về một dạng hội tụ của dãy và chuỗi

N d N d = {n = (n 1 , n 2 , , n d ) : n i ∈ N, i = 1, 2, , d}. {X(n), n ∈ N d } d {X(n), n ∈ N d }  n∈N d X(n) max 1id n i = ∨n i → ∞ d N d = {n = (n 1 , n 2 , , n d ) : n i ∈ N, i = 1, 2, , d}. N d m = (m 1 , m 2 , , m d ), n = (n 1 , n 2 , , n d ) N d m  n ⇐⇒ m i  n i , i = 1, . . . , d. N d I d = {i = (i, i, . . . , i) : i ∈ N}. n = (n 1 , n 2 , . . . , n d ) ∨n i = max 1id n i ; ∧n i = min 1id n i |n| = n 1 .n 2 . . . n d . N d d d d ∨n i → ∞ ∧n i → ∞ d ∧n i → ∞ d ∨n i → ∞ N d (d > 1) d d {x(n), n ∈ N d } d {x(n), n ∈ N d } x ∈ R ∨n i → ∞ ∧n i → ∞) ε > 0 n 0 ∈ N n 0 ∈ N d n = (n 1 , n 2 , . . . , n d ) ∈ N d ∨n i ≥ n 0 n ≥ n 0 |x(n) − x| < ε lim ∨n i →∞ x(n) = x lim ∧n i →∞ x(n) = x). {x(n), n ∈ N d } d {x(n), n ∈ N d } x ∈ R | n| → ∞ ε > 0 n 0 ∈ N n = (n 1 , n 2 , . . . , n d ) ∈ N d |n| ≥ n 0 |x(n) − x| < ε. lim |n|→∞ x(n) = x. n ≥ n 0 |n| ≥ |n 0 | ∨n i → ∞ |n| → ∞ lim |n|→∞ x(n) = x ⇔ lim ∨n i →∞ x(n) = x ⇒ lim ∧n i →∞ x(n) = x. {x(n), n ∈ N d } d  n∈N d x(n) (1) d S(n) =  mn x(m) S(n) n ∨n i → ∞ {S(n), n ∈ N d } ∨n i → ∞ S = lim ∨n i →∞ S(n) =  n∈N d x(n) r(n) = S − S(n) =  m:mn x(m) n lim ∨n i →∞ r(n) = 0 {x(n), n ∈ N d } {y(n), n ∈ N d } d x(n) → x y(n) → y ∨n i → ∞ (∧n i → ∞) x(n) + y(n) → x + y ∨n i → ∞ (∧n i → ∞). x(n) → x ∨n i → ∞ (∧n i → ∞) |x(n)| → |x| ∨n i → ∞ (∧n i → ∞). x(n) → x ∨n i → ∞ (∧n i → ∞) λx(n) → λx ∨n i → ∞ (∧n i → ∞) λ ∈ C.  n∈N d x(n) x ∈ R  n∈N d y(n) y ∈ R ∨n i → ∞  n∈N d (x(n) + y(n)) x + y ∨n i → ∞.  n∈N d λx(n), λ ∈ C λx ∨n i → ∞. {x(n), n ∈ N d } d {x(n), n ∈ N d } M > 0 |x(n)|  M, n ∈ N d . {x(n), n ∈ N d } d {x(n), n ∈ N d } x ∈ R ∨n i → ∞ {x(n), n ∈ N d } {x(n), n ∈ N d } x ∈ R ∧n i → ∞ {x(n), n ∈ N d } {x(m, n)} x(m, n) =  m, n = 1 0, n = 1. x(m, n) → 0 m ∧ n → ∞ d {x(n), n ∈ N d } {x(n), n ∈ N d } x(n) ≥ x(m) x(n)  x(m) ∨n i ≥ ∨m i {x(n), n ∈ N d } ∨n i → ∞ lim ∨n i →∞ x(n) = lim i→∞ x(i) (i ∈ I d ). {x(n), n ∈ N d } {x(i), i ∈ I d } x(n), n ∈ N d } {x(n), n ∈ N d } {x(i), i ∈ I d } x i → ∞ {x(n), n ∈ N d } x ∨n i → ∞ ε > 0 n 0 ∈ N 0  x − x(i) < ε (i ∈ I d ) i ≥ n 0 n ∨n i = k > n 0 x(n 0 )  x(n)  x(k), (n 0 , k ∈ I d ) 0  x − x(k)  x − x(n)  x − x(n 0 ) < ε. lim ∨n i →∞ x(n) = x = lim i→∞ x(i) (i ∈ I d ). {x(n), n ∈ N d } {−x(n), n ∈ N d }  {x(n), n ∈ N d } ∨n i → ∞ ε > 0 n o ∈ N m, n ∈ N d ∨m i ≥ n 0 , ∨n i ≥ n 0 |x(m) − x(n)| < ε. {x(n), n ∈ N d } x ∈ R, ∨n i → ∞, {x(n), n ∈ N d } {x(n), n ∈ N d } x ∈ R ∨n i → ∞ ε > 0 n 0 ∈ N n ∈ N d ∨n i ≥ n 0 |x(n) − x| < ε 2 . m, n ∈ N d ∨m i ≥ n 0 , ∨n i ≥ n 0 |x(m) − x(n)| = |(x(m) − x) − (x(n) − x)|  |x(m) − x| + |x(n) − x| < ε 2 + ε 2 = ε {x(i), i ∈ I d } {x(n), n ∈ N d }. {x(i), i ∈ I d } {x(i), i ∈ I d } x ∈ R i → ∞ {x(n), n ∈ N d } ε > 0 n 0 ∈ N n ∈ N d , i ∈ I d ∨n i ≥ n 0 , i ≥ n 0 |x(n) − x|  |x(n) − x(i)| + |x(i) − x| < ε {x(n), n ∈ N d } x ∈ R ∨n i → ∞.  d {X(n), n ∈ N d } d (Ω, F, P) {X(n)} {X(n), n ∈ N d } I d , {X(i), i ∈ I d } {X(i)}. {X(n)} X ∨n i → ∞, ε > 0 lim ∨n i →∞ P(|X(n) − X| > ε) = 0 X(n) P −→ X ∨n i → ∞. {X(n)} X ∨n i → ∞ A 0 X(n)(ω) → X(ω) ∨n i → ∞ ω /∈ A. X(n) −→ X, ∨n i → ∞. {X( n)} p (0 < p < ∞) ∨n i → ∞ lim ∨n i →∞ E|X(n) − X| p = 0. X(n) L p → X , ∨n i → ∞. {A(n)} d {A(n)} A(m) ⊂ A(n) ∨m i  ∨n i P(  n∈N d A(n)) = lim ∨n i →∞ P(A(n)). {A(n)} A(m) ⊃ A(n) ∨m i  ∨n i P(  n∈N d A(n)) = lim ∨n i →∞ P(A(n)). {A(n)} {A(n)} I d {A(i)}. {A(i)} {A(i)} P(  i∈I d A(i)) = lim i→∞ P(A(i)).  n∈N d A(n) =  i∈I d A(i).  n∈N d A(n) ⊃  i∈I d A(i) ω ∈  n∈N d A(n) n 0 = (n 01 , n 02 , . . . , n 0d ) ∈ N d ω ∈ A(n 0 ). i 0 = (m 0 , m 0 , . . . , m 0 ) ∈ I d m 0 = max 1id n 0i . ∨n 0i  ∨m o = m 0 A(n 0 ) ⊂ A(i 0 ), ω ∈ A(i 0 ) ω ∈  i∈I d A(i)  n∈N d A(n) ⊂  i∈I d A(i  n∈N d A(n) =  i∈I d A(i). P(  n∈N d A(n)) = lim i→∞ P(A(i)). {P(A(n))} lim ∨n i →∞ P(A(n)) = lim i→∞ P(A(i)) = P(  n∈N d A(n)).  d {A(n), n ∈ N d } d  n∈N d P(A(n)) < ∞ P( lim ∨n i →∞ sup n A(n)) = 0.  n∈N d P(A(n)) = ∞ {A(n), n ∈ N d } P( lim ∨n i →∞ sup n A(n)) = 1. lim ∨n i →∞ sup n A(n) =  n∈N d  mn A(m). B(1) =  m1 A(m), . . . , B(k) =  mk A(m), . . . {B(k), k ∈ N d } P( lim ∨n i →∞ sup n A(n)) = lim k→∞ P(  nk A(n))  lim k→∞  nk P(A(n)) = 0. {A(n), n ∈ N d } { ¯ A(n), n ∈ N d } P(  nk ¯ A(n)) =  nk P( ¯ A(n)) =  nk (1 − P(A(n)))   nk e −P(A(n)) = e − P nk P(A(n)) = e −∞ = 0 P(  nk A(n)) = 1 P( lim ∨n i →∞ sup n A(n)) = 1  X(n) L p −→ X, X(n) −→ X ∨n i → ∞ X(n) P −→ X ∨n i → ∞. X(n) L p −→ X, ∨n i → ∞ ε > 0 0  P(|X(n) − X| > ε)  E|X(n) − X| p ε p lim ∨n i →∞ E|X(n) − X| p = 0 lim ∨n i →∞ P(|X(n) − X| > ε) = 0 X(n) P −→ X ∨n i → ∞ X(n) −→ X ∨n i → ∞ P( lim ∨n i →∞ |X(n) − X| = 0) = 1. ε > 0 D(n)(ε) =  m∈N d :∨m i ≥∨n i (|X(m) − X| ≥ ε). D(n) c (ε) =  m∈N d :∨m i ≥∨n i (|X(m) − X| < ε). ( lim ∨n i →∞ |X(n) − X| = 0) = ∞  k=1  n∈N d D(n) c ( 1 k ). X(n) −→ X ∨ n i → ∞ ⇔ P( lim ∨n i →∞ |X(n) − X| = 0) = 1 ⇔ P( ∞  k=1  n∈N d D(n) c ( 1 k )) = 1 ⇔ P(  n∈N d D(n) c ( 1 k )) = 1 ⇔ P(  n∈N d D(n)( 1 k )) = 0. {D(n)( 1 k ), n ∈ N d } lim ∨n i →∞ P(D(n)( 1 k )) = P(  n∈N d D(n)( 1 k )) = 0. ε > 0 k 1 k < ε D(n)(ε) ⊂ D(n)( 1 k ). (|X(n) − X| > ε) ⊂ (|X(n) − X| ≥ ε) ⊂ D(n)(ε) ⊂ D(n)( 1 k ). 0  P(|X(n)−X| > ε)  P(D(n)( 1 k )) → 0, ∨n i → ∞ lim ∨n i →∞ P(|X(n)− X| > ε) = 0 X(n) P −→ X ∨n i → ∞  lim ∨n i →∞ P(X(n) = Y (n)) = 0 X(n) P −→ X ∨n i → ∞ Y (n) P −→ X ∨n i → ∞. X(n) −→ X ∨n i → ∞ ε > 0 lim ∨n i →∞ P( sup {m:∨m i ≥∨n i } |X(m) − X| ≥ ε) = 0.  n∈N d P(|X(n) − X| > ε) ∨n i → ∞ ε > 0 X(n) −→ X, ∨n i → ∞. ε > 0 P( sup {m:∨m i ≥∨n i } |X(m)−X| > ε) = P(  m,∨m i ≥∨n i |X(m)−X| > ε)   m,∨m i ∨n i P(|X(m)−X| > ε). ∨m i  ∨n i m  n  m,∨m i ∨n i P(|X(m) − X| > ε)   m:mn P(|X(m) − X| > ε) = r(n) → 0 ∨n i → ∞.  X(n) −→ X, Y (n) −→ Y, ∨n i → ∞ X(n) + Y (n) −→ X + Y, ∨n i → ∞. {X(n)} d  n∈N d E|X(n)| p < ∞ p > 0 X(n) −→ 0 X(n) L p −→ 0, ∨n i → ∞. {X(n)}, n ∈ N d d (Ω, F, P) {X(n)} P( lim ∨m i →∞,∨n i →∞ |X(m) − X(n)| = 0) = 1. {X(n)} ∀ε > 0 : lim ∨m i →∞,∨n i →∞ P(|X(m) − X(n)| ≥ ε) = 0. {X(n)} p (0 < p < ∞) lim ∨m i →∞,∨n i →∞ E|X(m) − X(n)| p = 0. {X(n)} ε > 0 lim ∨n i →∞ P( sup {k,l:∨k i ,∨l i ≥∨n i } |X(k) − X(l)| ≥ ε) = 0. lim ∨n i →∞ P( sup {k:∨k i ≥∨n i } |X(k) − X(n)| ≥ ε) = 0. {X(n)} ∆(n)(ε) =  k,l,∨k i :∨l i ≥∨n i (|X(k) − X(l )| ≥ ε) ⊂ ( sup {k,l:∨k i ,∨l i ≥∨n i } |X(k) − X(l )| ≥ ε). {∆(n)} ( lim ∨k i →∞,∨l i →∞ |X(k) − X(l)| = 0) = ∞  m=1  n∈N d ∆(n) c ( 1 m ). {X(n)} ⇔ P( ∞  m=1  n∈N d ∆(n) c ( 1 m )) = 1 ⇔ P(  n∈N d ∆(n) c ( 1 m )) = 1 ⇔ P(  n∈N d ∆(n)( 1 m )) = 0 ⇔ lim ∨n i →∞ P(∆(n)( 1 m )) = 0. {∆(n)( 1 m )} ε > 0 m ∈ N 1 m < ε ( sup {k,l:∨k i ,∨l i ≥∨n i } |X(k) − X(l )| ≥ ε) ⊂  k,l:∨k i ,∨l i ≥∨n i (|X(k) − X(l )| ≥ 1 m ) ⊂ ∆(n)( 1 m ). 0  P( sup {k,l:∨k i ,∨l i ≥∨n i } |X(k) − X(l )| ≥ ε)  P(∆(n)( 1 m )) −→ 0 ∨n i → ∞. {X(n)} P(  n∈N d ∆(n)( 1 m )) = lim ∨n i →∞ P(∆(n)( 1 m ))  lim ∨n i →∞ P( sup {k,l:∨k i ,∨l i ≥∨n i } |X(k)−X(l)| ≥ 1 m ) = 0, ∈ N. P( ∞  m=1  n∈N d ∆(n) c ( 1 m )) = 1, m ∈ N. P( lim ∨n i →∞ |X(k) − X(l )| = 0) = 1 k, l ∈ N d ∨ k i , ∨l i ≥ ∨n i . {X(n)}  {X(n)}, n ∈ N d } d X(n) X(n) {X(n)} {X(n)} I d {X(j)} {X(j)} {X(j)} X X(n) P −→ X ∨n i → ∞ P(|X(k) − X| > ε)  P(|X(n) − X(j| > ε 2 ) + P(|X(j) − X| > ε 2 ) −→ 0 ∨n i , j → ∞ X( n) P −→ X ∨n i → ∞  {X(n)}, n ∈ N d d EX(n) = 0 n ∈ N d  n∈N d DX(n) ∨n i → ∞  n∈N d X(n) ∨n i → ∞. ε > 0 0  P( sup {k:∨k i ≥∨n i } |S(k) − S(n)| > ε)  1 ε 2  k:∨k i ≥∨n i DX(k)  1 ε 2  k:kn DX(k) = r(n) → 0 ∨n i → ∞ lim ∨n i →∞ P( sup {k:∨k i ≥∨n i } |S(k) − S(n)| > ε) = 0 ε > 0 {S(n)} ∨n i → ∞  n∈N d X(n) ∨n i → ∞.  {X(n)} d  n∈N d DX(n) ∨n i → ∞  n∈N d (X(n) − EX(n)) ∨n i → ∞. {X(n), n ∈ N d } d c > 0 X c (n) = X(n)I(|X(n)|  c).  n∈N d P(|X(n)| > c),  n∈N d E(X c (n),  n∈N d D(X c (n)) ∨n i → ∞  n∈N d X(n) ∨n i → ∞.  n∈N d X c (n) ∨n i → ∞  n∈N d P(X(n) = X c (n)) =  n∈N d P(|X(n)| > c) ∨n i → ∞  n∈N d X(n)  n∈N d X c (n)  n∈N d X c (n) ∨n i → ∞  n∈N d X(n) ∨n i → ∞  X(i, j) =  (−1) j i i ≥ 1, j  2, 0 n ≥ 2 S(m, n) =  im  jn X(i, j) =  im  j2 X(i, j) = (−1) 1 .1 + (−1) 1 .2+ [...]... = 0 Vậy S(m, n) hội tụ h.c.c tới 0 khi m n Nhưng chuỗi P(|X(i, j)| c) () i1 j1 phân kỳ với mọi 0 |n|) 0 khi... = X(i)I(|X(i)| |n|) Khi đó nếu P(|X(i)| > |n|) 0 khi ni , (i) i n (ii) (iii) thì 1 |n| 1 |n| EX(ni) 0 khi ni , i n 1 |n|2 DX(ni) 0 khi ni i n i n X(i) hội tụ theo xác suất tới 0 khi ni Chú ý rằng mệnh đề 3.17 là dạng nhiều chiều của luật yếu số lớn kinh điển quen biết (xem [7] trang 278) Tài liệu tham khảo [1] Nguyễn Quốc Thắng (2005), Luận văn thạc sĩ Toán học, Đại học Vinh [2] Nguyễn