In probability theory, a telescoping Markov chain (TMC) is a vectorvalued stochastic process that satisfies a Markov property and admits a hierarchical format through a network of transition matrices with cascading dependence.
For any
N
>
1
consider the set of spaces
{
S
ℓ
}
ℓ
=
1
N
. The hierarchical process
θ
k
defined in the productspace
θ
k
=
(
θ
k
1
,
.
.
.
.
.
,
θ
k
N
)
∈
S
1
×
.
.
.
.
.
.
×
S
N
is said to be a TMC if there is a set of transition probability kernels
{
Λ
n
}
n
=
1
N
such that

θ
k
1
is a Markov chain with transition probability matrix
Λ
1
P
(
θ
k
1
=
s

θ
k
−
1
1
=
r
)
=
Λ
1
(
s

r
)
 there is a cascading dependence in every level of the hierarchy,

θ
k
satisfies a Markov property with a transition kernel that can be written in terms of the
Λ
's,
P
(
θ
k
+
1
=
s
→

θ
k
=
r
→
)
=
Λ
1
(
s
1

r
1
)
∏
ℓ
=
2
N
Λ
ℓ
(
s
ℓ

r
ℓ
,
s
ℓ
−
1
)
where
s
→
=
(
s
1
,
…
,
s
N
)
∈
S
1
×
⋯
×
S
N
and
r
→
=
(
r
1
,
…
,
r
N
)
∈
S
1
×
⋯
×
S
N
.