Hoeffding's Tail Bound is well-known for subgaussian variables. It can be written in the following way:
Assume $X_i$ for $1\leq i\leq n$ satisfies: $$ \mathbb{P}(|X_i-\mu|>t)\leq 2\exp\left(\frac{-t^2}{2\sigma_i^2}\right) $$ Then for $t\geq 0$, we have that: $$ \mathbb{P}(|\frac{\sum_i X_i}{n}-\mu|>t)\leq 2\exp\left(\frac{-n^2t^2}{2\sum_i \sigma^2}\right) $$ I am wondering if this can be generalized to the case when the constant is not the mean. i.e. assume that for $X_i$ we have the bounds: $$ \mathbb{P}(|X_i-a_i|>t)\leq 2\exp\left(\frac{-t^2}{2\sigma_i^2}\right) $$ Where $\mathbb{E}(X_i)\neq a_i$. Then can we bound: $$ \mathbb{P}(|\frac{\sum_i X_i-a_i}{n}|>t) $$ similarly to Hoeffding's bound? I feel like I am missing something obvious but I am quite stuck.