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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)?

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This is worked out in some detail in the paper of Fan, Grama and Liu, J. Math Anal. Appl. 448 (2017), 538-566 (see in particular Theorem 2.1 there, and the references). Unfortunately I do not have an open access to an electronic copy, and it doesn't seem to exist on arXiv.

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  • $\begingroup$ Thanks -- I got a copy from a library. Apparently I was wondering if there exists a general way to embed the martingale into a random walk, so we do not need to handle these extensions again. $\endgroup$
    – Nikolayevich
    Commented Jan 10, 2018 at 6:16

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