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Дронов С.В. Конспект лекций по теории случайных процессов
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Академическая и специальная литература
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Теория вероятностей и математическая статистика
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(
∀
k
)(
∀
c
1
,
...,
c
k
∈
C
)
(
∀
t
1
,
...,
t
k
)
k
X
i
=1
k
X
j
=1
c
i
¯
c
j
K
(
t
i
−
t
j
)
≥
0
.
K
(0)
(
∀
t
)
K
(
t
)
=
K
(
−
t
);
(
∀
t
)
|
K
(
t
)
|
≤
K
(0)
.
k
=
1
, c
1
=
1
, t
1
=
0
,
K
(0)
≥
0
k
=
2
,
t
1
=
t,
t
2
=
0
c
1
,
c
2
(
|
c
1
|
2
+
|
c
2
|
2
)
K
(0)
+
c
1
¯
c
2
K
(
t
)
+
¯
c
1
c
2
K
(
−
t
)
≥
0
.
(
∀
a
∈
C
)
aK
(
t
)
+
¯
aK
(
−
t
)
∈
R
.
a
=
a
1
+
ia
2
,
K
(
t
)
=
K
1
+
iK
2
,
K
(
−
t
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=
K
3
+
iK
4
.
=
m
(
aK
(
t
)
+
¯
aK
(
−
t
))
=
0
,
a
1
,
a
2
a
1
(
K
2
+
K
4
)
+
a
2
(
K
1
−
K
3
)
=
0
,
K
2
=
−
K
4
,
K
1
=
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3
k
=
2
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t
1
=
t,
t
2
=
0
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c
1
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|
K
(
t
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|
,
c
2
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−
K
(
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=
K
(
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.
2
|
K
(
t
)
|
2
K
(0)
−
2
|
K
(
t
)
|
3
=
2
|
K
(
t
)
|
2
(
K
(0)
−
|
K
(
t
)
|
)
≥
0
.
K
(
t
))
=
0
2
|
K
(
t
)
|
K
(
t
)
F
(
λ
)
(
∀
t
)
K
(
t
)
=
K
(0)
∞
Z
−∞
e
itλ
dF
(
λ
)
.
K
(
n
)
, n
∈
Z
F
(
λ
)
[
−
π
,
π
]
n
K
(
n
)
=
K
(0)
π
Z
−
π
e
inλ
dF
(
λ
)
,
K
(0)
=
1
˜
K
(
n
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=
K
(
n
)
/K
(0)
X
j,k
c
j
¯
c
k
K
(
n
j
−
n
k
)
=
π
Z
−
π
X
j,k
c
j
¯
c
k
e
iλ
(
n
j
−
n
k
)
dF
(
λ
)
=
=
π
Z
−
π
X
j
c
j
e
iλn
j
2
dF
(
λ
)
≥
0
,
ρ
∈
(0
,
1)
λ
∈
[
−
π
,
π
]
f
ρ
(
λ
)
=
1
2
π
X
s
∈
Z
K
(
s
)
ρ
|
s
|
e
−
iλs
.
|
K
(
s
)
|
≤
K
(0)
=
1
,
λ
t
K
(
t
)
ρ
|
t
|
=
π
Z
−
π
e
itλ
f
ρ
(
λ
)
dλ
⇒
π
Z
−
π
f
ρ
(
λ
)
dλ
=
1
.
f
ρ
c
n
=
ρ
n
e
iun
,
t
n
=
−
n
∞
0
≤
∞
X
m
=0
∞
X
n
=0
ρ
n
e
iun
ρ
m
e
−
ium
K
(
m
−
n
)
.
m
−
n
z
m
=
n
+
z
≥
0
z
>
0
n
z
≤
0
n
≥
−
z
I
1
+
I
2
≥
0
I
1
=
X
z
≤
0
∞
X
n
=
−
z
ρ
m
+
n
e
−
iuz
K
(
z
)
=
X
z
≤
0
∞
X
m
=0
ρ
|
z
|
+2
m
e
−
iuz
K
(
z
)
,
I
2
=
∞
X
z
=1
∞
X
n
=0
ρ
z
+2
n
e
−
iuz
K
(
z
)
.
I
1
0
≤
X
z
∈
Z
∞
X
n
=0
ρ
|
z
|
+2
n
e
−
iuz
K
(
z
)
=
2
π
f
ρ
(
u
)
∞
X
n
=0
ρ
2
n
=
2
π
f
ρ
(
u
)
1
−
ρ
2
,
f
ρ
F
ρ
(
λ
)
=
λ
Z
−
π
f
ρ
(
u
)
du.
ρ
|
t
|
K
(
t
)
ρ
→
1
K
(
t
)
F
ρ
⇒
F
F
K
(
t
)
K
(
t
)
=
π
Z
−
π
e
itλ
dF
(
λ
)
.
A
[
−
π
,
π
]
ξ
(
t
)
M
ξ
(
t
)
=
0
.
H
ξ
L
2
(Ω)
=
L
2
(
<
Ω
,
F
,
P
>
)
P
j
α
j
ξ
(
t
j
)
P
ξ
µ
:
A
→
H
ξ
m
(
A
)
=
K
(0)
P
ξ
(
A
)
(
∀
t
)
ξ
(
t
)
=
Z
e
itu
dµ
(
u
)
,
[
−
π
,
π
]
(
−∞
,
∞
)
I
:
L
2
(
m
)
→
H
ξ
.
I
k
X
j
=1
c
j
e
it
j
u
=
k
X
j
=1
c
j
ξ
(
t
j
)
.
<
I
(
e
itu
)
,
I
(
e
isu
)
>
L
2
(
<
Ω
,
F
,
P
>
)
=
M
ξ
(
t
)
¯
ξ
(
s
)
=
K
(
t
−
s
)
,
<
e
itu
,
e
isu
>
L
2
(
m
)
=
Z
e
itu
e
isu
dm
(
u
)
=
K
(
t
−
s
)
.
e
itu
L
2
(
m
)
I
L
2
(
m
)
L
2
(
m
)
µ
(
A
)
=
I
(
1
A
)
,
A
∈
A
,
I
A,
B
1
A
∪
B
=
1
A
+
1
B
µ
(
A
∪
B
)
=
µ
(
A
)
+
µ
(
B
)
.
(
∀
A
∈
A
)
M
|
µ
(
A
)
|
2
=
<
µ
(
A
)
,
µ
(
A
)
>
L
2
(Ω)
=
Z
1
2
A
dm
=
m
(
A
)
.
c
j
t
j
∈
T
M
I
X
c
j
e
it
j
u
=
M
X
c
j
ξ
(
t
j
)
=
0
I
f
∈
L
2
(
m
)
M
I
(
f
)
=
0
M
µ
(
A
)
=
M
I
(
1
A
)
=
0
.
A,
B
M
µ
(
A
)
µ
(
B
)
=
<
µ
(
A
)
,
µ
(
B
)
>
L
2
(
<
Ω
,
F
,
P
>
)
=
=
<
1
A
,
1
B
>
L
2
(
m
)
=
R
1
A
1
B
dm
=
m
(
A
∩
B
)
=
0
.
µ
m
−
π
=
a
0
<
a
1
<
...
<
a
N
=
π
max
1
≤
j
≤
N
{
a
j
−
a
j
−
1
}
→
0
N
→
∞
S
N
(
t,
u
)
=
N
X
j
=1
e
ita
j
1
[
a
j
−
1
,a
j
)
(
u
)
,
S
N
(
t,
u
)
→
e
itu
N
→
∞
u
I
t
I
(
S
N
(
t,
u
))
=
N
X
j
=1
e
ita
j
µ
([
a
j
−
1
,
a
j
))
→
Z
e
itu
dµ
(
u
)
I
(
S
N
(
t,
u
))
→
I
(
e
itu
)
=
ξ
(
t
)
.
I
ϕ
(
u
)
=
P
k
j
=1
c
j
e
it
j
u
I
(
ϕ
)
=
Z
ϕ
(
u
)
dµ
(
u
)
.
I
ϕ
∈
L
2
(
m
)
R
f
(
u
)
dµ
(
u
)
L
2
(
m
)
H
ξ
e
iut
7→
ξ
(
t
)
.
Z
(
u
)
=
µ
((
−∞
,
u
))
.
ξ
(
t
)
=
Z
e
itu
d
Z
(
u
)
(
−∞
,
∞
)
Z
(
u
)
a
<
b
<
c
<
d
M
(
Z
(
b
)
−
Z
(
a
))(
Z
(
d
)
−
Z
(
c
))
=
0
.
A,
η
ϕ
[
−
π
,
π
]
A,
η
ξ
(
t
)
=
A
cos(
η
t
+
ϕ
)
=
1
2
Ae
iη
t
e
iϕ
+
1
2
Ae
−
iη
t
e
−
iϕ
.
(
−∞
,
−
η
)
,
(
−
η
,
η
)
,
(
η
,
∞
)
−
η
1
2
Ae
−
iϕ
η
1
2
Ae
iϕ
µ
ξ
(
t
)
±
η
ξ
(
t
)
,
t
∈
Z
a
≥
2
m
[
−
π
a
,
π
a
]
X
z
∈
Z
α
z
e
iz
ua
=
e
iut
,
α
z
(
t
)
=
sin
π
t
a
−
π
z
π
t
a
−
π
z
−
L
2
ξ
(
t
)
=
π
Z
−
π
e
iut
dµ
(
u
)
=
X
z
∈
Z
α
z
(
t
)
π
Z
−
π
e
iz
au
dµ
(
u
)
,
ξ
(
t
)
=
X
z
∈
Z
α
z
(
t
)
ξ
(
az
)
.
ξ
a
A
ξ
(
t
)
=
Z
e
itλ
dµ
(
λ
)
,
[
−
π
,
π
]
dξ
dt
(
t
)
=
h
→
0
ξ
(
t
+
h
)
−
ξ
(
t
)
h
=
=
h
→
0
Z
e
i
(
t
+
h
)
λ
−
e
itλ
h
dµ
(
λ
)
.
Z
lim
h
→
0
e
i
(
t
+
h
)
λ
−
e
itλ
h
dµ
(
λ
)
=
Z
iλe
itλ
dµ
(
λ
)
.
‹
1
2
3
4
5
6
7
8
9
10
›