Chapter 2: Stochastics

[jensen]

Can anyone confirm my understanding, please:

1) Independent increments => Markov process

2) iid => Stationary => Markovian

are these correct?

Thanks.

 

[maryam]

My Idea

The first one seem to be correct as independent Increments imply Markov process. If you want to check whether or not a stochastic process is Markov, the following rules can be used:
1. Check if it has independent increments, if yes then it is markov
2. if it does not have independent increments, it may still be if it satisfy the markov definition.

For the second one,White noise is strictly stationary but white noise is a set of iid random variable. But am sure that a stationary process does not imply markov.

My opinion

Can anyone confirm my understanding, please:

1) Independent increments => Markov process

2) iid => Stationary => Markovian

are these correct?

Thanks.

 

[jensen]

The first one seem to be correct as independent Increments imply Markov process. If you want to check whether or not a stochastic process is Markov, the following rules can be used:
1. Check if it has independent increments, if yes then it is markov
2. if it does not have independent increments, it may still be if it satisfy the markov definition.

For the second one,White noise is strictly stationary but white noise is a set of iid random variable. But am sure that a stationary process does not imply markov.

My opinion

Thanks Maryam.

So if I understand you, stationary does not mean Markov, but iid (which is happens to be also stationary) implies Markov.

Is it possible to have a process that is stationary but not iid?

Thanks.

 

[Robert Chadburn]

Yes I agree - iid increments must mean markov because the future state will only depend on the current state and the value of the increrment. So, because the increment is independent of everything that's happened before, then the future value of the state must also be independent of what's happened before - ie Markov.
Stationarity just means that the statistical properties of the process are the same at all time points. A non-Markov process could easily be stationary (many time-series processes fall into this description).
Robert

 

[jensen]

ok, so:

1) iid means Markov
2) stationary does not necessarily mean Markov, and
3) stationarity has got nothing to do with iid, or vice versa.

Thanks!