Дронов С.В. Конспект лекций по теории случайных процессов
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T R
T
T
< Ω, F, P >
ξ : T × Ω → R
∀t ∈ T ξ(t) = ξ(t, ·)
ξ(t)
ξ(·, ω) ∈ R
T
T ⊂ R t
ω T
Z N
ω ∈ Ω
ξ(ω, ·)
K(t, s) = cov(ξ(t), ξ(s)) = Mξ(t)ξ(s) − Mξ(t) Mξ(s)
ξ
k
(∀ c
1
, ..., c
k
∈ R) (∀t
1
, ..., t
k
)
k
X
i=1
k
X
j=1
c
i
c
j
K(t
i
, t
j
) ≥ 0.
m(t) = Mξ(t)
X
i
X
j
c
i
c
j
K(t
i
, t
j
) = M
X
i,j
c
i
c
j
(ξ(t
i
) −m(t
i
))(ξ(t
j
) − m(t
j
)) =
= M
k
X
j=1
c
j
(ξ(t
j
) − m(t
j
))
2
= D
k
X
j=1
c
j
ξ(t
j
)
≥ 0.
¯
t =
¯
t(n)
T
¯
t = {t
1
, t
2
, ..., t
n
} ⊂ T
¯
t(n) R
¯
t
n
R
¯s
⊂ R
¯
t
¯s ⊂
¯
t
R
T
= {f | f : T → R}.
R
¯
t
= {(f(t
1
), ..., f(t
n
)) | f ∈ R
T
}−
f R
T
ξ
¯
t
= (ξ(t
1
), ..., ξ(t
n
))
¯
t ξ : T → R
ξ(t)
ξ
¯
t
¯
t
T
ξ
∀B ∈ B(R
¯
t
)
P
¯
t
(B) = P(ξ
¯
t
∈ B).
¯
t ⊃ ¯s π
¯
t,¯s
R
¯
t
R
¯s
(∀A ∈ B(R
¯s
)) P
¯s
(A) = P
¯
t
(π
−1
¯
t,¯s
(A)).
P
¯
t
¯
t
T
< Ω, F, P > ξ(t) P
¯
t
ξ
¯
t
¯
t ⊂ T
Ω = R
T
, ξ(t, ω) = ω(t). B ⊂ R
T
{ω|(ω(t
1
), ..., ω(t
n
)) ∈ A}
t
1
, ..., t
n
T A ∈
B(R
n
) C
R
T
, F = σ(C) = B(R
T
) B ∈ C
P(B) = P
¯
t
(A).
σ F
< Ω, F, P >
ξ(t)
¯
t T
< λ, x >
¯
t
=
X
{k:t
k
∈
¯
t}
λ(t
k
)x(t
k
), λ, x ∈ R
T
.
P
¯
t
ϕ
¯
t
(λ) =
Z
R
T
exp{i < λ, x >
¯
t
}dP
¯
t
(x).
(∀λ ∈ R
T
) (¯s ⊂
¯
t) ⇒
ϕ
¯s
(λ) = ϕ
¯
t
(π
¯
t,¯s
(λ))
.
(π
¯
t,¯s
(λ))(u) = λ(u) u ∈ ¯s u ∈
¯
t \¯s
m(t) K(t, s)
ϕ
n
(t), n ∈ N ∪ {0}
X
n
, n ∈ N
ξ(t)
ξ(t) = ϕ
0
(t) +
∞
X
n=1
X
n
ϕ
n
(t)
Mξ(t) = ϕ
0
(t); K(t, s) =
∞
X
n=1
σ
2
n
ϕ
n
(t)ϕ
n
(s),
σ
2
n
= DX
n
, n ∈ N.
“
“
ξ(t)
ϕ(
~
λ) = exp{i <
~
λ,~a > −
1
2
< B
~
λ,
~
λ >},
~a B
~a
B
a(t)
K(t, s)
a(t)
K(t, s)
a(t)
¯
t = {t
1
, ..., t
n
} P
¯
t
R
n
B = B
¯
t
B
i,j
= K(t
i
, t
j
)
ϕ
¯
t
(
~
λ) = exp{−
1
2
< B
~
λ,
~
λ >} −