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?
There isn't a "the" for multimodal distributions. There's an infinite number of different ones.
In general, no, adding modes doesn't necessarily make the result tend toward uniformity. You need to specify how the modes are being added, but in general they might look like almost any distribution.
Consider, for example, that I place a first mode at 1, and a second mode (say 3/4 as high, 2/3 as wide) at 2 and continue in similar fashion putting "smaller" modes at 3, 4, 5, .... each with half the contribution to total area of the one before it.
Then this would never look uniform no matter how many modes were added.
Multimodal probability density distribution is formed by the convex combination of independent normal probability density functions
Not in general, but they can be.