Given a probability space $(\Omega, \mathcal{ F}, \mathbb{P})$, let $\{X_n : n \ge 1\}$ be a sequence of identically distributed R-valued random variables (not necessarily independent) with $E[|X_n|] < \infty$. I am trying to Show that $$\lim_{n\to\infty} \frac{1}{n}\mathbb{E}\left[\max_{1 \le j \le n}|X_j|\right] = 0.$$
A coarse relation such as $\max_{1 \le j \le n}|X_j| \le \sum_{i=1}^n |X_i|$, does not give me $\mathbb{E}\left[\max_{1 \le j \le n}|X_j|\right] = o(n)$ that I need. As a hint I might need to use the identity $$\mathbb{E}[X] = \int_{0}^{\infty}\mathbb{P}(X\ge t)dt.$$
Fix $R\gt 0$ and let $Y_{j,r}:=\left\lvert X_j\right\rvert\mathbf 1\left\{\left\lvert X_j\right\rvert\leqslant R\right\}$ and $Z_{j,r}:=\left\lvert X_j\right\rvert\mathbf 1\left\{\left\lvert X_j\right\rvert\gt R\right\}$. Since $ \left\lvert X_j\right\rvert=Y_{j,R}+Z_{j,R}$, it follows that $$ \frac{1}{n}\mathbb{E}\left[\max_{1 \le j \le n}|X_j|\right]\leqslant \frac{1}{n}\mathbb{E}\left[\max_{1 \le j \le n}Y_{j,R}\right]+\frac{1}{n}\mathbb{E}\left[\max_{1 \le j \le n}Z_{j,R}\right]. $$ Observe that $Y_{j,R}\leqslant R$ and that $$\frac{1}{n}\mathbb{E}\left[\max_{1 \le j \le n}Z_{j,R}\right]\leqslant \frac{1}{n}\mathbb{E}\left[\sum_{1 \le j \le n}Z_{j,R}\right]=\mathbb E\left[Z_{1,R}\right]$$ hence for each positive $R$, $$ \limsup_{n\to +\infty} \frac{1}{n}\mathbb{E}\left[\max_{1 \le j \le n}|X_j|\right]\leqslant \mathbb E\left[Z_{1,R}\right]. $$