It is well known that, if I have $n$ couples $(\underline X_i,Y_i)$ with $\underline X_i\in \mathbb R^k$, $Y_i\in \mathbb R$ and $$Y_i = \beta^T \underline X_i+\epsilon_i \qquad \{\epsilon_i\}_{i=1}^n\overset{i.i.d.}\sim \mathcal N(0,1)$$ Then, the least squares estimation estimation $\hat \beta_n$ of $\beta$ is gaussian and converges is a way that $$\mathbb P(\|\hat \beta_n -\beta\|_2>\delta)=O(e^{-n})$$ for any $\delta>0$ (this is a result form standard Linear Regression that is also true if the noises are i.i.d. conditioned to $\{\underline X_i\}_{i=1}^n$).
In an AutoRegressive model, we have $$X_{n+1} = \sum_{j=1}^k a_j X_{n-j+1} + \epsilon_{n+1},\qquad \epsilon_n\sim \mathcal N(0,1)$$ where each $\epsilon_n$ is independent from $(X_1,...X_{n-1})$. We can use linear regression to esimate the $a_i's$, taking
$$\underline X_i =[X_{i-k}, ... X_{i-1}], \qquad Y_i=X_i$$ $$\beta = [a_1, ..., a_k]$$
but the previous assumptions are not met. Can we stilll give an estimate of $$\mathbb P(\|\hat \beta_n -\beta\|_2>\delta)?$$