The central limit theorem states that the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed. i.e.
$$ \lim_{n \to \infty} \sqrt{n}\left(\frac{1}{n}\sum_i X_i - \mu\right) = \mathcal{N}(0,\sigma^2). $$
My question is whether a similar result exists which allows for some combinations of samples to tend towards a normal distribution centred about the peak (i.e. mode) of the distribution of the $X_i$, this obviously corresponds to the mean for normal distributions but I am interested in non-symmetric distributions where this is not the case. So I am asking if there exists some function $H$ s.t.
$$ \lim_{n \to \infty} \alpha(H(X_1,...,X_n) - \phi) = \mathcal{N}(0,\sigma^2), $$
where $\alpha$ is some number which might depend on $n$ and $\phi$ is the mode of the distribution from which the $X_i$ are drawn.