For a simple case, assume $X_i$ i.i.d and it has moment generating function defined everywhere, i.e for $t\in\mathbb{R}$, $\mathbb{E}(e^{tX_1}) < \infty$. Denote $\mu = \mathbb{E}(X_1)$, then one implication from the Cramer's rule tells us for any $\delta > 0$, $$\mathbb{P}\Big(\Big|\frac{\sum_{i=1}^n X_i}{n} - \mu\Big| > \delta\Big)\sim e^{-nI(\delta)}$$i.e the probability of sample average deviates from the mean is exponential decay.
My question is, based on this, if I denote the sample in an ordered way as $X_{(1)},\cdots, X_{(n)}$ and I exclude the largest j(fixed integer) samples from the summation, what conclusion could we make about the following probability decay rate? $$\mathbb{P}\Big(\Big|\frac{\sum_{i=1}^{n-j} X_{(i)}}{n} - \mu\Big| > \delta\Big)$$
This may be not one specific question, so I guess what I am looking for is some textbook or paper studying this type of questions. I would be very interested in the more general multivariate case, where the samples are ordered by their modulus.
Thank you!