Show that stochastic process $X_n = max\{\xi_1, ..., \xi_n\}$, where $\xi_1$, ..., $\xi_n$ are equally distributed Poisson random variables with rate $\lambda$, is a Markov chain. Find it's transition matrix for one step.Is this chain homogeneous?

I understand how to prove that this process is a Markov chain. But I have struggled with calculating the matrix and the last question too.

Pls help ^^

  • If at time $n$ the chain is in state $x$ (meaning $x$ is the biggest $\xi$ seen so far) what is the chance that the next $\xi$ will equal $y$, for values of $y\ge x$, and for values of $y<x$. Does this chance depend on $n$? – kimchi lover Dec 27 '17 at 23:16
  • For $ y \geq x$ it is 1, and for $y < x$ it is 0. Chance does not depend on n. So in the matrix all values above and on the main diagonal are ones, and below are zeros? However, homogeneity is still unclear for me :( – Roman Nikitin Dec 27 '17 at 23:35
  • Yes about not depending on $n$, but wrong about $y$. Think: the sum of the probs of the next $\xi$ equalling $y$ should be $1$, so the individual $y$ values for $y\ge x$ cannot all equal to $1$. – kimchi lover Dec 27 '17 at 23:39

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