Convolution: telegram style notes

Shota Gugushvili

December 4, 2013

1 Definition and properties

Convolution F ∗ G of two distribution functions F and G is defined as Z F ∗ G(z) = F (z − x) dG(x), ∀z ∈ R. R Theorem 1. Let X and Y be independent random variables with distribution functions F and G. Then F ∗ G is the distribution function of X + Y. Proof. We want to show

P(X + Y ≤ z) = F ∗ G(z), ∀z ∈ R. Fix z ∈ R and define a function ( 1, if x + y ≤ z, h(x, y) = 0, otherwise. Then by independence of X and Y and Fubini ZZ P(X + Y ≤ z) = h(x, y)µX,Y ( dx, dy) R2 Z Z = µY ( dy) h(x, y)µX ( dx) R R Z Z = µY ( dy) µX ( dx) R (−∞,z−y] Z ∞ = F (z − y) dG(y). −∞ Since z is arbitrary, the result follows.

Corollary 2. Convolution is a commutative operation. Theorem 3. Convolution of absolutely continuous distribution functions F and G with densities f and g is absolutely continuous with density Z h(z) = f ∗ g(z) = f(z − x)g(x) dx. R

1 Proof. By Fubini Z z Z z Z ∞ h(u) du = du f(u − v)g(v) dv −∞ −∞ −∞ Z ∞ Z z  = f(u − v) du g(v) dv −∞ −∞ Z ∞ = F (z − v) dG(v) −∞ = F ∗ G(z).

This proves the theorem.

Theorem 4. Let F be an absolutely continuous distribution function with density f and let G be another distribution function. Then F ∗ G is absolutely continuous with density Z h(z) = f(z − t) dG(t). R Proof. By Fubini Z z Z z Z h(x) dx = f(x − t) dG(t) dx −∞ −∞ R Z Z z  = f(x − t) dx dG(t) R −∞ Z = F (z − t) dG(t) R = F ∗ G(z).

Let us now determine the probability law corresponding to the convolution of two distribution functions. For arbitrary Borel subsets A and B of R let A ± B = {x ± y : x ∈ A, y ∈ B}.

In particular, x ± B = {x} ± B and −B = 0 − B. Theorem 5. Let F and G be two distribution functions and µ and ν be the corresponding laws. Let µ ∗ ν be the law corresponding to F ∗ G. Then for each Borel set B, Z µ ∗ ν(B) = µ(B − y)ν( dy). R Proof. You check that µ ∗ ν is a probability law on (R, B(R)). To prove that its distri- bution function is F ∗ G, it is enough to show that for all B = (−∞, x], x ∈ R, µ ∗ ν(B) = F ∗ G(x).

But this is obvious from definitions.

2 Theorem 6. Let µ and ν be two probability laws on (R, B(R)). For each Borel measurable g that is integrable with respect to µ ∗ ν, we have Z Z Z g(u)µ ∗ ν( du) = g(x + y)µ( dx)ν( dy). R R R

Proof. Let g be an indicator function 1B. Then for each y the function gy(x) = g(x + y) is the indicator function of the set B − y. Thus Z g(x + y)µ( dx) = µ(B − y), R and the statement reduces to that given in the previous theorem. The general case follows by the standard machine.

As an easy exercise, use this theorem to compute the characteristic function φµ∗ν cor- responding to the convolution µ∗ν. Compare to what you already knew on characteristic functions of independent random variables.

3