Khả tích đều theo nghĩa casàro và luật mạnh số lớn

0✳ ❑❤✐ ➤ã P(A1 A2 An ) = P(A1 )P(A2 /A1 ) P(An /A1 An−1 ) ✶✳✶✳✹✳ ❚Ý♥❤ ➤é❝ ❧❐♣ ❝ñ❛ ❝➳❝ ❜✐Õ♥ ❝è ●✐➯ sö (Ω, F, P) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t✳ ị ĩ ế ố A B ợ ❣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 ➤Ò✉ ➤é❝ ❧❐♣✳ ❍ä ❝➳❝ ❜✐Õ♥ ❝è (Ai )i∈I ➤➢ỵ❝ ❣ä✐ ❧➭ ➤é❝ ❧❐♣ t♦➭♥ ❝ơ❝ ✭❣ä✐ t➽t ❧➭ ➤é❝ ❧❐♣✮ ♥Õ✉ ➤è✐ ✈í✐ ♠ä✐ ❤ä ❤÷✉ ❤➵♥ ❝➳❝ ❜✐Õ♥ ❝è Ai1 , Ai2 , , Ain ❝ñ❛ ❤ä ➤ã✱ t❛ ➤Ò✉ ❝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❡❧❧✐✮✳ ●✐➯ sö (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❤Ó ♥❤❐♥ ♠ét tr♦♥❣ ❤❛✐ ❣✐➳ trị tù t ỗ P(An ) ❤é✐ tô ❤❛② ♣❤➞♥ ❦ú✳ n=1 ✶✳✷✳ ➳♥❤ ①➵ ➤♦ ➤➢ỵ❝ ✈➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ✶✳✷✳✶✳ ➳♥❤ ①➵ ➤♦ ➤➢ỵ❝ ➜Þ♥❤ ♥❣❤Ü❛✳ ●✐➯ sư (Ω1 , F1 ) ✈➭ (Ω2 , F2 ) ❧➭ ❤❛✐ ❦❤➠♥❣ ❣✐❛♥ ➤♦✳ ➳♥❤ ①➵ X : Ω1 → Ω2 ❣ä✐ ❧➭ ➳♥❤ ①➵ F1 /F2 ợ ế ọ B F2 tì 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 ỉ ợ 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❝ ♥Õ✉ ♥ã ❧➭ ➳♥❤ ①➵ X −1 (B) G) ợ tì X st X : Ω → R G/B(R) G ❧➭ σ ➤➢ỵ❝ ❣ä✐ ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤♦ ➤➢ỵ❝ ✭tø❝ ❧➭ ✈í✐ ♠ä✐ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ➤➷❝ ❜✐Ưt✱ ❦❤✐ ✲ ➤➵✐ sè ❝♦♥ X B ∈ B(R) ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ F tì ợ ọ ột ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✳ ❍✐Ó♥ ♥❤✐➟♥✱ ❜✐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 ✳ ❝❤Ø ♥❤❐♥ ❤÷✉ ❤➵♥ ❣✐➳ trị tì ó ợ ọ ế ế ò ợ ọ ợ ♥❣➱✉ ♥❤✐➟♥✳ ❚Ý♥❤ ❝❤✃t ➜Þ♥❤ ❧ý ✶✳ X ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ♠ét tr♦♥❣ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ 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 ✷✳ ●✐➯ sư X1 , X2 , , Xn (Ω, F, P)✱ f : Rn → R ✈í✐ ♠ä✐ a ∈ R✳ ❧➭ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❝ï♥❣ ①➳❝ ➤Þ♥❤ tr➟♥ ❧➭ ❤➭♠ ➤♦ ➤➢ỵ❝ ✭❝ã ♥❣❤Ü❛ 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♥❣ ①➳❝ ➤Þ♥❤ 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➟♥ ❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❝ã t❤Ó ♠ë ré♥❣ ❝❤♦ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ G ✲ ➤♦ ợ t ỳ P ố st ị ĩ ●✐➯ sö (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 ❝ñ❛ ♠ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ♥➭♦ ➤ã✳ ❈❤ó ý✳ ❚➢➡♥❣ ø♥❣ ❣✐÷❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ✈➭ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝đ❛ ❝❤ó♥❣ ❦❤➠♥❣ ♣❤➯✐ ❧➭ t➢➡♥❣ ø♥❣ ✶✲✶✳ ◆❤÷♥❣ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❝ã ❝ï♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ st ợ ọ ữ ế ù ♣❤➞♥ ♣❤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✳ ✶✳ ✷✳ ◆Õ✉ FX 1✳ a 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✉ t❤× EX t❤× 0✳ EX = C ✳ ✷✹ gn : (0, ∞) → (0, ∞) ❈❤♦ ❧➭ ❤➭♠ sè t➝♥❣ t❤❡♦ x ✈í✐ ♠ä✐ n 1, gn (0) ❝ã t❤Ĩ ♥❤❐♥ ❣✐➳ trÞ tï② ý✳ ●✐➯ sư r➺♥❣✿ x/gn (x) ✈➭ gn (x) x2 ❣✐➯♠ t❤❡♦ x ∞ E(gn (|Xn |))/gn (an ) < ∞ ✈➭ ❞➲② {an /f (n)}n ❜Þ ❝❤➷♥✳ ❑❤✐ ➤ã n=1 (f (n))−1 [S(n) − E(S(n))] → h.c.c ❦❤✐ n → ∞ ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ sö ❞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♥❣ ♠✐♥❤ x {Xn }n t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❚❛ ❝ã t❤♦➯ ♠➲♥✳ ❚❛ x2 gn (x) 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))] → h.c.c ❦❤✐ n → ∞ ✷✺ ❍Ö q✉➯ ✷✳✷✳✺✳ ❈❤♦ {Xn } ❧➭ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣ ➤➠✐ ♠ét ✈➭ n E(|Xk |I(|Xk | sup[ n ak ))/n] < ∞ k=1 ∞ ế pn ỗ n npn E(|Xn |) < ∞ t❤× ✈➭ n=1 n−1 [S(n) − E(S(n))] → h.c.c ❦❤✐ n → ∞ ❈❤ø♥❣ ♠✐♥❤✳ ❙ö ❞ơ♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✹ ✈í✐ gn (x) = xpn , an = n ✈➭ f (n) = n✱ t❛ ➤➢ỵ❝ ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ➜Þ♥❤ ❧Ý ✷✳✷✳✻✳ ❈❤♦ {Xn }n ❧➭ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣ ➤➠✐ ♠ét ✈➭ n E(|Xk |I(|Xk | sup[ n ❈❤♦ ak ))/f (n)] < ∞ k=1 gn : (0, ∞) → (0, ∞) ❧➭ ❤➭♠ sè t➝♥❣ t❤❡♦ x ✈í✐ ♠ä✐ n 1, gn (0) ❝ã t❤Ĩ ♥❤❐♥ ❣✐➳ trÞ tï② ý✳ ●✐➯ sö r➺♥❣✿ x/gn (x) t➝♥❣ t❤❡♦ x ✈➭ E(Xn ) = 0, ∞ E(gn (|Xn |))/gn (an ) < ∞ ✈➭ {an /f (n)}n ❜Þ ❝❤➷♥✳ ❑❤✐ ➤ã n=1 (f (n))−1 [S(n) − E(S(n))] → 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 ❞ô♥❣ 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 )) ✷✻ ∞ ∞ −2 (f (n)) E(|Xn2 |I(|Xn | a2n E(gn (|Xn |)/(f (n)gn (an )) < ∞ an )) n=1 n=1 s✉② r❛ ➤✐Ị✉ ❦✐Ư♥ (c) t❤♦➯ ♠➲♥✳ ❚õ ❣✐➯ tết t ó ợ ề ệ (d) ủ ị í ✷✳✷✳✷✳ ❱❐② (f (n))−1 [S(n) − E(S(n))] → h.c.c ❦❤✐ n → ∞ ✷✳✸✳ ❑❤➯ tÝ❝❤ ➤Ò✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦ ✈➭ ❧✉❐t ♠➵♥❤ sè ❧í♥ ➤è✐ ✈í✐ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➯ tÝ❝❤ ➤Ị✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦ ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✸✳✶✳ ❉➲② ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ {Xn }n ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➯ tÝ❝❤ ➤Ò✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦ ♥Õ✉ n −1 E(|Xi |I(|Xi | > N ))] = lim sup[n N →∞ n ❚Ý♥❤ ❝❤✃t ✷✳✸✳✷✳ i=1 ◆Õ✉ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ {Xn }n ❦❤➯ tÝ❝❤ ➤Ò✉ t❤× ♥ã ❦❤➯ tÝ❝❤ ➤Ị✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦✳ ❚❤❐t ✈❐②✱ ❣✐➯ sö {Xn }n ❧➭ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➯ tÝ❝❤ ➤Ò✉✳ ❑❤✐ ➤ã lim sup E(|Xi |I(|Xi | > N )) = N →∞ i ❉♦ ➤ã✱ ✈í✐ ♠ä✐ > 0✱ ∃N0 > s❛♦ ❝❤♦ ✈í✐ ♠ä✐ N > N0 t❤× sup E(|Xi |I(|Xi | > N )) < i E(|Xi |I(|Xi | > N ))] < , ∀i ❉♦ ➤ã✱ ∀N > N0 ✈➭ n 1✱ t❛ ❝ã ❙✉② r❛ n −1 n E(|Xi |I(|Xi | > N )) n i=1 −1 n i=1 ❱× ✈❐② = n−1 n = < 2 n −1 E(|Xi |I(|Xi | > N ))] = 0, lim sup[n N →∞ n i=1 ✷✼ {Xn }n ♥❣❤Ü❛ ❧➭ ❦❤➯ tÝ❝❤ ➤Ị✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦✳ ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✸✳✸✳ ❉➲② ❝➳❝ sè ❞➢➡♥❣ {an }n {n−1 (a1 + · · · + an )}n ❈❡s➭r♦ ♥Õ✉ ❞➲② ➤➢ỵ❝ ❣ä✐ ❧➭ ❜Þ ❝❤➷♥ t❤❡♦ ♥❣❤Ü❛ ❜Þ ❝❤➷♥✳ ➜Þ♥❤ ❧Ý ✷✳✸✳✹✳ ❉➲② ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ {Xn }n ❦❤➯ tÝ❝❤ ➤Ò✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❝➳❝ ➤✐Ò✉ ❦✐Ư♥ s❛✉ ➤➢ỵ❝ t❤♦➯ ♠➲♥✿ (i) ❉➲② (ii) {E|Xn |}n ❜Þ ❝❤➷♥ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦❀ > 0✱ tå♥ t➵✐ δ > s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ❞➲② ❜✐Õ♥ ❝è {Ak }k ❱í✐ ♠ä✐ n n ➤✐Ị✉ ❦✐Ư♥ P(Ak )] < δ ✱ t❤× t❛ ❝ã sup[n−1 sup[n−1 n ❈❤ø♥❣ ♠✐♥❤✳ n k=1 {Xn }n ➜✐Ị✉ ❦✐Ư♥ ❝➬♥✿ ●✐➯ sư ❞➲② ❈❡s➭r♦✳ ❑❤✐ ➤ã✱ ✈í✐ = 1✱ tå♥ t➵✐ a0 t❤♦➯ ♠➲♥ E(|Xk |I(A))] < k=1 ❦❤➯ tÝ❝❤ ➤Ị✉ t❤❡♦ ♥❣❤Ü❛ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ a > a0 t❤× n −1 E(|Xi |I(|Xi | > a))] = c < sup[n n (1) i=1 ▼➷t ❦❤➳❝✱ ✈í✐ ♠ä✐ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ |X|dP = E|X| = X ✈➭ ♠ä✐ a > 0✱ t❤× |X|dP + Ω (|X| a) |X|dP (|X|>a) |X|dP adP + (|X| a) (|X|>a) |X|dP = a + E(|X|I(|X| > a)) a+ (|X|>a) n ❉♦ ➤ã✱ −1 n E(|Xi |) n i=1 ❚õ ✭✶✮ ✈➭ ✭✷✮✱ ❧✃② a n −1 n = a0 + > a+n i=1 a0 ✱ t❛ ➤➢ỵ❝✿ −1 E(|Xn |I(|Xi | > a)) (2) i=1 n −1 E(|Xi |) n a0 + + < ∞ i=1 ❙✉② r❛ ➤✐Ị✉ ❦✐Ư♥ (i)✿ ❉➲② {E|Xn |}n ❚✐Õ♣ t❤❡♦✱ ❝ị♥❣ ❞♦ ❞➲② {Xn }n ❜Þ ❝❤➷♥ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦✳ ❦❤➯ tÝ❝❤ ➤Ị✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦✱ ♥➟♥ ✈í✐ ♠ä✐ ✷✽ > 0✱ tå♥ t➵✐ a0 > s❛♦ ❝❤♦✿ n −1 sup n n (|Xk | a0 ) k=1 |Xk |dP < n ▲✃② δ= 2a0 (Ak )k ✱ ❦❤✐ ➤ã ✈í✐ ♠ä✐ E(|Xk |I(Ak )) = n Ak k=1 n n −1 −1 |Xk |dP + n =n Ak ∩(|Xk | a0 ) k=1 n n −1 k=1 < a0 δ + 2a0 |Xk |dP k=1 n Ak ∩(|Xk |>a0 ) k=1 (|Xk |>a0 ) P(Ak ) + n−1 a0 n ❉♦ ➤ã✱ ➤✐Ò✉ ❦✐Ö♥ k=1 a0 P(Ak ) + n−1 k=1 = a0 |Xk |dP Ak ∩(|Xk |>a0 ) n n−1 |Xk |dP + = (ii) t❤♦➯ ♠➲♥✳ ➜✐Ò✉ ❦✐Ư♥ ➤đ✿ n (i) s✉② r❛ M = sup(n −1 n ❱í✐ ♠ä✐ t❤× t❛ ❝ã |Xk |dP k=1 ❚õ P(Ak ) < δ n −1 −1 n k=1 n n s❛♦ ❝❤♦ −1 E(|Xk |)) < ∞ k=1 a > 0✱ P(|Xk | s✉② r❛ a) a−1 E(|Xk |) ✭❇✃t ➤➻♥❣ t❤ø❝ ▼❛r❦♦✈✮ n −1 n P(|Xk | n a) k=1 −1 a−1 E|Xk | n a−1 M k=1 n ❉♦ −1 (ii) ♥➟♥ ✈í✐ ♠ä✐ > 0✱ tå♥ t➵✐ δ > s❛♦ ❝❤♦ ♥Õ✉ n P(Ak ) < δ k=1 n −1 E(|Xk |I(Ak )) < n k=1 t❤× ✷✾ a0 = ➜➷t M ✳ ❑❤✐ ➤ã ✈í✐ ♠ä✐ a > a0 ✱ t❛ ❝ã δ n n −1 −1 E(|Xk |I(Xk > a)) n E(|Xk |I(Xk > a0 )) n k=1 k=1 n ◆Õ✉ ➤➷t −1 P(Ak ) < Ak = (|Xk | > a0 ) t❤× n k=1 M = δ ✳ ❉♦ ➤ã✱ a0 n −1 E(|Xk |I(Xk > a)) < n k=1 ❙✉② r❛ n −1 E(|Xk |I(Xk > a))] = lim sup n [ a→∞ n ❱❐② ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ k=1 {Xn }n ❦❤➯ tÝ❝❤ ➤Ị✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦✳ ➜Þ♥❤ ❧ý s❛✉ ➤➞② t➢➡♥❣ tù ➤Þ♥❤ ❧ý ❱❛❧❧Ð❡ P♦✉ss✐♥✳ ➜Þ♥❤ ❧Ý ✷✳✸✳✺✳ ❉➲② ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ {Xn }n ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ tå♥ t➵✐ ❤➭♠ ❧å✐ ➤♦ ➤➢ỵ❝ ❦❤➯ tÝ❝❤ ➤Ị✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦ φ : (0, ∞) → (0, ∞) s❛♦ ❝❤♦ (i) t−1 φ(t) t➝♥❣ ➤Õ♥ ∞ ❦❤✐ t ↑ ∞; n (ii) sup[n −1 n E(φ(|Xi |))] < ∞ i=1 ❈❤ø♥❣ ♠✐♥❤✳ ➜✐Ị✉ ❦✐Ư♥ ❝➬♥✿ ●✐➯ sư tå♥ t➵✐ ❤➭♠ tr➟♥✳ ❚❛ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ {Xn }n φ(t) t❤♦➯ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❧➭ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➯ tÝ❝❤ ➤Ò✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦✱ ♥❣❤Ü❛ ❧➭ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ n −1 E(|Xi |I(Xi > N ))] = lim sup n [ N →∞ n i=1 n ➜➷t −1 n ♠ä✐ E(φ(|Xi |))] < ∞ ❉♦ φ(t) t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ (i) ♥➟♥ ✈í✐ c = sup n [ > 0✱ i=1 tå♥ t➵✐ ♠ét sè ❞➢➡♥❣ N s❛♦ ❝❤♦ φ(t) t(c + 1) ✈í✐ t > N✳ ✸✵ ❙✉② r❛ t φ(t) c+1 ✳ ❉♦ ➤ã✱ ✈í✐ ♠ä✐ n n 1✱ t❛ ❝ã n E(|Xi |I(|Xi | > N )) = i=1 |Xi |dP i=1 n |Xi |>a i=1 |Xi |>a φ(|Xi |) c+1 dP n = φ(|Xi |)dP c+1 i=1 n Ω φ(|Xi |)dP c+1 = i=1 n |Xi |>a E(φ(|Xi |)) c+1 i=1 n = ❙✉② r❛ n c = i=1 c < c+1 n −1 E(|Xi |I(Xi > N ))] < sup n [ n ❱❐② c+1 E(φ(|Xi |)) sup c+1 {Xn }n i=1 ❦❤➯ tÝ❝❤ ➤Ị✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦✳ ➜✐Ị✉ ❦✐Ư♥ ➤đ✿ ●✐➯ sư {Xn }n ❧➭ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➯ tÝ❝❤ ➤Ò✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦✱ t❛ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ tå♥ t➵✐ ❤➭♠ ❧å✐ ➤♦ ➤➢ỵ❝ s❛♦ ❝❤♦ φ : (0, ∞) → (0, ∞)✱ φ(t)/t ↑ ∞ ❦❤✐ t → ∞ ✈➭ k −1 E(φ(|Xi |))] = c < ∞ sup n [ n i=1 ❚❤❐t ✈❐②✱ ✈í✐ ♠ä✐ ❞➲② sè t❤ù❝ ❦❤➠♥❣ ➞♠ {un }n ❜✃t ❦× ✈í✐ un ↑ ∞ ❦❤✐ n → ∞✱ ①Ðt ❤➭♠ g : (0, ∞) → (0, ∞) ①➳❝ ➤Þ♥❤ ❜ë✐ g(x) = un ❦❤✐ x ∈ [n−1, n) ✈í✐ n 1, n ∈ N✳ ➜➷t t φ(t) = g(x)dx, t > 0 ✸✶ φ(t) ❧➭ ❤➭♠ ❧å✐ ✭①❡♠ ❬➜Þ♥❤ ❧Ý ❆✱ ✾ ❪✮✳ ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ φ(t)/t ↑ ∞ ❦❤✐ t ↑ ∞✳ ❚❤❐t ✈❐②✱ t❛ ❝ã tr♦♥❣ ❦❤♦➯♥❣ β(t) := φ(t)/t (n − 1, n), n ✈✐Ö❝ ❝❤ø♥❣ ♠✐♥❤ ❱í✐ 1✱ (0, ∞) β(t) ♥➟♥ ➤Ĩ ❝❤ø♥❣ ♠✐♥❤ ✈í✐ ♠ä✐ t ∈ (n − 1, n)✱ n β (t) ✈➭ ❧➭ ❤➭♠ ❦❤➯ ✈✐ ❧➭ ❤➭♠ t➝♥❣✱ t❛ ❝❤Ø 1✳ ✈➭ t ∈ (n − 1, n)✱ t❛ ❝ã n φ(t) = g(x)dx + ❳Ðt ❧➭ ❤➭♠ ❧✐➟♥ tô❝ tr➟♥ n−1 g(x)dx + + g(x)dx + x ∈ [0, 1) t❤× g(x) = u1 t n−2 ♥➟♥ g(x)dx u1 dx = g(x)dx n−1 = (1 − 0).u1 = u1 ❚➢➡♥❣ tù g2 dx ✳✳ ✳ = u2 , n−1 n−2 gn−1 dx ❳Ðt = un−1 x ∈ [n − 1, t)✱ ❞♦ [n − 1, t) ⊂ [n − 1, n) ♥➟♥ g(x) = un ✱ s✉② r❛ t t un dx = [t − (n − 1)]un ux dx = n−1 n−1 ❉♦ ➤ã✱ n−1 ui + (t − n + 1)un , (u0 = 0), ∀t ∈ (n − 1, n] φ(t) = i=0 n ❙✉② r❛ (un − ui ) t β (t) = 0✳ ❙✉② r❛ β (t) ✈➭ ❞♦ ➤ã β(t) ❧➭ ❤➭♠ t➝♥❣✳ i=1 ➜Ó ❝❤ø♥❣ ♠✐♥❤ β(t) → ∞ ❦❤✐ t → ∞✱ t❛ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ β(n) → ∞ ❦❤✐ n n → ∞✳ ➜✐Ò✉ ♥➭② s✉② r❛ tõ ♥❤❐♥ ①Ðt r➺♥❣ −1 ❦❤✐ n → ∞✱ ✈× i=1 un → ∞ ❦❤✐ n → ∞✳ ❱❐② ui → ∞ n φ(t)/t ↑ ∞ ❦❤✐ t → ∞✳ ❚❛ sÏ ❝❤ø♥❣ r ó tể ọ ợ {un }n ị ♥❤➢ tr➟♥✱ s❛♦ ❝❤♦ n −1 E(φ(|Xi |))] sup[n n i=1 1 ✈➭ ❤➭♠ φ ➤➢ỵ❝ ①➳❝ ✸✷ ❚❤❐t ✈❐②✱ ❞♦ ❞➲② ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❈❡s➭r♦✱ ♥➟♥ tå♥ t➵✐ ❞➲② {Nj }j {Xn }n ❦❤➯ tÝ❝❤ ➤Ị✉ t❤❡♦ ♥❣❤Ü❛ ❝➳❝ sè ❞➢➡♥❣ s❛♦ ❝❤♦ ✈í✐ ỗ j tì n E(|Xi |)I(|Xi | sup[n n ✈➭ i=1 Nj → ∞ ❦❤✐ j → ∞✳ ❱í✐ n 1✱ ➤➷t un = card{j ✭❱× Nj → ∞ ❦❤✐ j → ∞ ♥➟♥ t❐♣ {j ❚❛ ❝ã✱ s❛♦ ❝❤♦ r➺♥❣ {un } ❧➭ ❞➲② t➝♥❣✳ ❱í✐ Nj < n ✈í✐ 2−j , Nj ))] M j = 1, , M ✳ : Nj < n} : Nj < n} ❤÷✉ ❤➵♥ ✈í✐ ♠ä✐ n ❜✃t ❦×✱ tå♥ t➵✐ ♠ét sè ♥❣✉②➟♥ ❑❤✐ ➤ã un M ✈í✐ ♠ä✐ n un → ∞ ❦❤✐ n → ∞✳ ▼➷t ❦❤➳❝ n−1 ui + (t − n + 1)un , (u0 = 0), ∀t ∈ (n − 1, n] φ(t) = i=0 n ui ✈í✐ t ∈ (n − 1, n], n i=1 ❙✉② r❛ n I(n1,n] (t) (t) n=1 ó ỗ k ui i=1 1✱ t❛ ❝ã ∞ Eφ(|Xk |) n I(n − < |Xk | E n=1 ∞ =E i=1 I(n − < |Xk | n=i ∞ ui P(|Xk | > i − 1), = i=1 ✈í✐ I(A) ❧➭ ❤➭♠ ❝❤Ø t✐➟✉ ❝ñ❛ A✳ ui ) ∞ ui i=1 n)( n) n0 ✳ 1✮ n0 ❙✉② r❛ ✸✸ n ❉♦ ➤ã✱ ✈í✐ ♠ä✐ t❤× ∞ n n Eφ(|Xk |) 1)P(|Xk | > m − 1) ( k=1 m=1 j:Nj m) = j=1 k=1 m=Nj ∞ n E(|Xk |I(|Xk | j=1 k=1 ∞ −j n Nj )) = n j=1 n ❱❐② −1 E(φ(|Xi |))] sup[n n 1 i=1 ❉ù❛ ✈➭♦ ➤Þ♥❤ ❧ý tr➟♥ ✈➭ ➜Þ♥❤ í t ứ ợ ị ý s ị ❧Ý ✷✳✸✳✻✳ tå♥ t➵✐ ❤➭♠ ❈❤♦ {Xn }n ❧➭ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣ ➤➠✐ ♠ét✳ ●✐➯ sö φ : (0, ∞) → (0, ∞) s❛♦ ❝❤♦ (i) t−1 φ(t) t➝♥❣ ➤Õ♥ ∞ ❦❤✐ t ↑ ∞; n −1 E(φ(|Xi |))] = c < ∞❀ (ii) sup[n n i=1 ∞ (φ(n))−1 ) < ∞ (iii) n=1 ❑❤✐ ➤ã✱ n−1 [S(n) − E(S(n))] → h.c.c ❦❤✐ n → ∞✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ❞ï♥❣ ❜ỉ ➤Ị s❛✉ ∞ ❇ỉ ➤Ị ✭❬✷❪✮ ◆Õ✉ {αn }n n=1 ❜✃t ❦× s❛♦ ❝❤♦ bn < ∞ ✈➭ {bn }n {nαn }n tì ọ ị ❧➭ ❞➲② t➝♥❣✱ t❛ ❝ã ∞ [nαn − (n − 1)αn−1 ]bn < ∞ n=1 ✸✹ ❈❤ø♥❣ ♠✐♥❤ ➤Þ♥❤ ❧Ý✳ ❚❛ sư ❞ơ♥❣ ➜Þ♥❤ ❧Ý ✷✳✷✳✷ ✈í✐ ➜➷t Bn = [−n, n] ✈í✐ n 1✳ n −1 E(φ(|Xi |)) ✈í✐ n αn = n i=1 ➜➬✉ t✐➟♥ t❛ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ị✉ ❦✐Ư♥ ∞ ∞ P(|Xi | > n) P(φ(|Xi |) n=1 ∞ E(φ(|Xi |))/φ(n) = n=1 {αn }n φ(n)) n=1 ∞ ❱× (a) [nαn − (n − 1)αn−1 ]/φ(n) n=1 ❧➭ ❞➲② ❜Þ ❝❤➷♥✱ {nαn }n ❧➭ ❞➲② t➝♥❣ ✈➭ {φ(n)}n ❧➭ ❞➲② t➝♥❣ ♥➟♥ t❤❡♦ ❇ỉ ➤Ị ♥➟✉ tr➟♥✱ t❛ ❝ã ∞ [nαn − (n − 1)αn−1 ]/φ(n) < ∞ n=1 ❙✉② r❛ ∞ P(|Xi | > n) < ∞ n=1 ❚✐Õ♣ t❤❡♦ t❛ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ị✉ ❦✐Ư♥ (b)✳ ❚õ ➤✐Ị✉ ❦✐Ư♥ {Xn }n ❦❤➯ tÝ❝❤ ➤Ò✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦✳ ❙✉② r❛✱ ✈í✐ ♠ä✐ ♥❣✉②➟♥ N1 > s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n (i) ✈➭ (ii) > 0✱ tå♥ t➵✐ sè t❤× n −1 E(|Xi |I(|Xi | > N1 ) < / n i=1 ❚✐Õ♣ t❤❡♦✱ tå♥ t➵✐ ♠ét sè ♥❣✉②➟♥ N > N1 s ỗ N E(|Xi |) < /2 n i=1 ❉♦ ➤ã✱ ✈í✐ ♠ä✐ n N ✱ t❛ ❝ã n E(|Xi |I(|Xi | > i) i=1 N1 n E(|Xi | > N1 )) < n E(|Xi |) + i=1 i=1 n s✉② r❛ N✱ ✸✺ ❚❛ ❝ã✱ ∞ ∞ −2 n−2 EXn2 I(|Xn | n var(Xn I(Xn ∈ [−n, n])) n=1 n)) n=1 ∞ ∞ −2 = n EXn2 I(|Xn | 1/4 0✳ ❉♦ ➤ã✱ ✈í✐ x ∈ [n1/4 , n] t❤× x 2nzn φ(x)/φ(xn ) φ(xn ) > φ(n1/4 ) ✈➭ zn x = n1/4 φ(n1/4 ) ✈í✐ n n1/4 ), 1✳ ❚❛ ❝ã ∞ ∞ −2 n E(Xn2 I(An )) n=1 E(φ(|Xn |))/tn n=1 ∞ [nαn − (n − 1)αn ]/tn =2 n=1 ✸✻ ∞ 1/tn < ∞✱ ♥➟♥ ➳♣ ❞ơ♥❣ ❇ỉ ➤Ị t❛ ➤➢ỵ❝ ❱× n=1 ∞ [nαn − (n − 1)αn ]/tn < ∞ n=1 ❙✉② r❛ ∞ n−2 E(Xn2 I(An )) < ∞ n=1 ❈✉è✐ ❝ï♥❣ t❛ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ị✉ ❦✐Ư♥ t0 > s❛♦ ❝❤♦ φ(t) t ✈í✐ t n n−1 E(|Xi |) t0 + c ✈í✐ t0 n ✈➭ (d)✳ ❱× t−1 φ(t) ↑ ∞ ❦❤✐ t ↑ ∞✱ tồ t |x| i=1 ị í ợ ❝❤ø♥❣ ♠✐♥❤✳ φ(|x|) ♥➟♥ |x| t0 + φ(|x|) s✉② r❛ ✸✼ ❑Õt ❧✉❐♥ ▲✉❐♥ ✈➝♥ ➤➲ t❤✉ ➤➢ỵ❝ ❝➳❝ ❦Õt q✉➯ ❝❤Ý♥❤ s❛✉✿ ✶✳ ❚r×♥❤ ❜➭② ❤Ư t❤è♥❣ ❝➳❝ ❦✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ✈Ò ❧ý t❤✉②Õt ①➳❝ s✉✃t ❧✐➟♥ q✉❛♥ ➤Õ♥ ➤Ị t➭✐ ❧✉❐♥ ✈➝♥✳ ✷✳ ❚r×♥❤ ❜➭② ✈Ị ♠ét sè ❧✉❐t ♠➵♥❤ sè ❧í♥ ➤è✐ ✈í✐ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➠♥❣ ➞♠✳ ✸✳ ❚r×♥❤ ❜➭② ✈Ị ♠ét sè ❧✉❐t ♠➵♥❤ sè ❧í♥ ➤è✐ ✈í✐ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣ ➤➠✐ ♠ét✳ ✹✳ ❚r×♥❤ ❜➭② ✈Ị ❦❤➳✐ ♥✐Ư♠ ❦❤➯ tÝ❝❤ ➤Ị✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦ ✈➭ ❧✉❐t ♠➵♥❤ sè ❧í♥ ➤è✐ ✈í✐ ❞➲② ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➯ tÝ❝❤ ➤Ị✉ t❤❡♦ ♥❣❤Ü❛ ❈❡s➭r♦✳ ❍➢í♥❣ ♣❤➳t tr✐Ĩ♥ ❝đ❛ ❧✉❐♥ ✈➝♥✿ ◆Õ✉ ❝ã ➤✐Ị✉ ❦✐Ư♥✱ sÏ t×♠ ❝➳❝❤ ♠ë ré♥❣ ❝➳❝ ❦Õt q✉➯ tr➟♥ ❝❤♦ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ♥❤❐♥ ❣✐➳ trÞ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❝ã t❤ø tù ✭❞➭♥ ❇❛♥❛❝❤✮✳ ✸✽ ❚➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ✶✳ ◆❣✉②Ơ♥ ❱➝♥ ◗✉➯♥❣ ✲ ◆❣✉②Ơ♥ ❱➝♥ ❍✉✃♥✱ ❈➡ së ①➳❝ s✉✃t ❤✐Ö♥ ➤➵✐✱ ◆❳❇ ➜➵✐ ❤ä❝ ❱✐♥❤ ✭✷✵✶✹✮✳ ✷✳ ❈❤❛♥❞r❛✱ ❚✳❑✳✱ ●♦s✇❛♠✐✱ ❆✳✱ ❈❡s➭r♦ ✉♥✐❢♦r♠ ✐♥t❡❣r❛❜✐❧✐t② ❛♥❞ t❤❡ str♦♥❣ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs✱ ❙❛♥❦❤②❛✱ ❙❡r ❆✳ ✺✹ ✭✶✾✾✷✮ ✷✶✺➊✷✸✶✳ ✸✳ ❈❤❛♥❞r❛✱ ❚✳❑✳✱ ●♦s✇❛♠✐✱ ❆✳✱ ❈❡s➭r♦ ❛✲✐♥t❡❣r❛❜✐❧✐t② ❛♥❞ ❧❛✇s ♦❢ ❧❛r❣❡ ♥✉♠✲ ❜❡rs ■✱ ❏✳ ❚❤❡♦r❡t✳ Pr♦❜❛❜✳ ✶✻ ✭✷✵✵✸✮ ✻✺✺➊✻✻✾✳ ✹✳ ❈❤❛♥❞r❛✱ ❚✳❑✳✱ ●♦s✇❛♠✐✱ ❆✳✱ ❈❡s➭r♦ ❛✲✐♥t❡❣r❛❜✐❧✐t② ❛♥❞ ❧❛✇s ♦❢ ❧❛r❣❡ ♥✉♠✲ ❜❡rs✳ ■■✱ ❏✳ ❚❤❡♦r❡t✳ Pr♦❜❛❜✳ ✶✾ ✭✷✵✵✻✮✱ ♥♦✳ ✹✱ ✼✽✾✲✽✶✻✳ ✺✳ ❈❤✉♥❣✱ ❑✳▲✳✱ ◆♦t❡ ♦♥ s♦♠❡ str♦♥❣ ❧❛✇s ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs✱ ❆♠❡r✳ ❏✳ ▼❛t❤✳ ✻✾✱ ✭✶✾✹✼✮✳ ✶✽✾✲✲✶✾✷✳ ✻✳ ❈s♦r❣♦✱ ❙✳✱ ❚❛♥❞♦r✐✱ ❑✳✱ ❚♦t✐❦✱ ❱✳✱ ❖♥ t❤❡ str♦♥❣ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❢♦r ♣❛✐r✇✐s❡ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✱ ❆❝t❛ ▼❛t❤✳ ❍✉♥❣❛r✳ ✹✷ ✭✶✾✽✸✮✱ ♥♦✳ ✸✲✹✱ ✸✶✾✲✸✸✵✳ ✼✳ ❊t❡♠❛❞✐✱ ◆✳✱ ❆♥ ❡❧❡♠❡♥t❛r② ♣r♦♦❢ ♦❢ t❤❡ str♦♥❣ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs✱ ❩✳ ❲❛❤rs❝❤✳ ❱❡r✇✳ ●❡❜✐❡t❡ ✺✺ ✭✶✾✽✶✮✱ ♥♦✳ ✶✱ ✶✶✾✲✶✷✷✳ ✽✳ ❊t❡♠❛❞✐✱ ◆✳✱ ❖♥ t❤❡ ❧❛✇s ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❢♦r ♥♦♥♥❡❣❛t✐✈❡ r❛♥❞♦♠ ✈❛r✐✲ ❛❜❧❡s✱ ❏✳ ▼✉❧t✐✈❛r✐❛t❡ ❆♥❛❧✳ ✶✸ ✭✶✾✽✸✮✱ ♥♦✳ ✶✱ ✶✽✼✲✶✾✸✳ ✾✳ ❘♦❜❡rts✱ ❆✳❲✳ ❛♥❞ ❱❛r❜❡r❣✱ ❉✳❊✱ ❈♦♥✈❡① ❋✉♥❝t✐♦♥s✱ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ◆❡✇ ❨♦r❦✱ ✶✾✼✸✳ ... ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥ ❤➡♥✳ ❱✐♥❤✱ t❤➳♥❣ ✽ ♥➝♠ ✷✵✶✻✳ ❚➳❝ ❣✐➯ ✸ ❈❤➢➡♥❣ ✶✳ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ị r ú t trì ột số ❦❤➳✐ ♥✐Ư♠ ✈➭ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t ✈➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✱ ❦ú ✈ä♥❣ ❝ñ❛ ❜✐Õ♥ ♥❣➱✉... s✉✃t✳ ❇é (Ω, F, P) ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t✳ ❚❐♣ σ Ω ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❜✐Õ♥ ❝è s số ỗ F A∈F ➤➢ỵ❝ ❣ä✐ ❧➭ σ ✲ ➤➵✐ sè ❝➳❝ ❜✐Õ♥ ❝è✳ ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❜✐Õ♥ ❝è✳ ❇✐Õ♥ ❝è Ω∈F ❇✐Õ♥ ❝è ∅∈F... ị ý ợ tỏ ề ❦✐Ư♥ s✉② r❛ ➤➢ỵ❝ ➤✐Ị✉ ❦✐Ư♥ ❍Ư q✉➯ ✷✳✶✳✸✳ varXi ✶✾ (d) 1✱ tå♥ t➵✐ ❞➲② ❦Ð♣ {ρij } ❝➳❝ số tự s n ỗ n n E |Sn − E(Sn )| ρij i=1 j=1 ✈➭ ∞ ∞ i=1 j=1 tr♦♥❣ ➤ã Bnc ρij < ∞, i ∨ j = max(i,

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