Sampling with and without replacement comparison for bounded products ($|U_N-V_N|\le 1-(N{)}_d/N^d\le (\genfrac{}{}{0ex}{}{d}{2})/N$)

For arrays of values in $[0,1]$, let $U_N$ average a product of $d$ entries over all ordered index tuples from $N$ coordinates, and let $V_N$ average it over the tuples with distinct indices. Decompose the first average as $U_N=\alpha V_N+(1-\alpha )W_N$, where $\alpha =(N{)}_d/N^d$ and the repeated-index average $W_N$ lies in $[0,1]$. This gives $|U_N-V_N|\le 1-\alpha$. For uniformly sampled indices, a union bound over the $\left(\genfrac{}{}{0ex}{}{d}{2}\right)$ pairs bounds the collision probability by $\left(\genfrac{}{}{0ex}{}{d}{2}\right)/N$. This deterministic estimate is the bridge between empirical-product limits and the De Finetti theorem.