A process converging to a certain distribution

Question

How is it possible to build a discrete-time stochastic process so that converges to a specific distribution at equilibrium, for example an exponential distribution or a gamma distribution ?

Answer 1

Fix a characteristic function $\varphi$. Define the operator $T$ by $$ T[\psi]=\frac{1}{2}\left(\varphi+\psi\right). $$ Note that $T$ maps characteristic functions to characteristic functions since any convex combination of characteristic functions is once again a characteristic function.

Define a Markov chain as follows: $X_{0}$ is given and $X_{n+1}$ has characteristic function $T[\varphi_{n}]$ where $\varphi_{n}$ is the characteristic function of $X_{n}$. Since $\varphi$ is a fixed point of $T$, it follows that $\varphi$ is an equilibrium of this Markov chain.

Note that $$ T^{k}[\psi]=\left(1-\frac{1}{2^{k}}\right)\varphi+\frac{1}{2^{k}}\psi $$ so that this Markov chain also has $\varphi$ as its limiting distribution.

Answering your question: you can take $\varphi$ to be the characteristic function of the gamma distribution to achieve your goal.

There isn't really anything special about this construction. It satisfies your requirements, but you could come up with a bunch of other ways to do so.