It is a common technique to obtain tail bounds from the MGF. But now suppose I have the other way round. Suppose it is given,
$$\mathbb P [X_i \geq t] \leq e^{-t^2/2\sigma^2}$$
then what can you say about,
$$\mathbb P [\sum_{i=1}^n X_i \geq t]$$
Clearly the ideal way would be to recovering the MGFs. But using standard CDF formulation,
$$\int \mathbb P[e^{\lambda X_i } \geq u] du$$
doesn't seem to work well i.e., I was not able to recover the MGF $\log \mathbb E \exp (\lambda X) \leq \lambda^2 \sigma^2/2$ which one intuitively expects. Any thoughts or ideas which I am missing?