Дронов С.В. Конспект лекций по теории случайных процессов
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P
∞
n=1
g(2
−(n+1)
)
P
∞
n=1
X
(n)
(t) − X
(n+1)
(t)
.
X
(n)
(t) = X
(1)
(t) −
n−1
X
j=1
X
(n)
(t) − X
(n+1)
(t)
ˆ
X(t) = lim
n→∞
X
(n)
(t).
ξ(t) =
ˆ
X(t), N < ∞
0, N = ∞
ξ(t) x(t) t = k/2
n
t ε > 0
w
h
= sup{|ξ(s) − ξ(q)|, |s − q| < h}.
ξ(t) w
h
→ 0 h → 0
t
m
=
k(m)
2
n(m)
−→ t, m → ∞.
P(|x(t) − ξ(t)| ≥ ε) ≤ P(|x(t) − x(t
m
)| ≥ ε/2)+
+P(|ξ(t) − ξ(t
m
)| ≥ ε/2).
P(w
t−t
m
≥ ε/2) → 0, m → ∞.
g(t − t
m
) → 0
m
P(|x(t) − x(t
m
)| ≥ ε/2) ≤ P(|x(t) − x(t
m
)| ≥ g(t − t
m
)) ≤
≤ q(t − t
m
) → 0, m → ∞,
P(|x(t) − ξ(t| ≥ ε) = 0
ε
A B
P(B) 6= 0
P(A/B) =
P(AB)
P(B)
.
B ξ
M(ξ/B) =
Z
Ω
ξdP(·/B).
B ξ
Z
B
M(ξ/B)dP = M(ξ/B) · P(B) =
Z
B
ξdP.
< Ω, F, P >
U = {B
1
, B
2
, ...} Ω
∪
j
B
j
= Ω, i 6= j ⇒ B
i
∩ B
j
= ∅, (∀j) B
j
∈ F, P(B
j
) 6= 0.
ˆ
ξ =
X
j
M(ξ/B
j
)1
B
j
ξ
U M(ξ/U)
B ∈ σ(U)
B = ∪
k
B
i
k
=
∪
k
C
k
.
Z
B
ξdP =
X
k
Z
C
k
ξdP =
X
k
Z
C
k
M(ξ/C
k
)dP,
B ∈ σ(U)
Z
B
ξdP =
Z
B
M(ξ/U)dP
B
η σ(U)
(∀B ∈ σ(U))
Z
B
ηdP =
Z
B
M(ξ/U)dP,
η = M(ξ/U) ( P).
G σ G
M(ξ/G)
ξ σ
B ∈ G
Z
B
ξdP =
Z
B
M(ξ/G)dP.
L
2
(∀B ∈ G) < ξ, 1
B
>
L
2
(<Ω,F,P>)
= < M(ξ/G), 1
B
>
L
2
(<Ω,F,P>)
,
G
< ξ −M(ξ/G), 1
B
>
L
2
(<Ω,F,P>)
= 0,
ξ L
2
(< Ω, F, P >)
G
g(B) σ
F µ
< Ω, F > µ
µ f
(∀B ∈ F) g(B) =
Z
B
f(x)dµ.
ξ F M(ξ/F) = ξ.
(∀B ∈ F)
Z
B
ξdP =
Z
B
ξ dP ⇒ ξ = M(ξ/F)
M M(ξ/F) = Mξ.
B = Ω
M M(ξ/F) =
Z
Ω
M(ξ/F)dP =
Z
Ω
ξdP = Mξ.
M(αξ + βη/F) = αM(ξ/F) + βM(η/F).
B ∈ F
F
R
B
(αM(ξ/F) + βM(η/F)) dP =
= α
R
B
ξdP + β
R
B
ηdP =
R
B
(αξ + βη)dP,
ξ F M(ξ/F) = Mξ.
(∀B ∈ F)(∀x) P(ξ < x, B) = P(ξ <
x)P(B), ξ 1
B
Z
B
ξdP = Mξ1
B
= MξM1
B
= Mξ
Z
B
dP =
Z
B
MξdP,
η F ξ
M(ξη/F) = ηM(ξ/F).
η = 1
C
, C ∈ F. B ∈ F
Z
B
ηM(ξ/F)dP =
Z
B∩C
M(ξ/F)dP =
=
Z
B∩C
ξdP =
Z
B
ξηdP,
F ⊂ F
1
M (M(ξ/F
1
)/F) = M(ξ/F).
B ∈ F
Z
B
M (M(ξ/F
1
)/F) dP =
Z
B
M(ξ/F
1
)dP =
=
Z
B
ξdP =
Z
B
M(ξ/F)dP,
F ⊂ F
1
, η F
1
M(ξη/F) = M (ηM(ξ/F
1
)/F) .
T = [a, b)
σ F
t
, t ∈ T
s < t F
s
⊂ F
t
a σ F
a
F
a+
= ∩{F
t
, t > a} σ
F
t+
, t ∈ T
T
t ξ(t) F
t
−
ξ(t)
ξ(t)
F
ξ
t
= σ(ξ(s), s ≤ t).
(∀t, s ∈ T ) (s < t) ⇒ M (ξ(t)/F
s
) = ξ(s)
M (ξ(t)/F
s
) ≥ ξ(s)
s < t
≥
≤ ξ(t)
ξ(t)
F
ξ
t
, t ∈ T s < t
M(ξ(t)/F
ξ
s
) = M((ξ(t) − ξ(s))/F
ξ
s
) + M(ξ(s)/F
ξ
t
),
M(ξ(t) − ξ(s)) = 0 ξ(s)
ξ σ
F
t
, t ∈ T X(t) = M(ξ/F
t
)
s < t
M(X(t)/F
s
) = M(M(ξ/F
t
)/F
s
) = M(ξ/F
s
) = X(s).
(∀s, t)(∀B ∈ F
s
) (s < t) ⇒
Z
B
ξ(s)dP ≤
Z
B
ξ(t)dP.
ξ(t) f(·)
Mf(ξ(t)) < ∞ f(ξ(t))
F
t
, t ∈ T
f(ξ(t))
(∀x
0
)(∃C)(∀x) f(x) ≥ f(x
0
) + C(x
0
) · (x − x
0
).
N ∈ N, B ∈ F
s
B
N
= B ∩ {|C(ξ(s))| ≤
N}.
Z
B
N
f(ξ(t))dP ≥
Z
B
N
f(ξ(s))dP +
Z
B
N
C(ξ(s))(ξ(t) − ξ(s))dP.
t < s B
N
∈ F
s
M(C(ξ(s)·(ξ(t)−ξ(s))/F
s
) = C(ξ(s))M((ξ(t)−ξ(s))/F
s
) = 0,
Z
B
N
f(ξ(t))dP ≥
Z
B
N
f(ξ(s))dP.
N → ∞
ξ(t) |ξ(t)|, ξ
2
(t), e
ξ(t)
η
b
= sup
a≤t≤b
ξ(t).
ξ(t)
P(η
b
≥ x) ≤
Mξ(b)
x
,
Mη
p
b
≤
p
p − 1
!
p
Mξ
p
(b), p > 1.
< Ω, F, P >
σ F
t
, t ∈ T = [a, b)
σ
M(T )
Mξ
2
(t) < ∞, t ∈ T.
M
C
(T ) M(T )
ξ(t), t ∈ [a, b]
ξ(b) M(T )
< ξ, η >
T
= Mξ(b)η(b).
M(T )
M
C
(T )
ξ
n
(t)
M(T ) ξ
n
(b)
L
2
(< Ω, F, P >)
µ
µ(t) = M(µ/F
t
).
F
t
, t ∈ T
|ξ
n
(t) −µ(t)|
M
sup
t
|ξ
n
(t) − µ(t)|
!
2
≤ 4M(ξ
n
(b) − µ(b))
2
→ 0.