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).

Chart 1. The boundary of the oval above the green line is described by u(x) and below - by l(x)
We can paint the oval with strokes along red lines from
and to the equality of integrals
(double 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
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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