Information Technology Reference
In-Depth Information
p
,
p
If we instead of
ε
use
∈
N
, then
η
n
(
ω
)
→
0 if and only if
1
p
,
1
p
))
ω
∈
η
−
1
∀
p
∃
k
∀
n
>
k
:
((
−
.
n
→
{ω
η
(
ω
)
→
}
And
η
n
0 almost everywhere, if the set
;
0
has measure 1.
(
)
Definition 4.1.
n
of observables
(i) converges in distribution to a function
F
:
R
A sequence
y
n
→
R
,if
(
((
−
∞
)) =
(
)
lim
n
m
y
n
,
t
F
t
→
∞
for every
t
∈
R
;
(ii) it converges to 0 in state
m
:
F→
[
0, 1
]
,if
lim
n
m
(
y
n
((
−
ε
,
ε
))) =
0
→
∞
ε >
for every
0;
(iii) it converges to 0
m
-almost everywhere, if
1
p
,
1
p
)) =
k
+
i
lim
p
lim
k
lim
i
m
(
∧
k
y
n
(
−
0.
=
n
→
∞
→
∞
→
∞
(
)
(
η
n
)
Theorem 4.4.
Let
y
n
n
be a sequence of observables,
n
be the sequence of corresponding
random variables. Then
(i)
(
y
n
)
n
converges to
F
:
R
→
R
in distribution if and only if
(
η
n
)
n
converges to
F
;
(ii)
y
n
)
n
converges to 0 in state
m
:
F→
[
0, 1
]
if and only if
(
η
n
)
n
converges to 0 in measure
P
:
S→
[
0, 1
]
(iii) if
(
η
n
)
n
converges
P
-almost everywhere to 0, then
(
y
n
)
n
m
-almost everywhere converges
to 0.
The details can be found in [66]. Many applications of the method has been described in [25],
[31], [35], [37], [39] , [52].
6. Conditional probability
Conditional entropy (of
A
with respect to
B
) is the real number
P
(
A
|
B
)
such that
(
∩
)=
(
)
(
|
)
P
A
B
P
B
P
A
B
.
(
|
)=
(
)
When
A
,
B
are independent, then
P
A
B
P
A
, the event
A
does not depend on the ocuring
of event
B
. Another point of view:
P
(
A
∩
B
)=
P
(
A
|
B
)
dP
.
B
