Take a large integer $R$, and let $(X_j)_{j\geq R}$ be a sequence of exponentially distributed random variables with parameters $\pi_j := j^{1+\alpha}$ ($\alpha>0$), so that $\sum_{j\geq R} \frac{1}{\pi_j} < +\infty$. Set $X_R := \sum_{j\geq R} X_j$. Assume that $t>0$ is small enough that $\mathbb{E}X_R \ll t$.
I am interested in good upper bounds for $\mathbb{P}(X_R>t)$. By using $\mathbb{P}(X_R>t) = \mathbb{P}(e^{\lambda X_r}> e^{\lambda t})$, applying Markov's inequality, and optimizing over $\lambda$, I think that I can get bounds like
\begin{equation*}\mathbb{P}(X_R>t)\leq C\exp(-ctR^{1+\varepsilon}) \end{equation*}
for $0<\varepsilon<\alpha$, but I am not sure if these are close to being optimal.
