Concentration of highest order statistic of bounded distribution

Question

I have a question about how the highest order statistic of a bounded distribution concentrates around the upper bound of the distribution. Intuitively, as sample size increases, the statistic should be "closer" to the upper bound in some sense. Does the statistic converge to upper bound in mean? Does it converge in probability? Is there any finite bound available for the mean difference between the upper bound and the statistic? I would appreciate it if anyone can provide some comments or point me to some references.

Answer 1

Let $X_n\overset{iid}{\sim}F_X$ with support on $[a,b]$ s.t. $F_X(x)<1$ on $[a,b)$ and let $Y_n=\max_{1\le i\le n}\{X_i\}$. Then $$ F_Y(x)=\mathsf{P}(Y\le x)=(F_X(x))^n $$ and for any $\epsilon>0$, $$ \mathsf{P}(b-Y>\epsilon)=\mathsf{P}(Y\le b-\epsilon)=(F_X(b-\epsilon))^n\to 0 \quad\text{as }n\to\infty, $$ which means that $Y\to b$ in probability. Since $Y$ is bounded, $Y\to b$ in $L_1$ by the dominated convergence theorem.