Finding Stationary Distribution for a Stochastic Process

[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 $$t \rightarrow \infty, Var(X_{n+1} = \rho \sigma^2 + t$$ ?

yeah I am very sure I am not doing it right... Can someone please help me out?

 

[mathman]

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

 

[Ardla]

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

Sorry i didn't realize, I fixed it now, can you please help?

 

[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(X_n)=1/(1-r²)

 

[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)?

 

[mathman]

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)?

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

 

[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 I am not sure about how to show it

 

[vineethbs]

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 I am not sure about how to show it

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.

 

[Ardla]

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

umm yeahh I am 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!