For a Markov process $\{X_n\}$ is there any inequality available for $$ E[\sup_{0 \leq n \leq k} X_{n}]$$ in terms of moments of $E[X_n], 0 \leq n \leq k$
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1$\begingroup$ certainly not - there are zero mean MC's. You need to assume more, either higher moments or something about the state space. $\endgroup$– ofer zeitouniCommented Feb 18, 2016 at 8:48
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$\begingroup$ @oferzeitouni: Thanks. That is ok. I am not asking for general case. Can you please give me some references ? $\endgroup$– SoshaCommented Feb 18, 2016 at 8:49
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$\begingroup$ You could benefit by looking at the Doob's martingale inequality. You can look at its proof to see if you can do something similar for Markov processes, though I am not sure if it will be fruitful. $\endgroup$– Samrat MukhopadhyayCommented Feb 18, 2016 at 9:19
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$\begingroup$ @SamratMukhopadhyay: Unlike submarttingale, Markov chain sampled at increasing stopping times may not be a Markov Chain. And the proof you told depends on this heavily. $\endgroup$– SoshaCommented Feb 18, 2016 at 11:13
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