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 $\lambda>0$ is a constant.

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.

Let $\varepsilon_1, \dots, \varepsilon_N$ be a martingale difference sequence in $R^d$ with $\|\varepsilon_n\|^2 \le B_n, a.s.$ for each $n=1,\dots,N$. Do we have some Azuma-type concentration inequality for the high-dimensional martingale like $$ P \left( \left\| \sum_{n=1}^N \varepsilon_n \right\| \ge \lambda \right) \le 2 \exp \left( -\frac{\lambda^2}{2 \sum_{n=1}^N B_n^2} \right) $$ where $\lambda>0$ is a constant.

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 $\lambda>0$ is a constant.