Convolutions smooth out hard sharp things into nice smooth things. (See previous post for why convolutions are important). Here are some convolutions between various distributions:

Separate convolutions of uniform, gamma, beta, and two-peaked distributions. Each column represents one convolution. The top row is $f$ , the middle row $g$ , and the bottom row $f * g$ .

(For all five examples, all the action in the functions $f$ and $g$ is in positive regions - i.e. $f (x)$ > 0 only when x > 0, and $g$ too - though this isn't necessary.)

Things to notice:

Everything is getting smoothed out! Even the bimodal case, in the column all the way to the right, has a convolution that looks like it's closer to something smooth than its two underlying distributions are. You might say the spike that uniform $*$ uniform gives doesn't seem more smooth, exactly, than the uniform distributions that led to it. But notice the second column, which convolves a third uniform distribution with the result of the convolution from the first column. That second convolution leads to something more intuitively smooth (or if not smooth, at least more hill-shaped). We can repeat this procedure, feeding the output of the first convolution to the next, and each time we get a black result distribution that is a little bit shorter and a little bit wider. Thus we get a smooth thing by repeatedly convolving two input sharp things (f and g):

The more we do this (for these particular shapes $f$ and $g$ ), the more the final distribution will resemble a Gaussian. In other words, the central limit theorem applies to sets of uniform distributions.

Even the more difficult bimodal distributions smooth out under repeated convolution, although it takes longer:

I'm not sure whether this goes into a Gaussian or not. But the shape is suggestive.

$f * g$ (for positive $f$ and $g$ ) is always to the right of $f$ and $g$ both. Because all these example $f$ and $g$ are positive, their means are positive. And it's a theorem that the mean $⟨ f * g ⟩$ is always equal to the sum of the underlying means $⟨ f ⟩ + ⟨ g ⟩$ . So the location of the distributions is always increasing. Of course, for $f$ and $g$ with negative first moments, the location of $f * g$ would go leftward; a

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The central limit theorems