Information Technology Reference
InDepth Information
In the discrete case we have
k
1
=
i
x
i
p
i
,
x
2
E
(
)=
in the continuous case we obtain
∞
x
2
t
2
E
(
)=
ϕ
(
t
)
dt
.
−
∞
5. Sequences
In the section we want to present a method for studying of limit properties of some
sequences
(
)
B
(
)
→F
of observables ([7], [25], [31], [32], [49]). The main idea
is a representation of the given sequence by a sequence of random variables
x
n
n
,
x
n
:
R
(
ξ
n
)
n
,
ξ
n
:
(
Ω
,
S
,
P
)
→
R
. Of course, the space
(
Ω
,
S
)
depends on a concrete sequence
(
x
n
)
n
, for different
sequences various spaces
(
Ω
,
S
,
P
)
can be obtained.
The main instrument is the Kolmogorov consistency theorem ([67]). It starts with a sequence
of probability measures
(
μ
n
)
n
,
μ
n
:
σ
(
J
)
→
[
0, 1
]
such that
n
μ
n
+
1

σ
(
J
n
)
×
R
=
μ
n
i. e.
μ
n
+
1
(
A
×
R
)=
μ
n
(
A
)
for any
A
∈
σ
(
J
)
(consistency condition). Let
C
be the family of
n
all cylinders in the space
R
N
, i. e. such sets
A
R
N
that
⊂
A
=
{
(
t
n
)
n
;
(
t
1
, ...,
t
k
)
∈
B
}
,
R
k
k
∈
∈B
(
)=
σ
(
J
)
where
k
N
,
B
. Then by the Kolmogorov consistency theorem there exists
exactly one probability measure
P
:
σ
(
C
)
→
[
0, 1
]
such that
(
)=
μ
k
(
)
P
A
B
.
(6)
π
n
:
R
N
R
n
,
→
If we denote by
π
n
the projection
t
i
)
i
=
1
)=(
((
)
π
n
t
1
,
t
2
, ...,
t
n
,
then we can formulate the assertion (6) by the equality
(
π
−
1
P
(
B
)) =
μ
n
(
B
)
,
(7)
n
for any
B
∈C
.
Theorem 4.1.
Let
m
be a state on a space
F
of all IFevents.
Let
(
x
n
)
n
be a sequence of
R
n
observables,
x
n
:
B
(
R
)
→F
, and let
h
n
:
B
(
)
→F
be the joint observable of
x
1
, ...,
x
n
,
n
=
R
n
1, 2, .... If we define
μ
n
:
B
(
)
→
[
0, 1
]
by the equality
μ
n
=
m
◦
h
n
,
then
(
μ
n
)
n
satisfies the consistency condition

(
σ
(
J
)
×
)=
μ
n
.
μ
n
+
1
R
n