Let $Y_0, Y_1,Y_2,... $ be a sequence of independent and identically distributed random variables. Then we define $X_n = \displaystyle\sum_{j=0}^{n} Y_j $ , $n=0,1,2,...$.

This is a Random Walk process. I would like to get help to prove that this is Time-homogeneous.

For the Markov property, I considered increments of this process and proved that they are independent and then used that to deduce the Markov property. But I am unable to prove that the transition probabilities do not depend on time. Could anyone help me with this please?

Thank You


Hi: For time homogeneity, you need $P(X_{n+1} = I_{n+1}| X_{n} = I_{n})$ is independent of the time, namely $n$.

The process is a random walk so

$P(X_{n+1} = I_{n+1}| X_{n} = I_{n}) = P(Y_{n+1} = I_{n+1} - I_{n})$.

But $Y_{n}$ is iid, so $P(Y_{n+1} = I_{n+1} - I_{n})$ is not dependent on $n$ which means that the chain is time homogeneous.

  • $\begingroup$ Thank you! This was very helpful. $\endgroup$ – Tejo Gedela Sep 20 '19 at 10:29
  • $\begingroup$ @Tejo Gedela: Your welcome but reading this again, I'm not 100 percent sure it's correct because $I_{n+1} - I_{n}$ is still a function of $n$. Could someone comment on whether this answer is correct ? or correct it. Thanks. $\endgroup$ – mlofton Sep 21 '19 at 13:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.