Does the multimodal probability distribution tend toward uniform distribution as number of modes becomes very large?

Question

Does the multimodal probability distribution tend toward uniform distribution as number of modes becomes very large? Multimodal probability density distribution is formed by the convex combination of independent normal probability density functions. When there are many such independent normal functions, we can see many maxima in the p.d.f., and it no longer looks like a normal distribution, and in fact, may look more like uniform distribution. Intuitively this looks logical, but is there a theorem/ lemma which states this?

Answer 1

Let's start with a mixture of two normal distributions:

For a 50:50 mixture of two normal distributions with the same SD σ, if you write the density $$f(x)=0.5g_1(x)+0.5g_2(x)$$ in full form showing the parameters, you will see that its second derivative changes sign at the midpoint between the two means when the distance between means increases from below 2σ to above.

Three cases are illustrated below. In each one the two normal curves that are 'mixed' have $\sigma=1.$ From left to right the distances between means are $3\sigma, 2\sigma,$ and $\sigma,$ respectively. The concavity of the mixture density at the midpoint (1.5) between means changes from $f^{\prime\prime}(1.5) < 0$ (dip) to $f^{\prime\prime}(1.5) = 0$ (flat) to $f^{\prime\prime}(1.5) > 0)$ (peak).

So if you mix two distributions, you may or may not be able to see bimodality (two distinct 'humps'). If you mix several normal distributions you may get something that looks like a normal distribution with a more-or-less flat top, but there will be tails at both ends.

In practice, many of the distributions modeled as normal are actually normal mixtures. For example, heights of men in the US are often described as normal, but are actually mixtures of heights of men from several different ethnic groups of moderately varying heights, but with means sufficiently close together that you don't see distinct modes.

In order to get a mixture distribution that looks uniform, you might combine uniform distributions various intervals. For example, a 50:50 mixture of the two distributions $\mathsf{Unif}(0,.5)$ and $\mathsf{Unif}(.5,1)$ should work.

m = 10^5
b = rbinom(m, 1, .5)
u1 = runif(m, 0, .5);  u2 = runif(m, .5, 1)
u = b*u1 + (1-b)*u2
hist(u, prob=T, xlim=c(-.1,1.1), col="skyblue2")
  curve(dunif(x), add=T, col="red", n=10001)