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May 18

Density of a sum of independent variables is given by convolution

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 X,Y be two independent variables with densities f_X,f_Y. Denote f_{X,Y} the joint density of the pair (X,Y).

By independence we have

(1) f_{X,Y}(x,y)=f_X(x)f_Y(y).

Let Z=X+Y be the sum and let f_Z,\ F_Z be its density and distribution function, respectively. Then

(2) f_Z(z)=\frac{d}{dz}F_Z(z).

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

(3) F_Z(z)=P(Z\le z)=P(X+Y\le z).

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

P(X+Y\le z)=\int\int_{x+y\le z}f_{X,Y}(x,y)dxdy.

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

(4) P(X+Y\le z)=\int\int_{x+y\le z}f_X(x)f_Y(y)dxdy=\int_R\int_{-\infty}^{z-x}f_X(x)f_Y(y)dxdy

=\int_Rf_X(x)\left(\int_{-\infty}^{z-x}f_Y(y)dy\right)dx.

The geometry is explained in Figure 1. The area x+y\le z is limited by the line y=z-x. In the repeated integral, we integrate first over red lines from -\infty to z-x and then in the outer integral over all x\in R.

Area of integration

Figure 1. Area of integration

(3) and (4) imply

F_Z(z)=\int_Rf_X(x)\left(\int_{-\infty}^{z-x}f_Y(y)dy\right)dx.

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

(5) f_Z(z)=\int_Rf_X(x)f_Y(z-x)dx.

This is the result. The integral on the right is called a convolution of functions f_X,f_Y.

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

Exercise. Convolution is usually denoted by (f*g)(z)=\int_Rf(x)g(z-x)dx. Prove that

  1. (f*g)(z)=(g*f)(z).
  2. \int_R|(f*g)(z)|dz\le \int_R|f(x)|dx\int_R|g(x)|dx.
  3. If X is uniformly distributed on some segment, then (f_X*f_X)(z) is zero for large z.

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