"Superposition principle" in probability: from one random variable to many I'm wondering if there is a principle like the superposition principle in the probability context.
For example, if I have a discrete random variable $X$, I'd like to know in which conditions it's possible to decompose $X$ as the sum of random variables.
 A: Yes, it's called "infinite divisibility":
A random variable $S$ is infinitely divisible if it can be expressed as the sum of an arbitrary number of i.i.d random variables.
Key examples are Normal, Poisson.
A (somewhat advanced/unhelpful) sufficient condition is that you can form a Levy process from it
A related concept is expressing the distribution of the random variable as a mixture of other distributions. Here, you are explicitly decomposing the distribution as the convex sum of other distributions -- but that will not mean the resulting random variable is the convex sum of random variables that are distributed as each of those mixture components.
Specifically, let $\sum_1^n \lambda_i = 1, \;\;\lambda_i>0$, then
$$f_X(x) = \sum_{i=1}^n\lambda_if_i(x)  \nRightarrow X = \sum_{i=1}^n\lambda_iX_i $$
A mixture model is a superposition of distributions but not random variables, whereas an infinitely divisible random variable can be decomposed into the sum of random variables.
A: To add to Bey's answer we might want a concrete example of a random variable (r.v.) as the sum of other r.v.s.
The normal distribution case is particularly neat: $N(\mu_1,\sigma^2_1)+N(\mu_2,\sigma^2_2)=N(\mu_1+\mu_2,\sigma^2_1+\sigma^2_2)$.
So, for instance, the r.v. $Z=N(0, 1)$ can be split into $N(0,0.5)+N(0,0.5)$ or $N(-1,0.6)+N(1,0.4)$ or anything of the form $N(\mu,\sigma^2)+N(-\mu,1-\sigma^2)$ for $\mu\in\mathbb{R},0<\sigma<1$. And there's no need to stop at two: you can, inductively, sum any number of independent normally distributed variables in the way you expect.
There's no need for infinite divisibility to satisfy the much looser condition you impose, though. For instance, the binomial r.v. $\text{Bin}(n, p)$ is effectively by definition the sum of $n$ i.i.d. Bernoulli r.v.s, $\text{Ber}(p)$.
Vacuously, the constant $0$ is a random variable and you can split any r.v. $X$ into $X=0+X$.
For a meaningful decomposition of a discrete r.v., you'll want it to take at least $3$ possible values. If it has $n$ possible values $x_1,\dots,x_n$ with probabilities $p_1,\dots,p_n$ then it's naturally a convex sum of $n$ indicator functions, "the $i$th outcome happens" (with distribution $x_i\text{Ber}(p_i)$), none of which are even pairwise independent.
