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Let $X = (X_n)_{n \in \mathbb{N}_0}$ be a homogeneous Markov-chain and $(n_r)_{r \in \mathbb{N}_0}$ a monotonic increasing sequence in $\mathbb{N}_0$. How can I show that $$Y = (Y_r)_{r \in \mathbb{N}_0} = (X_{n_r})_{r \in \mathbb{N}_0}$$ is as well a Markov-chain?

I could argue that the multiplication of the transition matrix is still a transition matrix, but this doesn't seem clean enough. I think you could also somehow prove this by induction, but I am unsure of the induction step as this is my first Markov-chain homework.

Hekri
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  • it is only true if $n_r$ is an arithmetic sequence – mercio Jun 30 '17 at 15:37
  • Does your definition of Markov chain allow for time-inhomogeneity? – kimchi lover Jun 30 '17 at 15:57
  • @kimchilover No. I edited the question – Hekri Jun 30 '17 at 16:10
  • I think mercio is correct in the sense that only if $(n_r)$ is an arithmetic sequence is $Y$ is a homogeneous Markov chain whenever $X$ is a homogeneous Markov chain. But if $X$ is an i.i.d. sequence then $Y$ is a homogeneous Markov chain for all subsequences $(n_r)$. – kimchi lover Jun 30 '17 at 17:24
  • Could you provide me with a counter example? Nothing comes to my mind – Hekri Jun 30 '17 at 18:45
  • @hekri, counter example to what? If you take "Markov chain" to mean possibly inhomogeneous Markov chain, then the answer to your original question is always "yes". Your intuition that products of transition matrices are transition matrices is of course correct; in the inhomogeneous case the number of matrices you are multiplying varies, and the products work out differently. Just for info, what is your precise definition of Markov chain? – kimchi lover Jun 30 '17 at 21:43
  • Counter example to my question i.e. an example for a monotonic increasing sequence that doesn't yield a homogenous markov chain. I use the following definition/property of homogenous markov chains $\mathbb{P}(X_{n+1} = y \vert ( X_n, X_{n-1}, \dots, X_0) = (x, x_{n-1}, \dots,x_0)) = \mathbb{P}(X_{n+1} = y \vert X_n = x) = \mathbb{P}(X_{1} = y \vert X_0 = x)$ – Hekri Jul 01 '17 at 11:18

1 Answers1

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This problem can be found in Grimmett, Stirzaker Probability and Random Processes, p. 219, 3rd edition.

For any sequence $i_0$, $i_1$, ... of states, consider

\begin{align} P(Y_{k+1}=i_{k+1} | Y_r=i_r \text{ for } 0\leq r \leq k)&=\frac{P(X_{n_s}=i_s \text{ for } 0 \leq s\leq k+1)}{P(X_{n_s}=i_s \text{ for }0\leq s \leq k)} \\ & \overset{(*)}=\frac{\prod \limits_{s=0}^{k}p_{i_s,i_{s+1}}(n_{s+1}-n_s)}{\prod \limits_{s=0}^{k-1}p_{i_s,i_{s+1}}(n_{s+1}-n_s)} \\ &=p_{i_k,i_{k+1}}(n_{k+1}-n_k)=P(Y_{k+1}=i_{k+1} | Y_k=i_k). \end{align}

Here $(*)$, we use that it the sequence is increasing.

Anton Steinfadt
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