Sub-Gaussian style concentration inequalities exist for the median, i.e. if $X$ is a random variable, then there exist positive constants $c_1, c_2$ such that $$ \Pr( |X - m_X | \ge t) \le c_1 e^{-c_2 t^2}. $$
My question is if $X_i$ are independent r.v.s indexed by $i = 1 \dots d$ each obeying the above inequality, then are there corresponding bounds like $$ \Pr( |\sum_i (X_i - m_{X_i}) | \ge t) \le c_3 e^{-c_4 t^2} $$ where $c_3, c_4$ are constants?
I know that bounds exist for $\sum X_i - \operatorname{median}(\sum_i X_i)$, and for sub-Gaussian variables in $\sum_i (X_i - E X_i)$ for independent r.v.s $X_i$.
It also seems that using Talagrand's convex inequality we can get bounds where $c_3 \propto 2^d$, (using the lower bound $\Pr( \sum_i (X_i - m_{X_i}) < 0) \ge \Pr(\forall i, X_i - m_{X_i} < 0) \ge 2^{-d} $) but this seems extremely weak and I wonder if it is possible to get something closer to a linear (in $d$) relationship between the constants $c_1, c_2$ and $c_3, c_4$.
It seems that this should be the case because concentration around the mean implies concentration around the median, but the fact that the median is non-linear seems to break the argument from e.g. Mean concentration implies median concentration. I would also be very interested if this is possible under some additional constraints, e.g. boundedness.
The context for this problem is trying to prove a result where the sum of the individual medians is tractable, but the median of the sum is not.
