Information Technology Reference
In-Depth Information
X
∞
and
X
∞
. It follows
is closed set in
ω
is a limiting point of
A
in
ω ∈ A
.We
X
∞
\ A
0
and
proved
B ⊆ A
0
.Then
B
=
A
0
. It follows that
A
0
are elements of
A
.Weusethat
P
0
is continuous. Then
n→∞
P
0
(
lim
B
n
)=1
.
X
∞
\ B
n
)=0,so
X
∞
\ B
n
can be taken as the set where
We cho ose
B
n
and
P
0
(
we refuse to accept
H
0
.Wehave
A
0
∩ A
1
(
θ
)=
∅
and
X
∞
\ B
n
)=
X
∞
\ A
0
.
lim
n→∞
(
Then for every
θ ∈ Θ
X
∞
\ B
n
)=
X
∞
\ A
0
)=1
n→∞
P
1
θ
(
lim
P
1
θ
(
.
We proved the following lemma.
Lemma 1.
If probability measures.
P
0
and
P
1
θ
,θ∈ Θ
are perpendicular on
X
∞
, A
themeasurablespace
(
)
and
A
0
(
θ
)
are closed for all
θ ∈ Θ
in Tychonoff
product, then exist consistent tests of
H
11
.
It can be proved that an arbitrary closed set can be represented as a limit of
a decreasing sequence of cylindrical sets.
H
0
versus alternatives
Lemma 1'.
If Probability Measures.
P
0
and
P
1
θ
,θ∈ Θ
are perpendicular on
X
∞
, A
the measurable space
(
)
can be represented as a limit of
a decreasing sequence of cylindrical sets, then exist consistent tests of
)
and every
A
0
(
θ
H
0
versus
alternatives
H
11
.
3
Wardens with Limited Resources
S
F
U
Assume that the link
possesses an additional interface
for the warden
S
who can see either the whole sequence of messages in
or a part of it, depending
on the properties of the new channel from
KA
to
U
which we denote
S
(
F
).
U
has the task to find out whether a covert transmission from agent
KA
to agent
KB
takes place or not. Both
U
and
KB
know
P
0
.
KB
knows
P
1
θ
as well, but
U
does not know
P
1
θ
.If
U
sees all messages in
S
, then lemma 1 shows when
U
KA
to
KB
.
can see the covert transmission from
Let us consider
S
(
F
) that does not transmit to
U
the whole trac of
S
,e.g.
low level protocol messages are omitted in
S
(
F
). Choose the set
X
1
,
X
1
⊆ X
and messages of
X
1
are unseen to
U
.Wehave
X
0
∪ X
1
=
X, X
0
∩ X
1
=
∅
.We
have that messages of
X
1
are taken away from the sequence of messages in
S
.
As a result
U
sees the reduced sequence and draw his conclusions from it. Let
X
0
A
1
be
σ
-algebra which is generated by cylindrical sets of
.
γ ∈ X
∞
.Denoteby
γ → α, β, γ ∈ X
∞
the unique decomposition, where
Let
α
is the subsequence of all elements of
X
0
in
γ
,and
β
is the subsequence of all
elements of
X
1
in
γ
. One of the sequences
α
or
β
may be empty.
