n k=1 ❙✐♥❝❡ t❤❡ ✇❡❛❦ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❤♦❧❞s ❢♦r s❡q✉❡♥❝❡ N (ε, δ ) n Qt (ξk Xk ) − Qt (E(ξ1 X1 )) > n ε n +P s✉❝❤ t❤❛t ❢♦r ❛❧❧ ε ε δ + 2 (Ut (ξn Xn ))✱ n ≥ N (ε, δ ) n Ut (ξk Xk ) − Ut (E(ξ1 X1 )) > k=1 ε ≤ δ ■t ❢♦❧❧♦✇s t❤❛t P ξk Xk − E(ξ1 X1 ) > ε ≤P P n n ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r k=1 ❝❤♦s❡♥ ✐♥ ✭✷✳✽✮ ❛♥❞ ✭✷✳✾✮ ❛♥❞ ❢♦r ❛❧❧ P ξk2 E Qt (E(ξ1 X1 )) < t E ξk Qt (Xk ) ❛♥❞ ❋r♦♠ ✭✷✳✼✮✱ ❢♦r t❤❡ ε n εn n n n ξk Xk − E(ξ1 X1 ) > ε < k=1 ✶✽ δ + δ =δ t❤❡r❡ ❡①✐sts ✷✳✷✳ ❲❡❛❦ ❧❛✇s ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ✇✐t❤ ❙❝❤❛✉❞❡r ❜❛s✐s n ≥ N (ε, δ )✳ ❢♦r ❛❧❧ ❙✐♥❝❡ δ ❝❛♥ ❜❡ ❛r❜✐tr❛r✐❧② s♠❛❧❧✱ ✇❡ ❤❛✈❡ n n P ξk Xk − → E(ξ1 X1 ) k=1 2.2.3 ❆❣❛✐♥✱ t❤❡ r❡s✉❧t ♦❢ t❤❡♦r❡♠ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ ❢♦r ❛❧❧ r❡❛❧ s❡♣❛r❛❜❧❡ ❇❛✲ ♥❛❝❤ s♣❛❝❡s ❜② ❡♠❜❡❞❞✐♥❣ ❡❛❝❤ s♣❛❝❡ ✐s♦♠♦r♣❤✐❝❛❧❧② ✐♥ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ C [0; 1]✳ ❚❤❡♦r❡♠ ✷✳✷✳✹ ✭❬✽❪✮✳ ▲❡t E ❜❡ ❛ r❡❛❧ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ❛♥❞ ❧❡t (Xn ) ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ r❛♥❞♦♠ ❡❧❡♠❡♥ts ✐♥ E s✉❝❤ t❤❛t E X1 < ∞✳ ▲❡t (ξn ) ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s s✉❝❤ t❤❛t n E ξk2 ≤ Γ n k=1 ❢♦r ❡❛❝❤ n ✇❤❡r❡ Γ ✐s ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t ❛♥❞ ❧❡t E(ξn Xn ) = E(ξ1 X1 ) ❢♦r ❡❛❝❤ n✳ ❋♦r ❡❛❝❤ f ∈ E ∗ t❤❡ ✇❡❛❦ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❤♦❧❞s ❢♦r t❤❡ s❡q✉❡♥❝❡ (f (ξn Xn )) ✐❢ ❛♥❞ ♦♥❧② ✐❢ n P n Pr♦♦❢✳ ❇② t❤❡♦r❡♠ 1.1.10✱ E ξk Xk − → E(A1 X1 ) k=1 C [0; 1]✳ ❍❡♥❝❡✱ E ✐♥t♦ C [0; 1]✳ ✐s ✐s♦♠❡tr✐❝ t♦ ❛ s✉❜s♣❛❝❡ ♦❢ ❡①✐sts ❛ ❜✐❥❡❝t✐✈❡✱ ❜✐❝♦♥t✐♥✉♦✉s✱ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ h ❢r♦♠ t❤❡r❡ 1.2.2✱ (h(ξn Xn )) ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ r❛♥❞♦♠ C [0; 1] ✇✐t❤ E h(Xi ) < ∞ ❛♥❞ ❇② ♣r♦♣♦s✐t✐♦♥ ❡❧❡♠❡♥ts ✐♥ Eξn h(Xn ) = Eh(ξn Xn ) = Eh(ξ1 X1 ) = Eξ1 h(X1 ) n✳ ❢♦r ❡❛❝❤ g (h(ξk Xk )) = k=1 ❚❤✉s✱ ❢♦r ❡❛❝❤ (g (h(ξn Xn )))✳ h ❙✐♥❝❡ g ∈ C [0; 1]∗ n n ▲❡t h g∈ n ❜❡ t❤❡ ❛❞❥♦✐♥t ❢✉♥❝t✐♦♥ ♦❢ h✱ ✇❡ ❤❛✈❡ h∗ (g (ξk Xk )) − → E[h∗ (g (ξ1 X1 ))] = E[g (h(ξ1 X1 ))] P k=1 C [0; 1]∗ ✱ t❤❡ ✇❡❛❦ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❤♦❧❞s ❢♦r t❤❡ s❡q✉❡♥❝❡ ❍❡♥❝❡✱ ❜② t❤❡♦r❡♠ h∗ n n ❛♥❞ ❧❡t n ξn Xk = k=1 2.2.3✱ n n P h(ξk Xk ) − → Eh(ξ1 X1 ) = hEX1 k=1 ✐s ❜✐❥❡❝t✐✈❡✱ ❜✐❝♦♥t✐♥✉♦✉s ❛♥❞ ❧✐♥❡❛r✱ n n P ξk Xk − → Eξ1 X1 k=1 ✶✾ ✷✳✸✳ ❲❡❛❦ ❧❛✇s ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ✐♥ ❘❛❞❡♠❛❝❤❡r t②♣❡ p ❇❛♥❛❝❤ s♣❛❝❡s ✷✳✸ ❲❡❛❦ ❧❛✇s ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ✐♥ ❘❛❞❡♠❛❝❤❡r t②♣❡ p ❇❛♥❛❝❤ s♣❛❝❡s ❉❡✜♥✐t✐♦♥ ✷✳✸✳✶✳ p ≤ 2✮ E ❆ ❇❛♥❛❝❤ s♣❛❝❡ ✐❢ t❤❡r❡ ✐s ❛ ❝♦♥st❛♥t C>0 s✉❝❤ t❤❛t ❢♦r ❡✈❡r② p n (εi ) p ✭1 ≤ x1 , , x n ∈ E ✱ xi p , i=1 i=1 ✇❤❡r❡ ❘❛❞❡♠❛❝❤❡r t②♣❡ n ≤C εi x i E ✐s s❛✐❞ t♦ ❤❛✈❡ ✐s ❛ ❘❛❞❡♠❛❝❤❡r s❡q✉❡♥❝❡✱ t❤❛t ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✇✐t❤ P[εi = 1] = P[εi = −1] = ❍♦✛♠❛♥♥✲❏ør❣❡♥s❡♥ ❛♥❞ P✐s✐❡r ✭❬✶✵❪✱ ❚❤❡♦r❡♠ ✷✳✶✮ ♣r♦✈❡❞ t❤❛t ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ✐s ♦❢ ❘❛❞❡♠❛❝❤❡r t②♣❡ p ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ (X n ) ♦❢ t❤ ♠♦♠❡♥t✱ t❤❡r❡ ❡①✐sts ❛ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ❡❧❡♠❡♥ts ✇✐t❤ ♠❡❛♥ ✵ ❛♥❞ ✜♥✐t❡ p ❝♦♥st❛♥t C>0 s✉❝❤ t❤❛t p n Xi p ≤C Xi E n i=1 i=1 ❯s✐♥❣ t❤❡ ❛❜♦✈❡ r❡s✉❧t✱ ✇❡ ❝❛♥ ❡st❛❜❧✐s❤ ❛ ✇❡❛❦ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❢♦r ❛ s❡✲ q✉❡♥❝❡ ♦❢ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ r❛♥❞♦♠ ❡❧❡♠❡♥ts t❛❦✐♥❣ ✈❛❧✉❡s p ✐♥ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ♦❢ t②♣❡ ✇✐t❤ ♠❡❛♥ ✵ ❛♥❞ ✜♥✐t❡ pt❤ ♠♦♠❡♥t✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ t❡❝❤♥✐❝❛❧ ❧❡♠♠❛ ✇✐❧❧ ❜❡ ♥❡❡❞❡❞ ❢♦r ♣r♦✈✐♥❣ t❤❛t✳ ▲❡♠♠❛ ✷✳✸✳✷✳ ▲❡t p ∈ [1; 2] ❛♥❞ ❧❡t k ❜❡ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳ ❚❤❡♥ ✭✐✮ ❋♦r ❛❧❧ r ∈ (0; p) p k ≤ r k p r ✭✐✐✮ ❋♦r ❛❧❧ i0 ∈ N✱ r ∈ p p i r −2 ≤ i=i0 r ∈ (0; p)✱ t❤❡ ❛❧❧ i ∈ {1, , k} ✭✐✮ ❙✐♥❝❡ ❍❡♥❝❡✱ ❢♦r i=1 ;p k Pr♦♦❢✳ p i r −1 ; p p p r k r −1 − (i0 − 1) r −1 p−r ❢✉♥❝t✐♦♥ i i r −1 = p f (x) = x r −1 i p i r −1 dx ≥ i−1 x r −1 dx i−1 ✷✵ p ✐s ✐♥❝r❡❛s✐♥❣ ♦♥ (0; ∞)✳ ✷✳✸✳ ❲❡❛❦ ❧❛✇s ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ✐♥ ❘❛❞❡♠❛❝❤❡r t②♣❡ p ❇❛♥❛❝❤ s♣❛❝❡s ■t ❢♦❧❧♦✇s t❤❛t k k i p −1 r ≥ x r∈ ❢♦r ❛❧❧ p ;p p −1 r k dx = i−1 i=1 i=1 ✭✐✐✮ ❙✐♥❝❡ i p f (x) = x r −2 ✱ t❤❡ ❢✉♥❝t✐♦♥ i ∈ {1, , k} i p ✐s ❞❡❝r❡❛s✐♥❣ ♦♥ i p i r −2 = p r p x r −1 dx = k r p (0; ∞)✳ ❍❡♥❝❡✱ p i r −2 dx ≤ x r −2 dx i−1 i−1 ■t ❢♦❧❧♦✇s t❤❛t k k i p −2 r i ≤ x i=i0 i=1 = p −2 r k p x r −2 dx dx = i0 −1 i−1 p p r k r −1 − (i0 − 1) r −1 p−r ❚❤❡♦r❡♠ ✷✳✸✳✸ ✭❬✶✶❪✮✳ ▲❡t E ❜❡ ❛ r❡❛❧ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ♦❢ t②♣❡ p ✭1 ≤ p ≤ 2✮ ❛♥❞ ❧❡t (Xn ) ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ ✐♥❞❡♣❡♥❞❡♥t ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ r❛♥❞♦♠ ❡❧❡♠❡♥ts ✇✐t❤ ♠❡❛♥ ✵ ❛♥❞ ✜♥✐t❡ pt❤ ♠♦♠❡♥t✳ ❋♦r r ∈ (0; p)✱ ✐❢ X1 > n r → nP t❤❡♥ n Pr♦♦❢✳ n P Xi − → r i=1 n ▲❡t Yk = Xk I ✱ Xk ≤n r n Zn = Yk ✱ S n = k=1 ❋♦r ❡❛❝❤ ❛♥❞ Bn = [Sn = Zn ]✳ k=1 ε > 0✱ P Xk n− r Sn > ε ≤ P(Bn )P ≤P ≤ε −p n− r Sn > ε Bn + P(Bnc )P n− r Sn > ε Bnc n− r Zn > ε + P(Bnc ) E n − r1 n p Zn + P Xk > n r k=1 n ≤ε −p − pr n p n E Xk I k=1 Xk ≤n r + P k=1 ✷✶ Xk > n r ✷✳✸✳ ❲❡❛❦ ❧❛✇s ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ✐♥ ❘❛❞❡♠❛❝❤❡r t②♣❡ p ❇❛♥❛❝❤ s♣❛❝❡s p ≤ ε−p n1− r p Xk dP + nP X1 > n r Xk ≤n r ▲❡t p In = ε−p n1− r ❲❡ ♥❡❡❞ t♦ s❤♦✇ t❤❛t In → n → ∞ ❛s Xk Xk ≤n r p dP ■♥❞❡❡❞✱ n In = ε −p 1− pr X1 p dP k r P [k − < X1 r ≤ k] n k−1< X1 k=1 n ≤k p p ≤ ε−p n1− r k=1 n ≤ε r k p −p 1− pr i r −1 n P [k − < X1 p p ≤ k] ✭▲❡♠♠❛ i=1 k=1 n r i r −1 P [i − < X1 ≤ ε−p n1− r r ≤ n] i=1 n p p ≤ ε−p n1− r i r −2 iP [ X1 r > i − 1] i=1 ❋♦r r ∈ 0; ❢♦r ❛❧❧ p ✱ ✇❡ ❤❛✈❡ p p i r −2 ≤ n r −2 i ∈ {1, , n}❀ ❛♥❞ s✐♥❝❡ lim nP [ X1 r > n − 1] = 0, n→∞ ❜② ❙t♦❧③✲❈❡sàr♦ t❤❡♦r❡♠ ✭❬✹❪✱ ❚❤❡♦r❡♠ ✷✳✼✳✷✮ ✇❡ ♦❜t❛✐♥ n r i P [ X1 lim > i − 1] i=1 ❍❡♥❝❡✱ = n n→∞ n In ≤ ε −p −1 iP [ X1 n r > i − 1] → i=1 ❋♦r r∈ p ;p ✳ ❙✐♥❝❡ lim iP [ X1 i→∞ ❢♦r ❡❛❝❤ δ > 0✱ r > i − 1] = 0, ✇❡ ❝❛♥ ✜♥❞ ❛♥ ✐♥t❡❣❡r i0 s✉❝❤ t❤❛t ❢♦r ❛❧❧ iP [ X1 r > i − 1] < δ ✷✷ i > i0 ✱ 2.3.2(i)) ✷✳✸✳ ❲❡❛❦ ❧❛✇s ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ✐♥ ❘❛❞❡♠❛❝❤❡r t②♣❡ p ❇❛♥❛❝❤ s♣❛❝❡s ❍❡♥❝❡✱ i0 In ≤ ε −p 1− pr n i n p −1 r P [ X1 r > i − 1] + ε −p 1− pr n i=1 i=i0 +1 i0 ≤ε p −p 1− pr i r −1 P [ X1 n p i r −2 δ r p > i − 1] + ε−p n1− r δ i=1 p p r −1 n r −1 − i0r p−r (▲❡♠♠❛ 2.3.2(ii)) i0 ≤ε −p 1− pr i n p −1 r P [ X1 i=1 ❛s → ε−p δr p−r n → ∞✳ ❙✐♥❝❡ δ ❚❤❡r❡❢♦r❡✱ ❢♦r ❡❛❝❤ r p p −1 ε−p δr > i − 1] + − n1− r i0r p−r ❝❛♥ ❜❡ ❛r❜✐tr❛r✐❧② s♠❛❧❧✱ ✇❡ ❤❛✈❡ In → ε > 0✱ lim P n→∞ n− r S n > ε = ✷✸ ❛s n → ∞✳ ❈❍❆P❚❊❘ ✸ ❲❊❆❑ ▲❆❲❙ ❖❋ ▲❆❘●❊ ◆❯▼❇❊❘❙ ❋❖❘ ❇❆◆❆❈❍ ❙P❆❈❊✲❱❆▲❯❊❉ ▼❆❘❚■◆●❆▲❊❙ ✸✳✶ ❙♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ ❇❛♥❛❝❤ s♣❛❝❡ ✈❛❧✉❡❞ ♠❛r✲ t✐♥❣❛❧❡s ❚❤❡♦r❡♠ ✸✳✶✳✶ ✭❬✶✷❪✮✳ ▲❡t (Xn ) ⊂ L2 (A, E ) ❜❡ ❛ ♠❛rt✐♥❣❛❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ (An ) ♦❢ s✉❜ σ✲❛❧❣❡❜r❛s ♦❢ A✳ ❚❤❡♥✱ ❢♦r ❡✈❡r② ♣♦s✐t✐✈❡ ♥✉♠❜❡r δ ✱ ✇❡ ❤❛✈❡ P sup Xn > δ ≤ n Pr♦♦❢✳ δ2 sup Xn 22 n ▲❡t Bn = ω∈Ω Xi (ω ) ≤ δ, i ∈ {1, , n − 1}, Xn > δ ❛♥❞ Ωδ = ω ∈ Ω sup Xn (ω ) > δ n ∞ ❚❤❡ s❡t Bn ❛r❡ ♣❛✐r✇✐s❡ ❞✐s❥♦✐♥t✱ Bn ∈ An ❢♦r ❛❧❧ n Bn = Ωδ ✳ ❛♥❞ n=1 ❲❡ ❤❛✈❡ ∞ ∞ δ P(Ωδ ) = δ P =δ Bk k=1 P(Bk ) k=1 ∞ n ≤ Xk k=1 dP = sup n Bk Xk k=1 n Ak = sup E Xn k=1 n Xn k=1 2 EAk IBk Xn dP = sup n Bk ≤ sup n dP Bk n n k=1 dP = sup n Bk ✷✹ ∞ k=1 Ω Xn Bk dP dP ✸✳✶✳ ❙♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ ❇❛♥❛❝❤ s♣❛❝❡ ✈❛❧✉❡❞ ♠❛rt✐♥❣❛❧❡s ≤ sup Xn n dP = sup Xn 22 n Ω ■t ❢♦❧❧♦✇s t❤❛t P sup Xn > δ ≤ n δ2 sup Xn 22 n ❈♦r♦❧❧❛r② ✸✳✶✳✷ ✭❬✶✷❪✮✳ ❆ss✉♠❡ t❤❛t t❤❡ ❤②♣♦t❤❡s✐s ✐♥ t❤❡♦r❡♠ 3.1.1 ✐s ❤♦❧❞✳ ❚❤❡♥✱ ❢♦r ❡✈❡r② m ∈ N✱ PAm sup Xn > δ dP ≤ n Ω Pr♦♦❢✳ sup Xn 22 δ2 n ❚❤❡ ♣r♦♦❢ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t PAm sup Xn > δ dP = P (Ω ∩ Ωδ ) = P(Ωδ ) n Ω ❈♦r♦❧❧❛r② ✸✳✶✳✸ ✭❬✶✷❪✮✳ ▲❡t (Xn ) ⊂ L2 (A, E ) ❜❡ ❛ ♠❛rt✐♥❣❛❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ (An ) ♦❢ s✉❜ σ✲❛❧❣❡❜r❛s ♦❢ A✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡ ✭✐✮ ■❢ t❤❡ s❡q✉❡♥❝❡ (Xn ) ✐s ❜♦✉♥❞❡❞✱ t❤❡♥ ✐t ✐s ❜♦✉♥❞❡❞ ✐♥ t❤❡ ♥♦r♠ ❛❧♠♦st s✉r❡❧②❀ ✭✐✐✮ ■❢ t❤❡ s❡q✉❡♥❝❡ (Xn ) ❝♦♥✈❡r❣❡s ✐♥ ❛❧♠♦st s✉r❡❧②✳ Pr♦♦❢✳ ✭✐✮ ❙✐♥❝❡ t❤❡ s❡q✉❡♥❝❡ t✐✈❡ ♥✉♠❜❡r C 0✱ L2 (A, E )✱ s✉❝❤ t❤❛t sup Xn ❋♦r ❡✈❡r② t❤❡♥ ✐t ❝♦♥✈❡r❣❡s ✐♥ t❤❡ ♥♦r♠ ❜② t❤❡♦r❡♠ 3.1.1✱ 2 < C ✇❡ ❤❛✈❡ P sup Xn > δ ≤ n δ2 C ■t ❢♦❧❧♦✇s t❤❛t P sup Xn = +∞ = n ❙♦✱ t❤❡r❡ ❡①✐sts M >0 s✉❝❤ t❤❛t sup Xn ≤ M n ✷✺ ❛✳s✳ t❤❡r❡ ❡①✐sts ❛ ♣♦s✐✲ ✸✳✶✳ ❙♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ ❇❛♥❛❝❤ s♣❛❝❡ ✈❛❧✉❡❞ ♠❛rt✐♥❣❛❧❡s (Xn ) ✐s ❛ ❈❛✉❝❤② s❡q✉❡♥❝❡ s❡q✉❡♥❝❡ ✐♥ n ∈ N✱ t❤❡♥ t❤❡ s❡q✉❡♥❝❡ (Xm − Xn )m≥n ✐s ❛ ♠❛rt✐♥❣❛❧❡ (Am )m≥n ✳ ❋♦r ❡✈❡r② ♣♦s✐t✐✈❡ ♥✉♠❜❡r δ ✱ ✉s✐♥❣ t❤❡♦r❡♠ 3.1.1✱ ✭✐✐✮ ❲❡ ♥❡❡❞ t♦ s❤♦✇ t❤❛t t❤❡ s❡q✉❡♥❝❡ E✳ ▲❡t ✉s ✜① ❛♥ ✇✐t❤ r❡s♣❡❝t t♦ ✇❡ ♦❜t❛✐♥ P sup Xm − Xn > δ ≤ m≥n δ2 sup Xm − Xn 22 m≥n ■t ❢♦❧❧♦✇s t❤❛t lim P sup Xm − Xn ≤ δ = n→∞ ❍❡♥❝❡✱ t❤❡ s❡q✉❡♥❝❡ (X n ) m≥n E✳ ✐s ❛❧♠♦st s✉r❡❧② ❈❛✉❝❤② ✐♥ ❚❤❡♦r❡♠ ✸✳✶✳✹ ✭❬✶✷❪✮✳ ▲❡t E ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❛♥❞ (An ) ✐s ❛♥ ✐♥❝r❡❛s✐♥❣ s❡✲ q✉❡♥❝❡ ♦❢ s✉❜ σ✲❛❧❣❡❜r❛s ♦❢ A✳ ❆ss✉♠❡ t❤❛t (Xn ) ⊂ L2 (A; E ) ✐s ❛ ♠❛rt✐♥❣❛❧❡ ✇✐t❤ r❡s♣❡❝t t♦ (An )✳ ❚❤❡♥✱ ❢♦r ❡✈❡r② ♣♦s✐t✐✈❡ ♥✉♠❜❡r δ✱ ✇❡ ❤❛✈❡ P sup n n Pr♦♦❢✳ n 1 Xi > δ ≤ δ2 i=1 sup Xn 22 n ▲❡t Ωδ = n ω ∈ Ω sup n n X i (ω ) > δ i=1 ❛♥❞ Bk = m ω∈Ω Xi (ω ) ≤ δ, m ∈ {1, , k − 1}, m k X i (ω ) > δ k i=1 i=1 ∞ ❚❤❡ s❡ts Bk Bn = Ωδ ✳ ❛r❡ ♣❛✐r✇✐s❡ ❞✐s❥♦✐♥t ❛♥❞ n=1 ❲❡ ❤❛✈❡ δ P sup n n ∞ ∞ n 2 Xi > δ = δ P =δ Bn P(Bn ) n=1 n=1 i=1 ✭✸✳✶✮ ❛♥❞ ❜② ❈❛✉❝❤②✲❙❝❤✇❛r③ ✐♥❡q✉❛❧✐t②✱ δ < n2 n ≤ Xi i=1 n ≤ Xi n2 i=1 n n Xi i=1 ❈♦♠❜✐♥❡ ✭✸✳✶✮ ❛♥❞ ✭✸✳✷✮✱ ✇❡ ♦❜t❛✐♥ ∞ δ P(Ωδ ) ≤ n=1 Bn n n n Xi i=1 dP = sup n ✷✻ k=1 Bk k k Xi i=1 dP ✭✸✳✷✮ ✸✳✷✳ ❲❡❛❦ ❧❛✇s ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❢♦r ❇❛♥❛❝❤ s♣❛❝❡✲✈❛❧✉❡❞ ♠❛rt✐♥❣❛❧❡s n = sup n Bk k=1 n ≤ sup n n Ai E Xn k Xn n n k=1 n k=1 Bk dP Xn k dP = sup dP = sup Bk k=1 i=1 Xn ≤ sup n k k EAi IBk Xn dP i=1 Bk dP = sup Xn 22 n Ω ❍❡♥❝❡✱ P sup n n n Xi > δ ≤ i=1 δ2 sup Xn 22 n ✸✳✷ ❲❡❛❦ ❧❛✇s ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❢♦r ❇❛♥❛❝❤ s♣❛❝❡✲ ✈❛❧✉❡❞ ♠❛rt✐♥❣❛❧❡s ❋✐rst✱ ✇❡ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛✳ ▲❡♠♠❛ ✸✳✷✳✶ ✭❬✶✷❪✮✳ ▲❡t (Xn ) ⊂ L2 (A, E ) ❜❡ ❛ ♠❛rt✐♥❣❛❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ (An ) ♦❢ s✉❜ σ✲❛❧❣❡❜r❛s ♦❢ A✳ ❚❤❡♥✱ t❤❡ s❡q✉❡♥❝❡ ( Xn ) ✐s ❛ s✉❜♠❛rt✐♥❣❛❧❡ ✇✐t❤ r❡s♣❡❝t t♦ (An )✳ Pr♦♦❢✳ ❋♦r n ≥ m✱ ✐t ❢♦❧❧♦✇s ❢r♦♠ ♣r♦♣♦s✐t✐♦♥ 1.4.4(vii) t❤❛t EAm Xn ≤ EAm Xn ❍❡♥❝❡✱ Xm ≤ EAm Xn ❚❤❡r❡❢♦r❡✱ ( Xn ) ✐s ❛ s✉❜♠❛rt✐♥❣❛❧❡ ✇✐t❤ r❡s♣❡❝t t♦ ●✐✈❡♥ ❛ s❡q✉❡♥❝❡ (X n ) (An )✳ ♦❢ r❛♥❞♦♠ ❡❧❡♠❡♥ts✱ ✇❡ ❞❡♥♦t❡ Xn∗ = max i∈{1, ,n} Xi ❚❤❡♦r❡♠ ✸✳✷✳✷ ✭❬✶✷❪✮✳ ▲❡t E ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❛♥❞ (An ) ✐s ❛♥ ✐♥❝r❡❛s✐♥❣ s❡✲ q✉❡♥❝❡ ♦❢ s✉❜ σ✲❛❧❣❡❜r❛s ♦❢ A✳ ❆ss✉♠❡ t❤❛t (Xn ) ⊂ L2 (A, E ) ✐s ❛ ♠❛rt✐♥❣❛❧❡ ✇✐t❤ r❡s♣❡❝t t♦ (An ) ❛♥❞ Xn > ❢♦r ❛❧❧ n✳ ■❢ ❢♦r ❡✈❡r② ♣♦s✐t✐✈❡ ♥✉♠❜❡r ε✱ ✇❡ ❤❛✈❡ lim E I[X ≥ε] Xn = 0✱ t❤❡♥ t❤❡ s❡q✉❡♥❝❡ ( Xn ) ♦❜❡②s t❤❡ ✇❡❛❦ ❧❛✇ ♦❢ ❧❛r❣❡ n→∞ ♥✉♠❜❡rs ✐♥ t❤❡ ♠❛①✲❝♦♥✈♦❧✉t✐♦♥✱ ✐✳❡✳✱ ∗ n lim P n→∞ max i∈{1, ,n} ✷✼ Xi < ε = ✸✳✷✳ ❲❡❛❦ ❧❛✇s ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❢♦r ❇❛♥❛❝❤ s♣❛❝❡✲✈❛❧✉❡❞ ♠❛rt✐♥❣❛❧❡s Pr♦♦❢✳ (An )✳ ❇② ❧❡♠♠❛ 3.2.1✱ t❤❡ s❡q✉❡♥❝❡ ❋♦r ❡✈❡r② ♣♦s✐t✐✈❡ ♥✉♠❜❡r ❛♥❞ s❡t k=n ε✱ ❧❡t ( Xn ) τ ✐❢ t❤❡r❡ ❡①✐sts ♥♦ s✉❝❤ ♥✉♠❜❡r s✉❜♠❛rt✐♥❣❛❧❡✱ ❢♦r ❛❧❧ ✐s ❛ s✉❜♠❛rt✐♥❣❛❧❡ ✇✐t❤ r❡s♣❡❝t t♦ ❜❡ t❤❡ ✜rst ♥✉♠❜❡r k✳ k ❙✐♥❝❡ t❤❡ s❡q✉❡♥❝❡ n≥τ Xτ ≤ EAτ Xn ❙♦✱ Xτ I[Xτ∗ ≥ε] ≤ EAτ Xn I[Xn∗ ≥ε] ■t ❢♦❧❧♦✇s t❤❛t P [ X τ ≥ ε] ≤ E ε ❙✐♥❝❡ Xτ I[Xτ∗ ≥ε] ≤ ε E Xn I[Xn∗ ≥ε] [ X τ ≥ ε] = [ X n ≥ ε] ✱ P [ X n ≥ ε] ≤ E ε ❍❡♥❝❡✱ Xn I[Xn∗ ≥ε] P [ Xn < ε] = − P [ Xn ≥ ε] ≥ − E ε ❯s✐♥❣ t❤❡ ❤②♣♦t❤❡s✐s lim E I[Xn∗ ≥ε] Xn n→∞ = 0✱ lim P [Xn∗ < ε] ≥ n→∞ ❙✐♥❝❡ P [Xn∗ < ε] ≤ ❢♦r ❛❧❧ n✱ lim P n→∞ max i∈{1, ,n} ✷✽ Xn I[Xn∗ ≥ε] ✇❡ ♦❜t❛✐♥ Xi < ε = Xn ≥ ε ( Xn ) ✐s ❛ s✉❝❤ t❤❛t ❇■❇▲■❖●❘❆P❍❨ ❬✶❪ N N Vahani , V I Tarieladze, S A Qoban n✱ Vero tnostnye Raspredeleni v Banahovyh Prostranstvah✱ Moskva «Nauka»✱ ✶✾✽✺✳ ❬✷❪ ❱✳ ■✳ ❇♦❣❛❝❤❡✈✱ ▼❡❛s✉r❡ ❚❤❡♦r② ✭❱♦❧✉♠❡ ■✮✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ❇❡r❧✐♥ ❍❡✐❞❡❧✲ ❜❡r❣✱ ✷✵✵✼✳ ❬✸❪ ◆✳ ▲✳ ❈❛r♦t❤❡rs✱ ❆ ❙❤♦rt ❈♦✉rs❡ ♦♥ ❇❛♥❛❝❤ ❙♣❛❝❡ ❚❤❡♦r②✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✲ ✈❡rs✐t② Pr❡ss✱ ✷✵✵✺✳ ❬✹❪ ❆✳❉✳❘✳ ❈❤♦✉❞❛r②✱ ❈♦♥st❛♥t✐♥ P✳ ◆✐❝✉❧❡s❝✉✱ ❘❡❛❧ ❆♥❛❧②s✐s ♦♥ ■♥t❡r✈❛❧s✱ ❙♣r✐♥❣❡r ■♥❞✐❛✱ ✷✵✶✹✳ ❬✺❪ ❇✳ ❱✳ ●♥❡❞❡♥❦♦✱ ❬✻❪ ▼✳ ▲♦❡✈❡✱ ❚❤❡ ❚❤❡♦r② ♦❢ Pr♦❜❛❜✐❧✐t②✱ ▼✐r P✉❜❧✐s❤❡rs✱ ▼♦s❝♦✇✱ ✶✾✼✽✳ Pr♦❜❛❜✐❧✐t② ❚❤❡♦r② ■✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ◆❡✇ ❨♦r❦✱ ✶✾✼✼✳ ❬✼❪ ❘♦❜❡rt ❊✳ ▼❡❣❣✐♥s♦♥✱ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❇❛♥❛❝❤ ❙♣❛❝❡ ❚❤❡♦r②✱ ❙♣r✐♥❣❡r✲ ❱❡r❧❛❣ ◆❡✇ ❨♦r❦✱ ✶✾✾✽✳ ▲❛✇s ♦❢ ▲❛r❣❡ ◆✉♠❜❡rs ❢♦r ◆♦r♠❡❞ ▲✐♥❡❛r ❙♣❛❝❡s ❛♥❞ ❈❡rt❛✐♥ ❋r❡❝❤❡t ❙♣❛❝❡s✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤❡♠❛t✐❝s ✸✻✵✱ ❙♣r✐♥❣❡r✲ ❬✽❪ ❲✳ ❏✳ P❛❞❣❡tt✱ ❘✳ ▲✳ ❚❛②❧♦r✱ ❱❡r❧❛❣ ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣ ◆❡✇ ❨♦r❦✱ ✶✾✼✸✳ ❬✾❪ ❲❛❧t❡r ❘✉❞✐♥✱ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ✭✷♥❞ ❡❞✳✮✱ ▼❝●r❛✇✲❍✐❧❧✱ ✶✾✾✶✳ ❚❤❡ ▲❛✇ ♦❢ ▲❛r❣❡ ◆✉♠❜❡rs ❛♥❞ ❚❤❡ ❈❡♥✲ tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠ ✐♥ ❇❛♥❛❝❤ ❙♣❛❝❡s✱ ❚❤❡ ❆♥♥❛❧s ♦❢ Pr♦❜❛❜✐❧✐t②✱ ❱♦❧✳ ✹✱ ◆♦✳ ❬✶✵❪ ❏✳ ❍♦✛♠❛♥♥✲❏ør❣❡♥s❡♥✱ ●✳ P✐s✐❡r✱ ✹ ✭✶✾✼✻✮✱ ✺✽✼✲✺✾✾✳ ❙t❛❜❧❡ ♠❡❛s✉r❡s ❛♥❞ ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠s ✐♥ s♣❛❝❡s ♦❢ st❛❜❧❡ t②♣❡✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✷✺✶ ✭✶✾✼✾✮✱ ✼✶✲✶✵✶✳ ❬✶✶❪ ▼❛r❝✉s✱ ▼✳✱ ❲♦②❝③②♥s❦✐✱ ❲✳✱ ❬✶✷❪ ❈❛♦ ❱❛♥ ◆✉♦✐✱ ▲❡ ❚✉ ◆❛♠ ▲♦♥❣✱ ❚❤❡ ▲❛✇ ♦❢ ▲❛r❣❡ ◆✉♠❜❡rs ❢♦r ▼❛rt✐♥❣❛❧❡s ✐♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡ ✐♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙❝✐❡♥❝❡✱ ❱♦❧✳ ✷✱ ■ss✉❡ ✶ ✭✷✵✶✺✮✱ ✶✵✻✲✶✶✹✳ ✷✾ ...THE UNIVERSITY OF DANANG UNIVERSITY OF EDUCATION FACULTY OF MATHEMATICS DUONG PHUOC LUAN WEAK LAWS OF LARGE NUMBERS IN BANACH SPACES UNDERGRADUATE THESIS Supervisor:... s♣❛❝❡s p ≤ ε−p n1− r p Xk dP + nP X1 > n r Xk ≤n r ▲❡t p In = ε−p n1− r ❲❡ ♥❡❡❞ t♦ s❤♦✇ t❤❛t In → n → ∞ ❛s Xk Xk ≤n r p dP ■♥❞❡❡❞✱ n In = ε −p 1− pr X1 p dP k r P [k − < X1 r ≤ k] n k−1< X1 k=1... ❙t♦❧③✲❈❡sàr♦ t❤❡♦r❡♠ ✭❬✹❪✱ ❚❤❡♦r❡♠ ✷✳✼✳✷✮ ✇❡ ♦❜t❛✐♥ n r i P [ X1 lim > i − 1] i=1 ❍❡♥❝❡✱ = n n→∞ n In ≤ ε −p −1 iP [ X1 n r > i − 1] → i=1 ❋♦r r∈ p ;p ✳ ❙✐♥❝❡ lim iP [ X1 i→∞ ❢♦r ❡❛❝❤ δ > 0✱ r >