= Sampling with and without replacement comparison for bounded products
{title2=$|U_N-V_N|\leq1-(N)_d/N^d\leq\binom d2/N$}
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 $|U_N-V_N|\leq1-\alpha$. For uniformly sampled indices, a <union bound> over the $\binom d2$ pairs bounds the collision probability by $\binom d2/N$. This deterministic estimate is the bridge between empirical-product limits and the <De Finetti theorem>.
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