Let $X$ be a positive random variable with pdf $f$, and for a given $n$ let $x_1,\dots,x_n$ be i.i.d. instances of $X$. Suppose we have a good understanding of how $$\left(\frac1n\sum_{i=1}^n x_i-\mathbb E[X]\right)$$ behaves as $n\to\infty$.
Can we leverage this information to bound the same quantity but zeroing out all instances above a certain threshold $t$: $$\left(\frac1n\sum_{i=1}^n \mathbb1_{x<t}\left( x_i\right)-\mathbb E[\mathbb1_{x<t}\left(X\right) ]\right)?$$
Let $p$ be the probability that $X<t$. I believe it is true that
$$\left(\frac1n\sum_{i=1}^n x_i-\mathbb E[X]\right)= \left(\frac1n\sum_{x_i<t} x_i -p\mathbb E[\mathbb1_{x<t}\left(X\right) ]\right) +\left(\frac1n\sum_{x_i\ge t} x_i -(1-p)\mathbb E[\mathbb1_{x\ge t}\left(X\right) ]\right),$$ but I'm not sure how to continue. This seems like it should be easy, but I'm struggling to formulate things in an effective way.
EDIT: To be very clear, I am concerned with leveraging the rate at which the sample mean converges over all the samples to understanding the rate at which the sample mean of the smallest samples converges to its mean.