Дронов С.В. Конспект лекций по теории случайных процессов
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(z
i
)
−1
z
i
P (z)
Q(z)
= c
0
z
k
(z − z
1
)(z − ¯z
−1
1
)...(z − z
n
)(z − ¯z
−1
n
)
(z − w
1
)(z − ¯w
−1
1
)...(z − w
m
)(z − ¯w
−1
m
)
,
c
0
k
P Q z
i
, w
j
(z − z
i
)(z − ¯z
−1
i
) =
(z
i
− z)(¯z
i
− z
−1
)
−z
−1
¯z
i
.
P (z)
Q(z)
= cz
k−n+m
(z
1
− z)(¯z
1
− z
−1
)...(z
n
− z)(¯z
n
− z
−1
)
(w
1
− z)( ¯w
1
− z
−1
)...(w
m
− z)( ¯w
m
− z
−1
)
.
z = e
iλ
z
−1
= ¯z
(z
j
− z)(¯z
j
− z
−1
) = |z
j
− z|
2
≥ 0.
c k + n −m = 0
f
1
(λ) =
√
c
(¯z
1
− e
−iλ
)...(¯z
n
− e
−iλ
)
( ¯w
1
− e
−iλ
)...( ¯w
m
− e
−iλ
)
.
f
1
, f
−1
1
∈ C
≤0
a
a − e
iλ
−1
∈ C
≤0
.
1
a − e
−iλ
=
1
a
1 −
e
−iλ
a
=
1
a
1 +
∞
X
j=1
e
−iλ
a
j
.
C
≤0
e
−iλ
a
< 1.
f
1
f
2
= f/f
1
ξ Mξ = 0, Mξ
2
= 1
s ∈ R
s ∈ [−π, π]. g ∈
C
ξ
t
= gξe
ist
, t ∈ Z.
K(n) = |g|
2
e
isn
n ∈ Z
K(n) =
π
Z
−π
e
iλn
dm(λ),
m s g
s
gξ
z
0
= e
is
, N = 1
z
1
, ..., z
N
Mz
i
= 0, M|z
i
|
2
= σ
2
i
, Mz
i
z
k
= 0
i k 6= i s
k
, k = 1, ..., N
[−π, π]
ξ
n
=
N
X
k=1
z
k
e
is
k
n
, n ∈ Z.
K(n) =
N
X
k=1
σ
2
k
e
is
k
n
s
1
, ..., s
N
σ
2
1
, ..., σ
2
N
[−π, π]
e
is
k
n
, s
k
∈ [−π, π]
ξ
k
, k ∈ Z
M|ξ
k
|
2
= 1, k ∈ Z
ξ
n
, n ∈ Z
K(n) n = 0
n
K(n) =
Z
π
−π
e
iλn
dF (λ),
F [−π, π]
f(λ) =
1
2π
, f
1
(λ) =
1
√
2π
, g(λ) = 0,
f(λ) = 5 + 4 cos λ.
f(λ) = 5 + 2e
iλ
+ 2e
−iλ
.
K(t) =
π
Z
−π
e
iλt
f(λ)dλ =
10 sin πt
t
+
4 sin π(t + 1)
t + 1
+
4 sin π(t − 1)
t − 1
.
t ±1 K(0) =
10π K(±1) = 4π
f(λ) =
2e
2iλ
+ 5e
iλ
+ 2
e
iλ
=
P (e
iλ
)
Q(e
iλ
)
.
f(λ) = 2
e
iλ
+
1
2
e
iλ
+ 2
e
iλ
=
2 + e
−iλ
e
iλ
+ 2
.
f
1
(λ) = 2 + e
−iλ
, f
2
(λ) = e
iλ
+ 2,
c
0
= 2, c
−1
=
1, c
−k
= 0 k ≥ 4
m = 1
g(λ) = c
−1
f
−1
1
(λ) =
1
2
1
1 +
1
2
e
−iλ
,
g(λ) =
1
2
1 +
∞
X
j=1
(−1)
j
e
−ijλ
2
j
.
ˆ
ξ(1)
≤0
=
1
2
ξ(0) +
∞
X
j=1
(−1)
j
ξ(−j)
2
j+1
.
σ
2
(1) = 2πc
2
0
= 8π,
σ
2
(2) = 2π(c
2
0
+ c
2
−1
) = 10π = Dξ(0),
ξ(t)
m
∗
=
ξ(−N) + ... + ξ(0)
N + 1
.
Mξ
n
= m
K(n) =
π
Z
−π
e
iλn
dF (λ).
N
x
0
, x
1
, ..., x
N−1
m m
∗
ˆ
K
N
(n) =
1
N−n
P
N−n−1
k=0
x
k+n
x
k
, n ≥ 0
1
N+n
P
N
k=−n
x
k+n
x
k
, n < 0.
(i, j) i − j = m
N −|m|
ˆ
K
N
(n)
K(n)
f
N
(λ) =
1
2πN
N−1
X
s=0
N−1
X
t=0
K(s − t)e
−iλ(s−t)
≥ 0.
f
N
(λ) =
1
2π
X
|m|<N
K(m)
1 −
|m|
N
e
−iλm
.
F (λ) f
N
π
Z
−π
e
iλn
dF
N
(λ) =
1 −
|n|
N
K(n), |n| < N
|n| ≥ N
1
√
2π
e
−iλn
n ∈ Z
F (λ) = lim
N→∞
F
N
(λ)
F
f(λ) = lim
N→∞
f
N
(λ)
ˆ
f
N
(λ) =
1
2π
X
|m|<N
ˆ
K
N
(m)
1 −
|m|
N
e
−iλm
,
f
N
(λ)
ˆ
f
N
(λ) =
1
2πN
N−1
X
s=0
x
s
e
−iλs
2
.
1
2πN
N−1
X
s=0
N−1
X
t=0
x
s
x
t
e
−iλ(s−t)
=
1
2πN
X
|m|<N
X
t
x
t+m
x
t
e
−iλm
,
N −|m|
1
2πN
X
|m|<N
(N − |m|)
ˆ
K(m)e
−iλm
=
ˆ
f
N
(λ).
M
ˆ
K(n) = K(n)
ˆ
f(λ) f
N
(λ)
f
N
(λ) =
1
2πN
N−1
X
s=0
N−1
X
t=0
K(s − t)e
−iλ(s−t)
=
=
1
2πN
π
Z
−π
N−1
X
k=0
e
i(µ−λ)k
2
f(µ)dµ.
Φ
N
(s) =
1
2πN
N−1
X
k=0
e
isk
2
=
1
2πN
sin(sN/2)
sin(s/2)
2
M
ˆ
f
N
(λ) = f
N
(λ) =
π
Z
−π
Φ
N
(µ − λ)f(µ) dµ → f(λ)
λ N → ∞
ˆ
f
N
(λ)
f(λ)
ξ
n
f(λ) ≡
1
2π
ˆ
f
N
(λ) =
1
2π
1
√
N
N−1
X
k=0
x
k
e
−iλk
2
.
λ = 0
ˆ
f
N
(0) =
1
2π
ξ
2
, ξ
M|
ˆ
f
N
(0) − f(0)|
2
=
1
4π
2
M|ξ
2
− 1|
2
6−→0.
W
N
f
W
N
(λ) =
π
Z
−π
W
N
(λ − ν)
ˆ
f
N
(ν)dν.
W
N
(0) W
N
R
π
−π
W
N
(λ)dλ = 1
(∀λ) lim
N→∞
M|
ˆ
f
W
N
(λ) − f(λ)|
2
= 0
a
N
= o(N) → ∞
W
N
(λ) = a
N
B(a
N
λ).
B(µ) =
1
2π
sin(µ/2)
µ/2
2
,
B(µ) =
3
8π
sin(µ/4)
µ/4
4
,
a
N
α ∈ (0, 2]
B(µ) =
α+1
2α
(1 − |µ|
α
), |µ| ≤ 1,
|µ| > 1
p
n
(t) t
n
λ
n
µ
n
(∀n) (µ
n
= 0)
(∀n) (λ
n
= 0)
∆t
o(∆t)
∆t