!"
!"# $%!%&' !"#( ' )*
#" * *) F ! + * !
$"% &'(
)*(
,%-'% " ,%.%/
0 ' (Ω, A, P) F !σ!1 A
+' 2 A1 , A2 , . . . , An F ! 1≤k≤n
1≤i1 <i2 <···<ik ≤n
F
E
k
s=1
IAis =
k
EF IAis .
s=1
F = (∅, Ω) 1 3 F = A A!
4 5 A1, A2, . . . , An Ak ⊂ A k = 1, . . . , n F !
5 A1 , A2, . . . , An Ai ∈ Ai i = 1, 2, . . . , n
5 F ! X : Ω −→ R FX σ !1 X FX
σ!1 * X X1 , X2, . . . , Xn F ! σ!1
F !
FX1 , FX2 , . . . , FXn
3 !
* 6/7 F ! !
6&7
, X Y F E F I[X<x,Y <y] = E F I[X<x] E F I[Y <y]
* !
X Y F Z F X + Z Y F , -.
Z Z=
Ak = {ω : Z = zk } 8
E F I[X+Z<x,Y <y] =
n
n
zk IAk ,
k=1
E F IAk I[X+zk <x,Y <y]
k=1
=
n
k=1
= EF
F
IAk E I[X<x−zk ,Y <y] =
n
n
IAk E F I[X<x−zk ] E F I[Y <y]
k=1
IAk I[X<x−zk ] E F I[Y <y] = E F I[X+Z<x]E F I[Y <y]
k=1
3 Z > 0 0 !* 5 !
Zn −→ Z n → ∞ E F I[X+Z<x,Y <y] = E F lim I[X+Zn <x,Y <y]
n→∞
F
= lim E I[X+Zn <x,Y <y] = lim E F I[X+Zn <x] E F I[Y <x]
=
n→∞
E F I[X+Z<x]E F I[Y <y]
n→∞
Z ≤ 0
. F ! Z Z = Z + −
−
Z D1 = {Z > 0} D2 = {Z ≤ 0} F
! E F I[X+Z<x,Y <y] I[D1 ∪D2 ] = E F I[X+Z<x,Y <y] ID1 + E F I[X+Z<x,Y <y] ID2
= ID1 E F I[X+Z + <x] E F I[Y <y] + ID2 E F I[X−Z − <x] E F I[Y <y]
= E F I[X+Z +<x] ID1 + E F I[X−Z − <x] ID2 E F I[Y <y]
= E F I[X+Z<x] E F I[Y <y]
* X Y F Z > 0
F XZ Y F , /
$ +' - X : Ω −→ R ϕF
X : R −→ LC F itX
ϕF
= E F cos(tX) + iE F sin(tX), t ∈ R,
X (t) = E e
LC C 9 5 1 0 C " |eitX | = 1 eitX F !* 8 0 t ϕF
X (t)
F itX
=E e
=
+∞
−∞
eitX dFXF (x)
,
FXF (x) 1 6:7 * * 6$7
, X ϕFX ' ϕFX (0) = 1 $ |ϕFX (t)| ≤ 1 & ϕFX (t) = ϕFX (−t) : ϕFaX+b (t) = eitbϕFX (at) b F *) *
! , X Y F F
F
ϕF
X+Y (t) = ϕX (t)ϕY (t) -.
+ F it(X+Y )
ϕF
= E F eitX eitY = E F eitX E F eitY
X+Y (t) = E e
F ! eitX eitY F
= ϕF
X (t)ϕY (t)
{Xn, n ≥ 1} ϕFX
ϕF
X (t) n → ∞ t ∈ R ϕX (t) −→ ϕX (t), n → ∞ t ∈ R.
, /
n
(t) −→
n
- *; * lim ϕX (t) = lim E(E F eitX ) = E( lim E F eitX ) n→∞
n→∞
n→∞
E( lim E F eitX ) = E(E F eitX ) = EeitX = ϕX (t) n→∞
-.
n
n
n
n
ϕX (t) −→ ϕX (t) Xn −→ X n → ∞ !
* 0 012 (Ω, A, P) Ω = [1, 2] P ({1}) =
P ({2}) = 12 ω = 1
1
X(ω) =
−1 ω = 2,
n = 2k,
X(ω)
Xn (ω) =
−X(ω) n = 2k + 1, k ∈ N,
F = A n
1
1
ϕX (t) = EeitX = eit + e−it = cos t,
2
2
n = 2k,
EeitX
ϕXn (t) = EeitXn =
−itX
n = 2k + 1, k ∈ N,
Ee
n = 2k,
n = 2k + 1, k ∈ N.
ϕX (t) −→ ϕX (t) n → ∞ +
n = 2k,
eitX
F itX
itX
ϕF
(t)
=
E
e
=
e
=
X
e−itX n = 2k + 1, k ∈ N.
5 ϕFX (t) * n → ∞
ϕXn (t) =
cos t
cos t
n
n
n
n
n
, X, Y : Ω −→ R F
ϕF
X (t) = ϕY (t) ,
ϕX (t) = ϕY (t).
-.
* 5
ϕX (t) = EeitX = E(E F eitX ) = E(E F eitY ) = EeitY = ϕY (t).
ϕX (t) = ϕY (t) FX (x) = FY (x)
ϕFX (t) = ϕFY (t) X Y E F I[X<x] = E F I[Y <x] x ∈ R.
, 3
-.
K = {f : f dF F (x) = f dGF (x)} " X Y
itx
F
e dF (x) = eitx dGF (x) eitx ∈ K t ∈ R eitx ∈ K e−itx ∈ K sin(tx) ∈ K cos(tx) ∈ K . f = (an cos(tx) + bn sin(tx)) f ∈ K g ∈ K f + g ∈ K fn ∈ K fn ↑ f f ∈ K 8 fn ∈ K 1, t < x,
lim fn =
n→∞
0, t ≥ x,
x
−∞
dF F (x) =
=
x
−∞
x
−∞
lim fn dF F (x) =
n→∞
dGF (x)
x
−∞
lim fn dGF (x)
n→∞
x ∈ R.
012 (Ω, A, P ) = ([0, 1], B([0, 1]), μ) μ *
F = A
E F I[X<x] = E F I[Y <x]
0, ω ∈ [0, 12 ),
X(ω) =
1, ω ∈ [ 12 , 1]
Y (ω) = 1 − X(ω) 8
ϕX (t) =
itX(ω)
e
Ω
dω =
1
2
0
it0
e dω +
1
1
2
1
eit dω = (1 + eit ),
2
1
ϕY (t) = (1 + eit ).
2
ϕX (t) = ϕY (t) F itX(ω)
ϕF
= eitX(ω)
X (t) = E e
F itY (ω)
ϕF
= eitY (ω)
Y (t) = E e
1, ω ∈ [0, 12 ),
=
eit , ω ∈ [ 12 , 1]
eit , ω ∈ [0, 12 ),
=
1, ω ∈ [ 12 , 1].
ϕFX (t) = ϕFY (t) ω ∈ Ω 8 $, X Y ϕFX (t) −→ ϕFX (t) ϕX
F−D
D
Xn −→ X Xn −→ X < 6:7=
n
n
(t) −→ ϕX (t)
& -
X Y
x ∈ R
E F I[X<x] = E F I[Y <x] + * Xi i ≥ 1 Xi + μ i ≥ 1 μ F F 4 -.
μ μ=
n
k=1
zk IAk ,
Ak = {ω : μ = zk } i = j i, j = 1, 2, . . .
F
E I[Xi +μ<x] =
=
=
n
k=1
n
k=1
n
F
E IAk I[Xi +zk <x] =
IAk E F I[Xi <x−zk ] =
n
k=1
n
IAk E F I[Xi +zk <x]
IAk E F I[Xj <x−zk ]
k=1
E F IAk I[Xj +zk <x] = E F I[Xj +μ<x]
k=1
3 μ > 0 0 !* 5 !
μn −→ μ n → ∞ E F I[Xi +μ<x] = E F lim I[Xi +μn <x] = lim E F I[Xi +μn <x]
n→∞
F
n→∞
F
= lim E I[Xj +μn <x] = E I[Xj +μ<x]
n→∞
μ ≤ 0
. F ! μ μ = μ+ − μ−
{Xn, n ≥ 1} F μ =
E F Xn n ≥ 1 σF2 = E F (Xn − E F Xn )2 n ≥ 1 Sn = X1 + X2 + · · · + Xn , Sn − nμ F−D
√ −→ N(0, 1).
σF n
F−D
D
√
√ −→ N(0, 1) <6:7=
3 Sσ −nμ
−→ N(0, 1) Sσ −nμ
n
n
-.
n
n
F
F
9 μ σF2 F ! Xi i ∈ N Xi = Xi − μ i ∈ N 5 F ! !
< '& &'= E F Xi = 0
n
Sn = Xk k=1
': S
n
F it σF √n
ϕF Sn√ (t) = E e
σF
= EF
n
=
n
F
Xk
it σ √
F n
E e
n
it σ
e
Xk
√
F n
k=1
X
k
F it σF √n
= E e
n
k=1
2 n
t2 X 2k
tXk
t
F
= E
1+i √ − 2 +o
σF2 n
σF n 2σF n
n
2
t2
t2
− t2
+ EF o
= 1−
−→
e
n → ∞,
2n
σF2 n
2 t
F
n→∞
lim E o
= 0 10 t
σF2 n
p.p.
−→
E F I Sn −nμ
Φ(x) n → ∞
√ <x
σF
n
! " # $ % &' (%) )
* +,
1 % &' 4 * 5%) 56
5
!
-./01!23 --3
30 2-3 --3
3 7 # )6
" 4) 56( 7 78 % (858)