= Solution
For a centered <random variable> $X$, being <Sub-Poisson random variable in the right tail>[sub-Poisson in the right tail] with variance parameter $\sigma^2$ means
$$
\log\mathbb E e^{\lambda X}
\leq\sigma^2(e^\lambda-\lambda-1)
\qquad(\lambda\geq0).
$$
It is <Sub-Gamma random variable in the right tail>[sub-Gamma in the right tail] with variance parameter $v$ and scale parameter $c$ when
$$
\log\mathbb E e^{\lambda X}
\leq\frac{v\lambda^2}{2(1-c\lambda)}
\qquad(0\leq\lambda<c^{-1}).
$$
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