Categories
Ask Mathematics

Use $\mathbb{P}(\vert \hat{s}_n-s\vert > x)\leq a(n,x)$ and $\mathbb{P}(\vert \hat{s}_n-s_n\vert > x)\leq b(n,x)$ to bound $\vert s_n – s\vert$

Let $s, s_n\in\mathbb{R}$ and $\hat{s}_n$ be a random variable. I have two concentration inequalities: $$\mathbb{P}(\vert \hat{s}_n-s\vert > x)\leq a(n,x)$$ for all $n\geq1$ and $x>0$; and $$\mathbb{P}(\vert \hat{s}_n-s_n\vert > x)\leq b(n,x)$$ for all $n\geq1$ and $x>0$. Is there a way to bound $\vert s_n – s\vert$?