Let $F$ and $G$ be a CDF with a support [0,1], strictly increasing and with finite pdf's.
What should be the limit of the following integration as $n$ approaches to $\infty$?
$$\int^{\frac{1}{2}}_0F^{n}(x+\frac{1}{2})dG(x)$$ My intuition tells me that the value should be zero because $F^n$ in the limit converges to a step function whose value is zero everywhere and 1 at $x=1$.
How can I prove that it indeed converges to zero?
We have that $F^n(x+\frac{1}{2})\rightarrow 0$ as $n \rightarrow \infty$
$$\int_0^{\frac{1}{2}}F^n(x+\frac{1}{2})dG(x)=\int_0^{\frac{1}{2}}F^n(x+\frac{1}{2})\frac{dG(x)}{dx}dx=\int_0^{\frac{1}{2}}F^n(x+\frac{1}{2})g(x)dx$$ where $g$ is the pdf for the cdf $G$
As $[0, \frac{1}{2}]$ is compact and in particular closed, then $g$ is bounded on $[0, \frac{1}{2}]$ as $g$ is finite (This is quite wavy so feel free to correct me)
Also, $F^n(x)\leq1$ hence by DCT we get
$$\int_0^{\frac{1}{2}}F^n(x+\frac{1}{2})dG(x) \rightarrow 0$$