Дронов С.В. Конспект лекций по теории случайных процессов
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P
0
0,k
(t) = λ
0
P
0,k
(t),
P
0
n,k
(t) = µ
n
P
n−1,k
(t) − (λ
n
+ µ
n
)P
n,k
(t) + λ
n
P
n+1,k
(t), n > 0.
k
n
i j
p
j
= lim
t→∞
P
i,j
(t),
P
j
p
j
= 1 p
0
, p
1
, ...
j
t → ∞ p
j
−λ
0
p
0
+ µ
1
p
1
= 0,
λ
k−1
p
k−1
− (λ
k
+ µ
k
)p
k
+ µ
k+1
p
k+1
= 0, k > 0.
π
0
= 1, π
j
=
λ
0
...λ
j−1
µ
1
...µ
j
, j ≥ 1.
p
j
=
π
j
P
k
π
k
, j ≥ 0.
p
1
=
λ
0
µ
1
p
0
= π
1
p
0
.
k ≤ j
p
k
= π
k
p
0
.
p
j+1
=
λ
j
π
j
µ
j+1
p
0
,
p
j+1
= π
j+1
p
0
.
k ≥ 0
P
k
p
k
= 1
1 = p
0
X
k
π
k
⇒ p
0
=
π
0
P
k
π
k
.
< Ω, F, P >
ξ : Ω → C
<eξ, =mξ
L
2
= L
2
(Ω, F, P) = {ξ | M|ξ|
2
< ∞}.
L
2
< ξ, η > = Mξ¯η, kξk
2
L
2
=< ξ, ξ > = M|ξ|
2
.
m(t) = Mξ(t), ξ
∗
(t) = ξ(t) − m(t).
K(t, s) ≡ Mξ
∗
(t)
¯
ξ
∗
(s),
(∀ c
1
, ..., c
k
∈ C) (∀t
1
, ..., t
k
∈ T )
k
X
i=1
k
X
j=1
c
i
¯c
j
K(t
i
, t
j
) ≥ 0.
(∀s, t) K(t, s) = K(s, t).
L
2
ξ(t) t → t
0
(∃m) m(t) → m
t → t
0
,
(∃K < ∞) lim
s,t→t
0
K(t, s) = K.
ξ(t)
L
2
−→ ξ.
m = Mξ, K = Mξ
∗
¯
ξ
∗
ξ
∗
= ξ −m
|m(t) − m| ≤ M|ξ(t) − ξ| ≤
q
M|ξ(t) − ξ|
2
→ 0,
|K(t, s) − K| ≤
≤
q
M|ξ
∗
(t)|
2
M|ξ
∗
(s) − ξ
∗
|
2
+
q
M|ξ
∗
|
2
M|ξ
∗
(t) − ξ
∗
|
2
≤
≤ c(t)kξ
∗
(s) − ξ
∗
k
L
2
+ ckξ
∗
(t) − ξ
∗
k
L
2
,
c t s
ε > 0 δ
ε/2 |t−t
0
| < δ
q
M|ξ
∗
(t)|
2
t
0
q
M|ξ
∗
(t)|
2
= kξ
∗
(t)k
L
2
≤ kξ(t) − ξk
L
2
+ kξ
∗
k
L
2
+ |m(t) − m|.
c(t)kξ
∗
(s)−ξ
∗
k
L
2
ε/2 s t
0
kξ(t) − ξ(s)k
2
L
2
= |m(t) − m(s)|
2
+ kξ
∗
(t) − ξ
∗
(s)k
2
L
2
.
kξ
∗
(t) − ξ
∗
(s)k
2
L
2
= K(t, t) + K(s, s) − 2<eK(t, s),
kξ(t) − ξ(s)k
2
L
2
−→ 0
(t − s) → 0
ξ(t) ξ
L
2
P(|ξ(t) − ξ| ≥ ε) ≤
M|ξ(t) − ξ|
ε
≤
kξ(t) − ξk
L
2
ε
ε > 0
ξ(t)
L
2
−→ ξ t → a
t→a
ξ(t) = ξ
h, a ∈ R
∆
h
ξ(a) =
ξ(a + h) − ξ(a)
h
.
(∃z ∈ L
2
) z =
h→0
∆
h
ξ(a),
ξ(t)
a z
z =
dξ
dt
(a).
ξ(t)
a
• m(t)
•
D
s
D
t
K(t, s)|
t=s=a
=
lim
h,q→0
K(a+h,a+q)−K(a,a+q)−K(a+h,a)+K(a,a)
hq
< ∞.
a ∈ [A, B]
D
s
D
t
K(t, s)|
t=s=a
ξ
∗
(t)
[A, B]
∂K
∂t
,
∂K
∂s
∂
2
K
∂t∂s
Mξ
∗
(t)
dξ
∗
dt
(s) =
∂K
∂s
(t, s), M
dξ
∗
dt
(t)
dξ
∗
dt
(s) =
∂
2
K
∂t∂s
(t, s).
K(t, s)
Mξ
2
< ∞ ∃
h→a
η(h) = η
lim
h→a
Mξ η(h) = Mξη.
|Mξη(h) − Mξη|
2
≤ Mξ
2
· kη(h) − ηk
2
L
2
→ 0.
Mξ
∗
= 0
∆
h
ξ
∗
(s) h → 0
lim
h→0
Mξ
∗
(s)∆
h
ξ
∗
(t) = lim
h→0
K(s, t + h) − K(s, t)
h
,
∂K
∂t
Mξ
∗
(t)
dξ
∗
dt
(s)
M
dξ
∗
dt
(s)
dξ
∗
dt
(t) = lim
h→0
M∆
h
ξ
∗
(s)
dξ
∗
dt
(t) =
= lim
h→0
1
h
Mξ
∗
(s + h)
dξ
∗
dt
(t) − Mξ
∗
(s)
dξ
∗
dt
(t)
=
= lim
h→0
K
0
t
(s+h,t)−K
0
t
(s,t)
h
=
∂
2
K
∂s∂t
(s, t).
ξ(t) [A, B] A = t
0
<
t
1
< ... < t
m
= B ∆t
j
= t
j
− t
j−1
S
m
=
m
X
j=1
ξ(θ
j
)∆t
j
, θ
j
∈ [t
j−1
, t
j
], j = 1, ..., m.
ξ(t)
I ∈ L
2
t
j
, θ
j
I =
∆→0
S
m
, ∆ = max
j
∆t
j
.
I
I =
B
Z
A
ξ(t) dt.
I
m(t) [A, B]
B
R
A
B
R
A
K(t, s)dtds ξ(t)
Mξ
∗
(t)
B
Z
A
ξ
∗
(t)dt =
B
Z
A
K(t, s)ds,
M
B
Z
A
ξ
∗
(t)dt
B
Z
A
ξ
∗
(t)dt =
B
Z
A
B
Z
A
K(t, s)dtds.
U A P(U)
U ˆm
σ(A) µ : A → L
2
(< Ω, F, P >)
ˆm
µ(A
1
∪ A
2
) = µ(A
1
) + µ(A
2
) A
1
∩ A
2
= ∅;
(∀A) Mµ(A) = 0, M|µ(A)|
2
= ˆm(A);
Mµ(A)µ(B) = 0 A ∩ B = ∅
L
2
( ˆm) =
ψ : U → C :
Z
U
|ψ(t)|
2
d ˆm < ∞
.
A
1
, ..., A
k
∈ A U
ϕ(t) =
k
X
j=1
c
j
1
A
j
(t) ∈ L
2
( ˆm)−
c
1
, ..., c
k
1
A
A
Z
U
ϕ(t)dµ =
k
X
j=1
c
j
µ(A
j
).
ϕ ∈ L
2
( ˆm)
ϕ
Z
U
ϕ(t)dµ =
n→∞
Z
U
ϕ
n
(t)dµ.
L
2
( ˆm)
ϕ(t), ψ(t)
ϕ(t) =
X
j
f
j
1
A
j
, ψ(t) =
X
j
g
j
1
A
j
A
j
, j = 1, ..., n
<
R
U
ϕ(t)dµ,
R
U
ψ(t)dµ >
L
2
= M
R
U
ϕ(t)dµ ·
R
U
ψ(t)dµ =
=
P
j,k
f
j
¯g
k
Mµ(A
j
)µ(A
k
) =
P
k
f
k
¯g
k
ˆm(A
k
) = < ϕ, ψ >
L
2
( ˆm)
.
ϕ, ψ ∈ L
2
( ˆm)
<
Z
U
ϕ(t)dµ,
Z
U
ψ(t)dµ >
L
2
(Ω,F,P)
= < ϕ, ψ >
L
2
( ˆm)
.
L
2
( ˆm) L
2
(< Ω, F, P >)
ξ(t)
[A, B]
Mξ(t) = 0
K(t, s) = Mξ(t)ξ(s) = Mξ(t − s + A)ξ(A) = K(t − s),