$\xi_1,\xi_2,\ldots,\xi_n$ are iid sub-Gaussian random variables (i.e, $P(\xi_1>t)\leq e^{-t^2/2}$ for $t>0$) with mean $0$ and $a_1,a_2,\ldots a_n$ are some real numbers.
Define $a_0:= \lim_{\delta\downarrow 0}\arg\min \frac1n\sum_{k=1}^n |a -a_k|+\delta a^2$ and $X_0:= \lim_{\delta\downarrow 0}\arg\min \frac1n\sum_{k=1}^n |x -X_k|+\delta x^2$ where $X_k:= a_k+\xi_k.$ What is an appropriate upper bound for $P(|X_0-a_0|>t)?$
I have tried doing that using McDiarmid type inequality for sub-Gaussian but the concentration inequality is for $P(|X_0-EX_0|>t)$ in that case. Any help would be appreciated.