## Sum of random variables and convolution

### Link between double and iterated integrals

Why do we need this link? For simplicity consider the rectangle The integrals

and

both are taken over the rectangle but they are not the same. is a double (two-dimensional) integral, meaning that its definition uses elementary areas, while is an iterated integral, where each of the one-dimensional integrals uses elementary segments. To make sense of this, you need to consult an advanced text in calculus. The difference notwithstanding, in good cases their values are the same. Putting aside the question of what is a "good case", we concentrate on geometry: how a double integral can be expressed as an iterated integral.

It is enough to understand the idea in case of an oval on the plane. Let be the function that describes the lower boundary of the oval and let be the function that describes the upper part. Further, let the vertical lines and be the minimum and maximum values of in the oval (see Chart 1).

We can paint the oval with strokes along red lines from

and to the equality of integrals

(**double integral**)**iterated integral**)

### Density of a sum of two variables

**Assumption 1** Suppose the random vector

From the definitions of the distribution function

we have

The integral on the right is a double integral. The painting analogy (see Chart 2)

suggests that

Hence,

Differentiating both sides with respect to

If we start with the inner integral that is with respect to

**Exercise**. Suppose the random vector

### Convolution theorem

In addition to Assumption 1, let

This is denoted as **convolution**.

The following may help to understand this formula. The function

The convolution of two densities is always a density because it is non-negative and integrates to one:

Replacing

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