# Exponentially Correlated Draws From Gamma Distribution

This is more about algorithms than math.

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 $$f =e^{-\lambda\Delta t}$$ 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):

$$f = e^{-\lambda \Delta t}$$
$$r_0 = \mu$$

for $$n > 0$$ :
$$\hspace{1cm} g_n = \sigma \times randn()$$
$$\hspace{1cm} r_n = f \times (r_{n-1} - \mu) + \sqrt{1 - f^2}\times g_n + \mu$$

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 $$\Delta t$$.

As can be seen here, 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?

EDIT

After some testing, I found that my original answer did not work well for $\rho \le 0.02$ or $\rho \ge 0.98$, so I did some digging and found this paper.

Using eqn(2.1) and eqn(4.3), I get very good results for all $\rho$.

Like below, a random draw, $r_t$ can be calculated as $r_t = \rho \times r_{t-1} + \zeta_t$, where $\rho$ is a correlation factor and $0 < \rho < 1$.

Again assuming a Gamma distribution with shape parameter $\alpha$ and rate parameter $\beta$, and noting that $Po(\lambda)$ is a draw from the Poisson distribution, apply the following:

$N = Po(\alpha\times \log{\frac{1}{\rho}})$

$\zeta_t = \sum\limits_{n=1}^{N}\rho^{U_n}Y_n$

where $U_n$ is an independent draw from the uniform distribution and $Y_n$ is an independent draw from the exponential distribution with parameter $\beta$

Edge cases:

if $\rho=0$, then $r_t = Ga(\alpha, \beta)$

If $\rho=1$, $r_t=r_{t-1}$

A coworker of mine was able to solve this problem, so I'm posting her solution here for completeness:

The solution to this problem is a stationary Gamma process; in particular an auto-regressive AR(1) process. The following algorithm comes from:

Walker, Stephen G., A note on the innovation distribution of a gamma distributed autoregressive process, Scand. J. Stat. 27, No.3, 575-576 (2000). ZBL0976.62091.

However, since that is behind a paywall, the algorithm is also presented here (section 2.1).

To summarize:

Assume a Gamma distribution with shape parameter $\alpha$ and rate parameter $\beta$ s.t. $\alpha,\beta > 0$. Assume each draw is correlated to the previous according to $\rho$ where $0 < \rho <1$.

A random draw, $r_t$ can be calculated as $r_t = \rho \times r_{t-1} + \zeta_t$

$\zeta_t$ can be calculated from the following, noting that $Ga(\alpha,\beta)$ is a random draw from the Gamma distribution and $Po(\lambda)$ is a random draw from the Poisson distribution:

$\lambda_t = Ga(\alpha,1)$

$N_t = Po(\frac{1-\rho}{\rho} \times \lambda_t)$

$\zeta_t = Ga(N_t, \frac{\beta}{\rho})$

An example can be seen here ($\alpha = 4, \beta = \frac{2}{3}$):

The resultant distribution (after a sufficient number of draws) is a Gamma distribution with approximately the same $\alpha$ and $\beta$ as the original distribution:

One final note: it's possible for $N_t=0$, in which case $\zeta_t = 0$