Probabilistic Performance Analysis of Fault Diagnosis Schemes

Proof. Let ϑ 0:l be a possible path. By Lemma 4.19, ϑ i−1 ≤ ϑ i , for i = 1,...,l, so the remainder
of the path ϑ 1:l makes at most m − ϑ 0 transitions from one state to another. If n such
transitions occur, then there are at most ( m−ϑ 0
)
n distinct sets of states that ϑ1:l may visit,
and there are no more than ( l
n)
combinations of times at which these transitions may occur.
Therefore, the total number of possible paths up to time l is upper-bounded by
C (l) :=
m∑
ϑ 0 =0
m−ϑ ∑ 0
n=0
( )( )
m − ϑ 0 l
.
n n
The bound ( )
l l(l − 1)···(l − n + 1)
:=
n
n!
< ln
n! ,
implies that
C (l) <
m∑
ϑ 0 =0
m−ϑ ∑ 0
n=0
(
m − ϑ 0
n
)
l n
n! = lm
m! +O(lm−1 ).
Of course, the structure of the transition probability matrices {Π k } depends on how
the states of the Markov chain are labeled. Since a relabeling of the states is affected by a
permutation, the following lemma analyzes the relationship between a Markov chain and its
permuted counterpart.
Lemma 4.21. Let θ ∼ ( Θ,{Π k },π 0
)
be a Markov chain, and let σ: Θ → Θ be a permutation.
Define
ˆπ 0 (i ) = π 0
(
σ(i )
)
, i ∈ Θ, (4.6)
and for all k ≥ 0 define
ˆΠ k (i , j ) = Π k
(
σ(i ),σ(j )
)
, i , j ∈ Θ, (4.7)
Then, the Markov chain ˆθ ∼ ( Θ,{ ˆΠ k }, ˆπ 0
)
has the same number of possible paths as θ.
Proof. Fix l > 0 and let ˆϑ 0:l be a path of ˆθ. For i = 0,1,...,l, define ϑ i := σ( ˆϑ i ). Then, the
equality
P( ˆθ 0:l = ˆϑ 0:l ) = ˆΠ( ˆϑ l−1 , ˆϑ l ) ··· ˆΠ( ˆϑ 0 , ˆϑ 1 ) ˆπ( ˆϑ 0 )
= Π ( σ( ˆϑ l−1 ),σ( ˆϑ l ) ) ··· Π ( σ( ˆϑ 0 ),σ( ˆϑ 1 ) ) π ( σ( ˆϑ 0 ) )
= Π(ϑ l−1 ,ϑ l ) ··· Π(ϑ 0 ,ϑ 1 ) π(ϑ 0 )
= P(θ 0:l = ϑ 0:l )
implies that ˆϑ 0:l is a possible path of ˆθ if and only if ϑ 0:l is a possible path of θ. Since the
permutation σ is a bijection, θ and ˆθ have the same number of possible paths.
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