Concentration inequality for median












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$begingroup$


$xi_1,xi_2,ldots,xi_n$ are iid sub-Gaussian random variables (i.e, $P(xi_1>t)leq e^{-t^2/2}$ for $t>0$) with mean $0$ and $a_1,a_2,ldots,a_ninmathbb{R}$.



Define $a_0:= lim_{deltadownarrow 0}argmin frac1nsum_{k=1}^n |a -a_k|+delta a^2$ and $X_0:= lim_{deltadownarrow 0}argmin frac1nsum_{k=1}^n |x -X_k|+delta x^2$ 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.






probability concentration-of-measure







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    $xi_1,xi_2,ldots,xi_n$ are iid sub-Gaussian random variables (i.e, $P(xi_1>t)leq e^{-t^2/2}$ for $t>0$) with mean $0$ and $a_1,a_2,ldots,a_ninmathbb{R}$.



    Define $a_0:= lim_{deltadownarrow 0}argmin frac1nsum_{k=1}^n |a -a_k|+delta a^2$ and $X_0:= lim_{deltadownarrow 0}argmin frac1nsum_{k=1}^n |x -X_k|+delta x^2$ 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.






    probability concentration-of-measure







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      $xi_1,xi_2,ldots,xi_n$ are iid sub-Gaussian random variables (i.e, $P(xi_1>t)leq e^{-t^2/2}$ for $t>0$) with mean $0$ and $a_1,a_2,ldots,a_ninmathbb{R}$.



      Define $a_0:= lim_{deltadownarrow 0}argmin frac1nsum_{k=1}^n |a -a_k|+delta a^2$ and $X_0:= lim_{deltadownarrow 0}argmin frac1nsum_{k=1}^n |x -X_k|+delta x^2$ 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.






      probability concentration-of-measure







      share|cite|improve this question









      $endgroup$




      $xi_1,xi_2,ldots,xi_n$ are iid sub-Gaussian random variables (i.e, $P(xi_1>t)leq e^{-t^2/2}$ for $t>0$) with mean $0$ and $a_1,a_2,ldots,a_ninmathbb{R}$.



      Define $a_0:= lim_{deltadownarrow 0}argmin frac1nsum_{k=1}^n |a -a_k|+delta a^2$ and $X_0:= lim_{deltadownarrow 0}argmin frac1nsum_{k=1}^n |x -X_k|+delta x^2$ 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.






      probability concentration-of-measure




      probability concentration-of-measure






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 3 at 16:02









      John_WickJohn_Wick

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