## Fixed point of a distributional recurrence equation: how to simulate the random variable?

Hello everybody! It's my first post on mathoverflow.

I have the following distributional recursion: $$h^{t+1} =^{d} B+\sum_{i=1}^K f(h_i^t)\ \ \ \text{for}\ t=0,1,2,..$$ $$h^0=B$$ where:

• $f$ is a deterministic function, precisely $f(x)=atanh(tanh(b)\;tanh(x))$
• $B>0$ and $b>0$ are deterministic constants
• $K$ is a random variable taking integer non negative values, with a chosen probability distribution
• $(h_i^t)_i$ are i.i.d. copies of the random variable $h^t$, and they're independent of $K$.

Now I know the recursion has a unique fixed point $h^*$ supported on $[0,\infty[$, i.e.

$$h^* =^{d} B+\sum_{i=1}^K f(h_i^*)$$

and I know that it is the distributional limit of the recursion, i.e.

$$h^t\xrightarrow[t\rightarrow\infty]{d}h^*$$

I would like to do some simulations of the random variable $h^*$ with mathlab. Do you know a method to do it?

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