✱ ✶✻✺✱ ♥♦✳ ✻✱ ✶✶✶✹✕✶✶✹✵✳ ❖✭◆✮ ❬✼❪ ▲❡ ❱❛♥ ❚❤❛♥❤ ❛♥❞ ◆❣✉②❡♥ ◆❣♦❝ ❚✉ ✭✷✵✶✾✮✱ ❊rr♦r ❜♦✉♥❞s ✐♥ ♥♦r♠❛❧ ❛♣♣r♦①✐✲ ♠❛t✐♦♥ ❢♦r t❤❡ sq✉❛r❡❞✲❧❡♥❣t❤ ♦❢ t♦t❛❧ s♣✐♥ ✐♥ t❤❡ ♠❡❛♥ ❢✐❡❧❞ ❝❧❛ss✐❝❛❧ N ✲✈❡❝t♦r ♠♦❞❡❧s✳ ❊❧❡❝tr♦♥✳ ❈♦♠♠✉♥✳ Pr♦❜❛❜✳ ✷✹✱ ♥♦✳ ✶✻✱ ✶✕✶✷✳ ✸✾ ❬✽❪ ◆❛s❡❧❧✱ ■ ✭✶✾✼✽✮✱ ❘❛t✐♦♥❛❧ ❜♦✉♥❞s ❢♦r r❛t✐♦s ♦❢ ♠♦❞✐❢✐❡❞ ❇❡ss❡❧ ❢✉♥❝t✐♦♥s✳ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ✾✱ ♥♦✳ ✶✱ ✶✕✶✶✳ ❙■❆▼ ❬✾❪ ◆❣✉②➵♥ ◆❣å❝ ❚ù✱ ◆❣✉②➵♥ ❈❤➾ ❉ô♥❣✱ ▲➯ ❱➠♥ ❚❤➔♥❤ ❛♥❞ ✣➦♥❣ ❚❤à P❤÷ì♥❣ ❨➳♥ ✭✷✵✶✾✮✱ ▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ ♠❡❛♥✲❢✐❡❧❞ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ✇✐t❤ ❡①t❡r♥❛❧ ♠❛❣♥❡t✐❝ ❢✐❡❧❞✳ ❚↕♣ ❝❤➼ ❦❤♦❛ ❤å❝ ✤↕✐ ❤å❝ ❱✐♥❤ ✭✤➣ ♥❤➟♥ ✤➠♥❣✮✳ ❬✶✵❪ ❙❤❛♦✱ ◗✳▼✳ ❛♥❞ ❩❤❛♥❣✱ ❩✳ ❙ ✭✷✵✶✾✮✱ ❇❡rr②✲❊ss❡❡♥ ❜♦✉♥❞s ♦❢ ♥♦r♠❛❧ ❛♥❞ ♥♦♥✲ ♥♦r♠❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r ✉♥❜♦✉♥❞❡❞ ❡①❝❤❛♥❣❡❛❜❧❡ ♣❛✐rs✳ ❆♥♥✳ Pr♦❜❛❜✳ ✹✼✱ ♥♦✳ ✶✱ ✻✶✕✶✵✽✳ ❬✶✶❪ ❙t❡✐♥✱ ❈❤❛r❧❡s ✭✶✾✼✷✮✱ ❆ ❜♦✉♥❞ ❢♦r t❤❡ ❡rr♦r ✐♥ t❤❡ ♥♦r♠❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❛ s✉♠ ♦❢ ❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❙✐①t❤ ❇❡r❦❡❧❡② ❙②♠♣♦s✐✉♠ ♦♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙t❛t✐st✐❝s ❛♥❞ Pr♦❜❛❜✐❧✐t②✱ ❱♦❧✉♠❡ ✷✳ ❯♥✐✈❡rs✐t② ♦❢ ❈❛❧✐❢♦r♥✐❛ Pr❡ss✳ ♣♣✳ ✺✽✸✕✻✵✷✳ ... ) + (E|W | + 1)a + E|R| 2λ sup |P (W ≤ z) − Φ(z)| ≤ E − z∈R ✷✵ ữỡ số tr ố ợ ổ N tr ợ trữớ tr ổ N tr ợ trữớ tr ởt số ❦➳t q✉↔ ❧✐➯♥ q✉❛♥ ●✐↔ sû N ≥ ✈➔ SN −1 ❧➔ ♠➦t ❝➛✉ ✤ì♥ ✈à tr♦♥❣ RN... ❜➧✱ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ ❜↕♥ tr ợ ỵ tt st t❤è♥❣ ❦➯ t♦→♥ ❤å❝ ✤➣ ❣✐ó♣ ✤ï✱ ✤ë♥❣ ✈✐➯♥ ✈➔ ỗ ũ t tr sốt q tr t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ❉♦ tr➻♥❤ ✤ë ✈➔ t❤í✐ ❣✐❛♥ ❤↕♥ ❝❤➳✱ ♠➦❝ ❞ị ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣✱ ♥❤÷♥❣... ①→❝ s✉➜t ♥û❛ ✤➛✉ t❤➳ ❦✛ ✷✵ ✤➣ ❝â ♥❤ú♥❣ t❤➔♥❤ tü✉ ✈÷đt ❜➟❝ tr♦♥❣ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ỵ ợ ữ t số ợ t rt ỵ ợ tr t ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✳✳✳ P❤÷ì♥❣ ♣❤→♣ ❝ê ✤✐➸♥ ự ỵ ợ tr t ỹ ✈➔♦ ❤➔♠