Please provide alternative suggestions with detailed steps to simplify this expression and well known results with references that might apply in this case.
If there are any mistakes, please let me know.
Expression,
$$ c=\int_{0}^{K}\left\{ F_{n}\left(t\right)-F\left(t\right)\right\} dt $$
Here, $K>0$ , $F_{n}\left(t\right)$ is the empirical cumulative distribution function and $F\left(t\right)$ is the true cumulative distribution function.
Steps Tried,
$$ \left|c\right|\leq\int_{0}^{K}\left|\left\{ F_{n}\left(t\right)-F\left(t\right)\right\} \right|dt $$ $$ \left|c\right|\leq \sup_{{x\in{\mathbb{R}}}}\left|\left\{ F_{n}\left(x\right)-F\left(x\right)\right\} \right|\int_{0}^{K}dt $$ $$ \left|c\right|\leq \sup_{{x\in{\mathbb{R}}}}\left|\left\{ F_{n}\left(x\right)-F\left(x\right)\right\} \right|K $$
$$ \frac{1}{K}\left|c\right|\leq \sup_{{x\in{\mathbb{R}}}}\left|\left\{ F_{n}\left(x\right)-F\left(x\right)\right\} \right| $$
Using the well known Kolmogorov-Smirnov / Dvoretzky, Kiefer and Wolfowitz results,
$$ \Pr{\Bigl(}\sup_{{x\in{\mathbb{R}}}}|F_{n}(x)-F(x)|<\varepsilon{\Bigr)}\geq1-2e^{{-2n\varepsilon^{2}}}\qquad{\text{for every }}\varepsilon>0 $$
$$ \Pr{\Bigl(}\frac{1}{K}\left|c\right|<\varepsilon{\Bigr)}\geq1-2e^{{-2n\left(\varepsilon\right)^{2}}}\qquad{\text{for every }}\varepsilon>0 $$
$$ \Pr{\Bigl(}|c|<K\varepsilon{\Bigr)}\geq1-2e^{{-2n\left(\varepsilon\right)^{2}}}\qquad{\text{for every }}\varepsilon>0 $$
Any suggestions on how to improve the bound because this seems to be a very similar bound as the Dvoretzky, Kiefer and Wolfowitz inequality.
$$ \Pr{\Bigl(}\left|c\right|<\delta{\Bigr)}\geq1-2e^{{-2n\left(\frac{\delta}{K}\right)^{2}}}\qquad{\text{for every }}\delta>0 $$ Putting, $\delta=\frac{\lambda}{\sqrt{n}}$, simplifying further and taking the limit as $n\rightarrow\infty$,
$$ \underset{n\rightarrow\infty}{\lim}\Pr{\Bigl(}\left|c\right|<\frac{\lambda}{\sqrt{n}}{\Bigr)}\geq1-2e^{{-2\left(\frac{\lambda}{K}\right)^{2}}}\qquad{\text{for every }}\lambda>0 $$
I assume you meant to say that $F_n$ was the empirical cumulative distribution of $X_1,\ldots,X_n$, drawn independently from the distribution $F$. Then by integration by parts, your $$c = EY - \frac 1 n \sum_{i=1}^n Y_i,$$ where $Y_i = X_i $ if $X_i\le K$ and $Y_i=0$ otherwise.
If $F$ has a finite second moment you can, for given $K$, work out the mean and variance of the $Y_i$ and apply the central limit theorem to find the limiting distribution for $c$. In non-trivial cases, the answer depends on $K$.