0
$\begingroup$

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

Improve this question
$\endgroup$

1 Answer 1

Reset to default
0
$\begingroup$

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.

Improve this answer
$\endgroup$
2
  • $\begingroup$ Thank you! This was very helpful. $\endgroup$
    – Tejo Gedela
    Sep 20, 2019 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, 2019 at 13:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

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