I'm struggling with the following expression in a statistics script: $$H(x) = \int_{-\infty}^\infty F(x-t) dG(t)$$ What does the dG(t) mean exactly? I've never seen that notation before.
Background: Random variables X, Y with CDFs F and G, we want to calculate H which is the CDF of (X + Y)
One remark on notation. Let $X$ and $Y$ be indep. r.v. with prob. distributions $f_X$ and $g_Y$. The r.v. $Z=X+Y$ has the following cumulative probability distribution.
$F_Z(z):=P(Z\leq z)=P(X+Y\leq z)=\int_{x+y\leq z} f_y(y)f_X(x)dxdy=$
$\int_{-\infty}^{+\infty} F_Y(z-x)f_X(x)dx=\int_{-\infty}^{+\infty} F_Y(z-x)dF_X(x)$,
i.e. the formula in your question, where $F_Y(z-x)=P(Y\leq z-x)=\int_{-\infty}^{z-x} f_Y(y)dy$.
To obtain the prob. distribution $f_Z$ of $Z$ one needs to derive w.r.t $z$ obtaining the convolution
$$f_Z(z)=\int_{-\infty}^{\infty} f_Y(z-x)f_X(x)dx $$