Дронов С.В. Конспект лекций по теории случайных процессов
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t
µ
0
(A) = iλµ(A)
F
dξ/dt
(λ) = M
λ
Z
−∞
is dµ(s)
2
=
λ
Z
−∞
|s|
2
dµ(s).
η(t) ≡ Aξ(t) =
Z
T
B(t − s)ξ(s)ds.
T
T
η(t) =
Z
T
Z
B(t − s)e
isλ
dµ(λ)ds =
Z
g(λ)e
iλt
dµ(λ),
g(λ) =
Z
T
B(−u)e
iλu
du.
η = Aξ(t) A
η(t) =
Z
g(λ)e
iλt
dµ(λ),
g(λ)
A e
itλ
t = 0
g(λ) = Ae
itλ
|
t=0
P
P
d
dt
!
η(t) = ξ(t).
η(t) =
Z
e
iλt
P (iλ)
dµ(λ).
Z
dm(λ)
|P (iλ)|
2
< ∞
m µ
P
P
d
dt
!
e
itλ
|
t=0
= P (iλ),
P
d
dt
!
η(t) =
Z
P
d
dt
!
e
iλt
|
t=0
e
iλt
dµ(λ)
P (iλ)
= ξ(t).
dη
dt
(t) = ξ(t)
Z
dm(λ)
λ
2
< ∞,
η(t) =
Z
e
itλ
iλ
dµ(λ) + ζ,
ζ
µ Z(λ)
ξ(t) = A cos(ηt+ϕ)
η(t) =
1
2
A
e
−iϕ
e
−iηt
−iη
+ e
iϕ
e
iηt
iη
+ ζ = A
sin(ηt + ϕ)
η
+ ζ.
ξ(t) η ∈ L
2
(< Ω, F, P >)
ξ(t) η
ˆη
≤t
η ξ(s), s ≤ t
M(t) = M |ˆη
≤t
− η|
2
ξ(t)
(∀η ∈ L
2
(< Ω, F, P >)) lim
t→−∞
M(t) = Dη,
η
t → +∞
M(t) ˆη
≤t
ξ(t), t ∈ Z
η = ξ(m)
ξ(t), t ≤ 0
m ˜m
ξ(t) L
2
( ˜m)
H
ξ
I(f) =
π
Z
−π
f dµ , −
g L e
izλ
, z ≤ 0
e
imλ
− g
⊥ L,
π
Z
−π
e
imλ
− g(λ)
e
−inλ
d ˜m(λ) = 0, n ≤ 0.
e
imλ
e
inλ
, n ≤ 0
[−π, π)
ξ(t)
ξ(t)
z e
−iλ
z
0
˜m z
0
6= 0
N z
|Nz
0
− z| ≤ N.
e
iλ
=
1
Nz
0
+ (z − Nz
0
)
=
∞
X
n=0
(Nz
0
− e
−iλ
)
n
(Nz
0
)
n+1
,
e
e
iλ
=
∞
X
n=0
n
X
k=0
(−1)
k
C
k
n
(Nz
0
)
n+1
(Nz
0
)
n−k
e
−ikλ
.
ˆ
ξ(1)
≤0
=
∞
X
n=0
1
(Nz
0
)
n+1
n
X
k=0
(−1)
k
C
k
n
(Nz
0
)
n−k
ξ(−k).
ξ(t), t ∈ Z
[−π, π]
c
1
, c
2
λ ∈ [−π, π]
c
1
< f(λ) < c
2
L
2
( ˜m) = L
2
[−π, π].
H
≤0
L L
2
[−π, π] H
>0
e
inλ
, n > 0
(∀g ∈ L
2
[−π, π]) (∃!h
1
∈ H
≤0
, h
2
∈ H
>0
) g = h
1
+ h
2
.
H
≤0
π
Z
−π
e
imλ
− g(λ)
e
−inλ
f(λ)dλ = 0, n ≤ 0.
g ∈ H
≤0
e
imλ
− g(λ)
f(λ) ∈ H
>0
.
C
≥0
e
inλ
, n ≥ 0
C
≤0
e
inλ
, n ≤ 0
g ∈ C
≤0
e
g
∈ C
≤0
(g
1
∈ H
≤0
, g
2
∈ C
≤0
) =⇒ g
1
g
2
∈ H
≤0
,
(g
1
∈ H
>0
, g
2
∈ C
≥0
) =⇒ g
1
g
2
∈ H
>0
.
f
1
∈ C
≤0
, f
2
∈
C
≥0
1
f
1
∈ C
≤0
,
1
f
2
∈ C
≥0
(∀λ) f(λ) = f
1
(λ)f
2
(λ).
f(λ)
f
1
= f
2
1/f
2
e
imλ
− g(λ)
f
1
(λ) ≡ h(λ) ∈ H
>0
,
e
imλ
f
1
(λ) = g(λ)f
1
(λ) + h(λ).
f
1
(λ) = c
0
+
∞
X
j=1
c
−j
e
−ijλ
.
e
imλ
f
1
(λ) =
m−1
X
k=0
c
−k
e
i(m−k)λ
+ c
−m
+
∞
X
j=1
c
−m−j
e
−ijλ
.
g(λ) =
c
−m
+
P
∞
j=1
c
−m−j
e
−ijλ
f
1
(λ)
.
g
g(λ) = b
0
+
∞
X
j=1
b
−j
e
−ijλ
,
m
ˆ
ξ
≤0
=
∞
X
j=0
b
−j
ξ(−j).
m
L
2
(Ω)
L
2
( ¯m)
σ
2
(m) =
π
Z
−π
e
imλ
− g(λ)
2
f(λ)dλ.
f(λ) = f
1
(λ)f
1
(λ)
σ
2
(m) =
π
Z
−π
e
imλ
− g(λ)
f
1
(λ)
2
dλ =
π
Z
−π
|h(λ)|
2
dλ.
L
2
[−π, π]
1
√
2π
e
inλ
, n ∈ Z
σ
2
(m) =
π
Z
−π
m−1
X
j=0
c
j
e
i(m−j)λ
2
dλ = 2π
m
X
k=1
|c
−m+k
|
2
.
m → ∞
σ
2
(m) → 2π
∞
X
j=0
|c
−j
|
2
=
π
Z
−π
|f
1
(λ)|
2
dλ = K(0),
ξ(m)
f
ln f(λ) =
X
z∈Z
a
z
e
izλ
.
f(λ)
a
0
∈ R, (∀z) a
−z
= a
z
.
R
1
(λ) =
a
0
2
+
X
z<0
a
z
e
izλ
, R
2
(λ) =
a
0
2
+
X
z>0
a
z
e
izλ
.
R
1
∈ C
≤0
, R
2
∈ C
≥0
, R
1
= R
2
.
f
1
(λ) = exp R
1
(λ), f
2
(λ) = exp R
2
(λ).
f
1
R
1
f
1
(λ) = 1 +
∞
X
j=1
R
j
1
(λ)
j!
= 1 +
∞
X
j=1
1
j!
a
0
2
+
X
z<0
a
z
e
izλ
j
.
c
0
= 1 +
∞
X
j=1
1
j!
a
0
2
!
j
= e
a
0
/2
,
σ
2
(1) = 2π|c
0
|
2
= 2πe
a
0
,
σ
2
(1) = 2π exp
1
2π
π
Z
−π
ln f(λ)dλ
.
f(λ) =
P (e
iλ
)
Q(e
iλ
)
,
P, Q
z
P (z)/Q(z)
z
i