Exponentially Correlated Draws From Gamma Distribution

I've posted this in the math SE, but since it's algorithm related, I figured it would fit here as well.

I want to generate a series of random numbers corresponding to a given distribution, but in such a way that each draw is correlated to the previous according to some relation.

Suppose I have a correlation and a Gaussian distribution with mean mu and standard deviation sigma. I know that the following algorithm will give me exponentially correlated draws from the Gaussian distribution (algorithm modified from here):

f = exp(-lambda * dt)
r[0] = mu

for n > 0:
    g = sigma * randn()
    r[n] = f * r[n-1] + sqrt(1 - f^2) * g + mu

An example can be seen in this figure. Here, lambda = 0.1, mu = 10, and sigma = 4.
I'm plotting with two different series, each with a different dt.

As can be seen in this figure, the resultant series of draws (after enough samples) is approximately Gaussian with the same mu and sigma.

How can I achieve this same effect with a gamma distribution?