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please prove there is no stationary distribution for {Xn}

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please prove there is no stationary distribution for {Xn}
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Answered 1 days After Sep 18, 2022

Solution

Ajay answered on Sep 19 2022
56 Votes
Discrete-Time Markov Chains.
 
Let {Zn}n≥1 be an IID sequence of geometric random variables: For k ≥0, P(Zn = k) = p(1 - p)^k where p∈ (0, 1). Let Xn = max(Z1,....,Zn) be the record value at time n, and suppose X0 is an N-valued random variable independent of the sequence {Zn}n≥1. 
 
Show that {Xn}n≥1 is an HMC and give its transition matrix.
 
 
This is a first-step analysis; a more complete analysis would give the equili
ium distribution of Xn.
The transition matrix for {Xn}n≥1 is given by:
 
P(Xn=k) = p(1-p)^(k-1)
 
P(Xn=0) = 1- p
 
P(Xn=N) = p(1-p)^N
 
P(Xn=k|Xn-1=k-1) = p
 
P(Xn=k|Xn-1=k) =...
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