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

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?