# Random variables stable by nonlinear function

Let $h$ a function and $X$ a random variable with CDF $F$.

We say that $X$ is stable by $h$ if $h(X)$ follows $F$.

I would like to know if there is a literature for those kind of random variables? (Note: Stable distributions are not closely related).

For example, existence and unicity under some conditions.

Examples: Here are three examples ($h_3$ is adapted from formula to sample Laplace distribution). Let:

$h_1 = \log |x|$

$h_2 = \text{sign}(x) \log |x|$

$h_3 = - \text{sign}(x) \log |1-2x|$

Density of a random variable stable by $h_1$

Density of a random variable stable by $h_2$

Density of a random variable stable by $h_3$

Code: Code in R.

h1 = function(x) {
log(abs(x))
}

h2 = function(x) {
sign(x)*log(abs(x))
}

h3 = function(x) {
-sign(x)*log(abs(1-2*x))
}

N = 1000000
x = runif(N, -10, 10)
# x = rnorm(N, -8, 9)
for(i in 1:200) {
print(i)
x = h3(x)
if(i %% 50 == 0) {
hist(x, breaks = 500, main = i, probability = TRUE, xlim = c(-10,10))
}
}


Note: I posted a related question in math.stackexchange first, see https://math.stackexchange.com/questions/2701525/solving-fx-fex-f-ex

• I fail to see how your examples illustrate your definition of "stable". In your third example you have $x \sim U(-10,10)$, and obviously $h(X)$ is not, so by your definition $X$ is not stable by $h$. Neither of your other two examples work either, AFAICT. – jbowman Mar 23 '18 at 3:29
• $x_0$ following $U(-10, 10)$ is the initial step, then we define $x_i = h(x_{i-1})$ and we hope for convergence to $x_{+\infty}$ stable by $h$. In presented examples, no matter initial distribution is uniform or normal, the resulting stable sample seems similar – ahstat Mar 23 '18 at 3:38
• Oh, the limiting distribution of a sequence of distributions $F_n$ generated by $x_{n+1} = h(x_n)$. That clarifies it, thanks. – jbowman Mar 23 '18 at 3:43
• You might find something useful in the Wikipedia pages for random dynamical system and measure-preserving dynamical system. – deasmhumnha Mar 23 '18 at 4:02

## 1 Answer

I looked for random dynamical system and measure-preserving dynamical system (from Dezmond Goff comment).

An accessible introduction can be found in the book Computational Ergodic Theory. A chapter of the book is available here: http://www.springer.com/gp/book/9783540231219 then click on "Download Sample pages 1 PDF (1.6 MB)".

In this chapter book:

• Fig. 2.2 is related with $h = 4 x(1-x)$,
• Fig. 2.5 is related with $h = 1/x - \left \lfloor{1/x}\right \rfloor$,
• Fig. 2.12 is related with $h = |1/x| - \left \lfloor{|1/x| + 1/2}\right \rfloor$,
• Fig. 2.13 is related with $h = 1/x - \left \lfloor{1/x + 1/2}\right \rfloor$.