Stationary distribution

1. Sep 29, 2009

Ardla

Hi, can someone please provide some guidance on how i should go about finding the stationary distribution of:

$$X_t =$$ $$\rho X_{t-1} + \epsilon_t$$, $$X_0 = 0$$and $$|\rho|<1$$
where $$\epsilon_1, \epsilon_2, \cdots$$ are all independent N(0,1)..

i have no idea what to do, so here's my attempt which i know to be completely wrong:
suppose,
$$Var(X_1) = \rho \sigma^2 < \infty$$
$$Var(X_2) = \rho\sigma^2 + 1$$
$$\vdots$$
$$Var(X_{n+1}) = \rho\sigma^2 + t$$
As $$t \rightarrow \infty, Var(X_{n+1} = \rho \sigma^2 + t$$ ????????

Last edited: Sep 30, 2009
2. Sep 29, 2009

mathman

the first line of your post is garbled. You need to fix it to get a response.

3. Sep 30, 2009

Ardla

4. Sep 30, 2009

mathman

Since X0=0, V(X1)=1.
Next V(X2)=r2+1.
V(X3)=r2(r2+1)+1.
etc. (r=rho).

The limit as n->oo of V(Xn)=1/(1-r2)

Last edited: Sep 30, 2009
5. Sep 30, 2009

Ardla

THank you!!

Sorry can i ask another question in this same post? Well is that MC aperiodic too because $$V(X_1) = 1$$? Is that the value of d(i)?

Last edited: Sep 30, 2009
6. Oct 1, 2009

mathman

I need to know what your terms mean. "MC aperiodic " - what is MC?, "d(i)" - what is d and what is i?

7. Oct 1, 2009

Ardla

Sorry I meant Markov Chain. Um the d(i) is the period defined as:
$$d(i) = gcd$${$$n:p_{ii}(n)>0$$}
i.e.the greatest common divisor
and something is aperiodic if d(i) = 1.

Coz like in my notes it says Markov is aperiodic if it can access all states. But im not sure about how to show it

8. Oct 2, 2009

vineethbs

Hi,
A Markov chain is aperiodic if all the states are in one class (as periodicity is a class property and the chain itself is called aperiodic in your case) and starting from state i, there is a non-zero probability of transition to state i (this is of course given by your definition of d(i)). I think if it can access all states the chain would be called irreducible (because it is a single communicating class and there are no other states so that it is also closed).

About your first question, I just want to confirm whether you are also getting the stationary distribution as N(0,1/1-rho^2).

Now as your state space is continuous I am not really very sure about the definition of periodicity. I guess that a reasonable defintion of aperiodicity would be when P(X(1) \in A | X(0) \in A) > 0 for A \in the state space. (Also there would be some restriction on what A could be). Because epsilon is Gaussian I think that the chain is aperiodic.

9. Oct 2, 2009

Ardla

hmm! well i think that the stationary distribution is right..

umm yeahh im not sure about the aperiodicity too, but i agree i think that its aperiodic. I think that i'll just say coz it can reach any subspace in one step? its ok, i think that i'll get the answer from my lecturer when uni starts again..

but thank you guys!