Solved – Prove that the mean value of a convolution is the sum of the mean values of its individual parts

Like many demonstrations involving convolutions, it comes down to applying Fubini's Theorem.

Let's establish notation and assumptions.

Let $f$ and $g$ be integrable real-valued functions defined on $mathbb{R}^n$ having unit integrals (with respect to Lebesgue measure): that is,

$$1=int_{mathbb{R}^n} f(x) dx = int_{mathbb{R}^n} g(x) dx.$$

(For convenience, let's drop the "$mathbb{R}^n$" subscript, because all integrals will be evaluated over this entire space.)

The convolution $fstar g$ is the function defined by

$$(fstar g)(x) = int f(x-y) g(y) dy.$$

(This is guaranteed to exist when $f$ and $g$ are both bounded or whenever $f$ and $g$ are both probability density functions.)

The mean of any integrable function is

$$E[f] = int x f(x) dx.$$

It might be infinite or undefined.

Solution

The question asks to compute $E[fstar g]$ (in the special case where $f$ and $g$ are nonnegative–but this assumption doesn't matter). Apply the definitions of $E$ and $star$ to obtain a double integral; switch the order of integration according to Fubini's Theorem (which requires assuming $E[fstar g]$ is finite), then substitute $x-yto u$ and exploit linearity of integration (which is a basic property established immediately whenever any theory of integration is developed). The result will appear because both $f$ and $g$ have unit integrals.

For those who want to see the details, here they are:

$$eqalign{ E[fstar g] &= int x (fstar g)(x) dx &text{Definition of }E\ &= int x left(int f(x-y) g(y) dyright) dx &text{Definition of convolution}\ &= int g(y) left(int x f(x-y) dxright) dy &text{Fubini}\ &= int g(y) left(int (x-y)f(x-y) + yf(x-y) dxright) dy&text{Expand }x=(x-y)+y \ &= int g(y) left(int (x-y)f(x-y) dx + yint f(x-y) dxright) dy &text{Linearity of integration}\ &= int g(y) left(int u f(u) du + y int f(u) duright) dy &text{Substitution } x-yto u\ &= int g(y) (E[f] + y(1)) dy &text{Assumptions about }f\ &= E[f]int g(y) dy + int y g(y) dy &text{Linearity of integration}\ &= E[f](1) + E[g] &text{Assumptions about }g\ &= E[f] + E[g]. }$$

These calculations are legitimate provided all three expectations $E[fstar g], E[f], E[g]$ are defined and finite. Fubini's Theorem requires only the finiteness of $E[fstar g],$ but the steps at the end (involving linearity) also need the finiteness of the other two expectations.