I have read that for null hypothesis $H\_0 : \beta = \beta\_0$, the likelihood ratio statistic from profile-likelihood is
$$LR = 2 (\log L\_p(\hat{\beta\_p}) - \log L\_p(\beta\_0))$$
where $\hat{\beta\_p}$ maximises the profiled likelihood $L\_p(\beta) = \max\_{\gamma} L(\beta, \gamma)$.

And that by assuming $LR$ has some distribution we can create a confidence interval for $\beta$. I've seen a lot of questions asking how to do this - I understand that - but I am confused as to why we can do this.

I.e. why (/when) can I assume LR has say a normal distribution given I know $\beta$ and $\gamma$'s distributions for example? I feel I am missing something obvious as I can't find any good resources on this.