I would like to know what is the best concentration inequality we can use for the estimated least squares regression coefficients. Let $\hat \beta_0, \hat \beta_1$ be the estimated regression coefficients when we solve the following simple linear regression model with ordinary least squares: $$ Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i, \quad \quad i=1,2,\dots,n, $$ where $E[\varepsilon_i|X] = 0$ and $\text{Var}[\varepsilon_i|X] = \sigma^2$.
Now consider $\hat \beta_1$ for example, Chebyshev's inequality gives us $$ P(|\hat \beta_1 - \beta| > t) \le \frac{\text{Var}(\hat \beta_1)}{t^2}. $$
Is this the only concentration inequality that we can use for $\hat \beta_1$? I was thinking that maybe we can exploit the fact that $\hat \beta_1$ is asymptotically normal, that is, $$ \beta_1 \stackrel{a}{\sim} \mathcal{N}\bigg(\beta_1,\frac{\sigma^2}{n} (X^TX)^{-1}\bigg). $$
Can we use this fact to state a concentration inequality that is tighter than Chebyshev's inequality in the case of a large number of samples?