How can I find the stationary distribution (as t goes to infinity) of stochastic difference equations in the form:

$x_{t+1} = a*x_t + b*N(0,1)$

where N(0,1) is a standard normal pdf

I have numerical results for a few examples I'm working through from notes, but I'm having trouble following the analytic derivation. There must be a simpler solution...

  • $\begingroup$ Are you sure it's not "as $t$ goes to infinity"? $\endgroup$ – Herr K. Jan 29 '18 at 14:35
  • $\begingroup$ Yes, that is what I meant. My apologies for the error. $\endgroup$ – user14631 Jan 29 '18 at 19:26

If a stationary distribution exists, the time index no longer matters as t goes to infinity. Replace x_{t+1} and x_{t} by x and solve for x. What you obtain is a random variable whose distribution you can infer. In your case you get x = b/(1-a) N(0,1) which has normal distribution with mean 0 and variance (b/(1-a))^2.

| improve this answer | |

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.