Let $\varepsilon_1, \dots, \varepsilon_N$ be a martingale difference sequence in $R^d$ with $\|\varepsilon_n\| \le B_n, a.s.$ for each $n=1,\dots,N$. Do we have some Azuma-type concentration inequality for the high-dimensional martingale like $$ P \left( \left\| \sum_{n=1}^N \varepsilon_n \right\| \ge \lambda \right) \le C \exp \left( -\frac{\lambda^2}{4 \sum_{n=1}^N B_n^2} \right) $$ where $\lambda$ is an arbitrary real positive number.
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1$\begingroup$ By a result of Kallenberg and Sztencel (PTRF 1991), popularized recently by Lee and Peres, it is enough to reduce to $d=2$, where the result is true (maybe with 2 replaced by 4 in the denominator). The point is that the bound does not depend on the dimension. $\endgroup$– ofer zeitouniCommented Nov 30, 2017 at 7:16
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1$\begingroup$ Thanks! I just found Pinelis' <Optimum bounds for the distributions of martingales in Banach spaces>, AOP 1994. Seems related. $\endgroup$– NikolayevichCommented Nov 30, 2017 at 8:54
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$\begingroup$ It seems indeed that Theorem 3.5 in Pinelis, Iosif Optimum bounds for the distributions of martingales in Banach spaces. (English summary) Ann. Probab. 22 (1994), no. 4, 1679–1706. is all what you need, as $\mathbb R^d$ endowed with the Euclidian is $(2,1)$-smooth. $\endgroup$– Davide GiraudoCommented Nov 30, 2017 at 11:30
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