cumulative distribution of sum of random variables

Question

Let $X, Y$ be two continuous indipendent random variables with cumulative distribution function $F_X(x)$ and $F_Y(x)$ respectively, and denote with $F_{X+Y}(x)$ the cumulative distribution of the sum $X+Y$. Is there a general inequality to relate the sum $F_X(x)+F_Y(x)$ (or the product $F_X(x) F_Y(x)$) with $F_{X+Y}(x)$?

Answer 1

Not really. $F_{X+Y}(x)$ can be expressed by $F_X$ and $F_Y$: $$ F_{X+Y}(x) = \int_{-\infty}^\infty dz \; F_X(x-z) \frac{d}{dz}F_Y(z) = \int_{-\infty}^\infty dz \; \Big(\frac{d}{dz}F_X(z)\Big) F_Y(x-z) $$ but it depends onother values than just $F_X(x)$ and $F_Y(x)$, so in general a relation does not exist.

However, in some special cases, some relation may exist. For example, if $X$ and $Y$ have only posivite values (that is $F_X(z)=F_Y(z)=0$ for $z<0$), then you have $$ F_{X+Y}(x) = \int_{0}^\infty dz \; F_X(x-z) \frac{d}{dz}F_Y(z) \le \int_{0}^\infty dz \; F_X(x) \frac{d}{dz}F_Y(z) = F_X(x)F_Y(x)$$

Answer 2

The pdf of a sum is the convolution of the individual pdf's.

Hence in the Fourier or Laplace domain, the ordinary product.