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