# Generalized CLT for any operation

When multiple random variables are summed up, their sum converges to a normal distribution. If the RVs are instead multiplied together, the product tends to a lognormal distribution. There are some lesser known theorems that describe the distribution of the $$\min\{X_1,X_2,...,X_n\}$$ or the $$\max\{X_1,X_2,...,X_n\}$$ as $$n\rightarrow \infty$$ (by Fisher–Tippett–Gnedenko).

My question is, is there a generalized version of the CLT that describes the asymptotic distribution for any function of random variables? As in, the distribution of $$g(X_1,X_2,...,X_n)$$ as n tends to infinity?

• There are various conditions that have to be satisfied, including that there is convergence in some sense. For example if you took the product of random variables only supported on the values $\pm1$ then you would not get your lognormal result. Commented Oct 18, 2022 at 13:01
• It's not really possible to define something that would apply to "any" function. For example, a constant function would be a likely counterexample of whatever it would state.
– Tim
Commented Oct 18, 2022 at 13:44
• Search this site for the tag delta-method. For the univariate case stats.stackexchange.com/questions/583376/…, multivariate: stats.stackexchange.com/questions/243510/… Commented Oct 18, 2022 at 14:39

There can only be such a theorem if $$g$$ is well-behaved in some sense: in particular, it should only depend on the set of values provided, not the order.
Here is a big class of such functions, to which the central limit theorem can be applied directly: functions $$g$$ for which there is an invertible function $$f$$ where:
$$g(X_1,X_2,...,X_n) = f^{-1}(f(X_1)+\dotsm +f(X_n))$$
Your example of the lognormal distribution is such case, where $$f$$ is $$\log$$. Your example of $$\max$$ is the limit as $$k \to \infty$$ of $$f(x)=x^k$$, and similarly for $$\min$$ with $$f(x)=x^{-k}$$.