Convergence , conditional distribution

Question

here my short question.

I saw the following conditional distribution, which converges, in a book:

$\lim_{s \to \infty}P\left(\frac{X-f(s)}{g(s)}\leq x\mid X>s\right)=G(x)\ \forall x$ in the set of continuity points of G

with the informations, that $f(s)>0,g(s)$ functions in real numbers and $G(x)$ distribution function. $X$ is a random variable with known distribution function.

my question is: what type of convergence is it???

my problem: we don t have a sequence of random variables $X_{n}$, because $f,g$ are functions in real numbers, so i think we can`t work with the known convergences like "weak convergence" or is it possible??

thanks for help

Answer 1

Weak convergence in this context means convergence in distribution, and that's excatly what you've got. Weak convergence to a c.d.f. $G$ means pointwise convergence to $G$ at all points at which $G$ is continuous. It doesn't matter whether it's convergence of a sequence or convergence of a function of a real variable or convergence of some other kind of net.