# Central limit theorem for distribution peak rather than mean

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.

• i couldn't get what you are asking -- the CLT does not depend on the underlying distribution of the $X_i$, it can be asymmetric or even discrete. – gt6989b Mar 16 '16 at 14:17
• The question is cumbersome because such a function trivially exists ($\alpha=\sqrt n$ and $H$ outputs $\phi+n^{-1}S_n-\mu$). Is your goal estimation of the mode? If so, you can do it since the empirical distribution converges to the actual distribution and you'll probably get some kind of a normal convergence for continuous single moded distributions. – A.S. Mar 16 '16 at 14:37
• @A.S., I think that the question is to construct a statistic. That is, it should not depend neither on $\mu$ nor on $\phi$. Now concerning the question, it is not easy. Estimating the mode consistently, i.e. constructing an estimator $H(X) = H(X_1,\dots,X_n)$ such that $H(X)\to \phi$, $n\to\infty$, is already a hard task, and there are many methods. Usually the idea is to use a kernel estimator to estimate the density and then find its maximum. But there are many caveats, and it is hard to explain everything here. You should search for mode estimators on Google Scholar. – zhoraster Mar 16 '16 at 14:49
• I would be surprised if there was a good general mode estimator which had an unbiased normal distribution as its limit, just by considering a sample drawn from an exponential distributed random variable where none of the observations will be below the mode. A random variable with a uniform distribution may not provide any convergence, while one with a discrete distribution may have part of the the distribution of the estimator with a positive probability of $0$. – Henry Mar 16 '16 at 14:50
• If I wanted a mode estimator from R, I might use something like the following den <- density(obs); mode_estimate <- den$x[den$y == max(den\$y)]  – Henry Mar 16 '16 at 14:52