Let $X_1^{(n)}, X_2^{(n)}, ..., X_J^{(n)}$ be independent and identically distributed random variables satisfying
\begin{align} \mathbb{E} \left [ X_i^{(n)} \right ] &\leq a_n \tag1\\ \mathbb{P} \left( X_i^{(n)} \geq a_n + \frac{\log(n)}{n} \right) &\leq e^{-n} \tag2 \end{align} for all $i \in \{1, ..., J \}$.
Intuition: As $n \to \infty$, the concentration of $X_i^{(n)}$ around its mean increases. Moreover, by the law of large numbers, the concentration of $$\frac{1}{J}\sum_{i=1}^J X_i^{(n)}$$ will doubly increase as $n$ and $J$ both become large. Therefore, one expects that
\begin{align} \mathbb{P} \left( \frac{1}{J}\sum_{i=1}^J X_i^{(n)} \geq a_n + \frac{\log(n)}{n} \right) &\leq e^{-n} \end{align} Furthermore, if I let $J = J(n)$ be an increasing function of $n$, I expect
\begin{align} \mathbb{P} \left( \frac{1}{J}\sum_{i=1}^J X_i^{(n)} \geq a_n + \frac{\log(n)}{n} \right) &\leq o\left( e^{-n} \right) \end{align} where $o(e^{-n})$ represents faster decay than $e^{-n}$.
Is my intuition correct? If yes, how to formalize this?