# Martingale version of Bernstein-type inequality for (slightly) heavy-tailed distributions?

It is known that for sub-exponentially distributed martingale difference sequence, the following Bernstein-type inequality holds: $$ℙ\left(\left| \sum_{i=1}^N a_i X_i \right| \ge t \right) \le 2\exp\left( −c \min\left( \frac{t^2}{K^2 \|a\|_2^2}, \frac{t}{K \|a\|_\infty} \right) \right)$$ where $K = \max \|X_1\|_{\psi_1}$ (in the conditional sense) and $a\in ℝ^d$.

Does similar concentration inequalities hold for heavy-tailed random variables where $X_i$ satisfies $ℙ(X_i > t) \le C\exp(ct^{-\alpha})$ for $\alpha \in (0,1)$?

In the independent increment (random-walk) case, Theorem 6.21 of Ledoux and Talagrand (Probability in Banach Spaces) gave a positive answer. What about a general martingale version (even just in one dimensional)?