## Density of a sum of independent variables is given by convolution

This topic is pretty complex because it involves properties of integrals that economists usually don't study. I provide this result to be able to solve one of UoL problems.

## General relationship between densities

Let be two independent variables with densities . Denote the joint density of the pair .

By independence we have

(1)

Let be the sum and let be its density and distribution function, respectively. Then

(2) .

These are the only simple facts in this derivation. By definition,

(3) .

For the last probability in (3) we have a *double integral*

.

Using (1), we replace the joint probability by the product of individual probabilities and the double integral by the *repeated one*:

(4)

.

The geometry is explained in Figure 1. The area is limited by the line . In the repeated integral, we integrate first over red lines from to and then in the outer integral over all .

(3) and (4) imply

.

Finally, using (2) we differentiate both sides to get

(5) .

This is the result. The integral on the right is called a **convolution** of functions .

**Remark**. Existence of density (2) follows from existence of , although we don't prove this fact.

**Exercise**. Convolution is usually denoted by . Prove that

- .
- .
- If is uniformly distributed on some segment, then is zero for large .

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