Set $n\le N$.
Suppose $x_1,...,x_n$ are uniformly random variables taking value in $[N]$
In addition Suppose ${y_1,...,y_n}$ are an $n-$subset of $[N]$ has been chosen uniformly random among all $N \choose n$ possibilities.
Is there any simple proof that shows $Y=\sum y_i$ is more tightly concentrated than $X=\sum x_i$ around their shared mean ?
The following is a theorem proved by Hoeffding in Probability Inequalities for Sums of Bounded Random Variables published in 1963 $$ \mathbb E f\left( \sum_{i = 1}^n X_i \right) \le \mathbb E f\left( \sum_{i = 1}^n Y_i \right) $$ $X_i$ are uniformly at random sampled without replacement, $Y_i$ are uniformly at random sampled with replacement, $f$ is convex and continuous.
This implies concentration results for sampling with replacement obtained using Chernoff bounds type methods (bounding moment generating function + Markov inequality) can be transferred to the case of sampling without replacement.