Stationary Increments & Markov Property - True or False?

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SUMMARY

A counting process with stationary increments is indeed time homogeneous, establishing the relationship: stationary increments → time homogeneous. Furthermore, independent increments do imply the Markovian property, confirming that independent increments → Markovian property. However, the converse implications are false, meaning that time homogeneity does not guarantee stationary increments, nor does the Markovian property ensure independent increments. The discussion highlights the nuanced connections between these concepts in stochastic processes.

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Hello

Is it true that if a counting process has stationary increments, then it is time homogeneous?
stationary increments → time homogeneous?

And is it also true that independent increments gives Markovian property?

that is :
independent increments → markovian property ?

The opposite implications are false?

And is there a connection between stationary increments and the markovian property? I know that the nonhomogeneous poisson process is a markov process. So we can not say that markov → stationary increments, but can the opposite implication hold?
 
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