The snake chain Let {Xn}n≥0 be
an hmc with state space E and transition matrix P. Define for L ≥ 1,...

Question

The snake chain Let {Xn}n≥0 be
an hmc with state space E and transition matrix P. Define for L ≥ 1, Yn
= (Xn, Xn+1,...,Xn+L). 

(a) The process {Yn}n≥0 takes
its values in F = EL+1. Prove that {Yn}n≥0 is an
hmc and give the general entry of its transition matrix. (The chain {Yn}n≥0
is called the snake chain of length L + 1 associated with {Xn}n≥0.)


(b) Show that if {Xn}n≥0 is
irreducible, then so is {Yn}n≥0 if we restrict the
state space of the latter to be F = {(i0,...,iL) ∈ EL+1; pi0i1
pi1i2 ··· piL−1iL > 0}. Show that if
the original chain is irreducible aperiodic, so is the snake chain. 

(c) Show that if {Xn}n≥0 has a
stationary distribution π, then {Yn}n≥0 also
has a stationary distribution. Which one?

Ajay · Accepted Answer

Explanation:
The snake chain Let {Xn}n≥0 be an hmc with state space E and transition matrix P. Define for L ≥ 1, Yn = (Xn, Xn+1,...,Xn+L).
Let A be the event that the chain is in a given set A at time n. Show that if A is an absorbing set, then for any x ∈ A,
limn→∞P(Yn = x) = 1.
 
(a) The process {Yn}n≥0 takes its values in F = EL+1. Prove that {Yn}n≥0 is an hmc and give the general entry of its transition matrix. (The chain {Yn}n≥0 is called the snake chain of length L + 1 associated with {Xn}n≥0.)
Yes, {Yn}n≥0 is an hmc.

The snake chain Let {X n } n≥0 be an hmc with state space E and transition matrix P. Define for L ≥ 1, Y n = (X n , X n+1 ,...,X n+L ). (a) The process {Y n } n≥0 takes its values in F = EL+1. Prove...

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