Harashima Senju
Ton up Member
Does an IID sequence imply that it statisfies the Markov property?
-
We are pleased to announce that the winner of our Feedback Prize Draw for the Winter 2024-25 session and winning £150 of gift vouchers is Zhao Liang Tay. Congratulations to Zhao Liang. If you fancy winning £150 worth of gift vouchers (from a major UK store) for the Summer 2025 exam sitting for just a few minutes of your time throughout the session, please see our website at https://www.acted.co.uk/further-info.html?pat=feedback#feedback-prize for more information on how you can make sure your name is included in the draw at the end of the session.
Markov Property
- Thread starter Harashima Senju
- Start date
Yes, in fact, this is a good example of processes that have the Markov property but DO NOT have independent increments.
A process with independent increments satisfies the Markov property. However, not the other way round.
Consider a IID sequence Zn = 1 or -1 on the toss of a coin (50/50)
It's trivially Markov because P[Z2=1|Z1] = P[Z2=1|F1]
(each coin toss is independent of the last, so knowing the history won't change our predictions of the future)
Now, what is P[Z2 - Z1] = 2?
0.25, since we must have Z1 = -1 and Z2 = 1
But what if we know that Z1 - Z0 = 2. Now, what is P[Z2 - Z1] = 2?
0 - it's impossible because we must have Z0 = -1 and Z1 = 1.
So, the increment Z1 - Z0 is NOT independent of Z2 - Z1
I'd arm myself with this example just in case they ask for one in the CT4 exam (though they never do!)
Good luck!
John
A process with independent increments satisfies the Markov property. However, not the other way round.
Consider a IID sequence Zn = 1 or -1 on the toss of a coin (50/50)
It's trivially Markov because P[Z2=1|Z1] = P[Z2=1|F1]
(each coin toss is independent of the last, so knowing the history won't change our predictions of the future)
Now, what is P[Z2 - Z1] = 2?
0.25, since we must have Z1 = -1 and Z2 = 1
But what if we know that Z1 - Z0 = 2. Now, what is P[Z2 - Z1] = 2?
0 - it's impossible because we must have Z0 = -1 and Z1 = 1.
So, the increment Z1 - Z0 is NOT independent of Z2 - Z1
I'd arm myself with this example just in case they ask for one in the CT4 exam (though they never do!)
Good luck!
John