7
$\begingroup$

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)?

$\endgroup$
5
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Koltchinskii Jan 10 '18 at 6:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.