# Stochastic process difference equation: stationary distribution

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...

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

## 1 Answer

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.