Let $\{a_n\}$ be a positive sequence of numbers such that $\displaystyle \frac1n\sum_{i=1}^n a_i \to a$ where $a>0$. Then can we say anything about the order of $\displaystyle b_n=\max_{i\in n} a_i$.
Note that $b_n =O(n)$ since $\displaystyle b_n < \sum_{i=1}^n a_i = \Theta(n)$.
Is it always true that $b_n= o(n)$, i.e., $\frac{b_n}{n} \to 0$ ?
No, you can have occasional spikes in the $a_i$ as long as they are rare enough. Let $a_i=1$ unless $i$ is a cube, then $a_i=i$ Since the cubes are rare, you still have $\displaystyle \frac1n\sum_{i=1}^n a_i \to 1$ but $b_n \to \infty$