On the dependence structure and quality of scrambled $(t,m,s)-$nets

In this paper we develop a framework to study the dependence structure of scrambled $(t,m,s)$-nets. It relies on values denoted by $C_b(\mathbf{k};P_n)$, which are related to how many distinct pairs of points from $P_n$ lie in the same elementary $\mathbf{k}-$interval in base $b$. These values quantify the equidistribution properties of $P_n$ in a more informative way than the parameter $t$. They also play a key role in determining if a scrambled set $\tilde{P}_n$ is negative lower orthant dependent (NLOD). Indeed this property holds if and only if $C_b(\mathbf{k};P_n) \le 1$ for all $\mathbf{k} \in \mathbb{N}^s$, which in turn implies that a scrambled digital $(t,m,s)-$net in base $b$ is NLOD if and only if $t=0$. Through numerical examples we demonstrate that these $C_b(\mathbf{k};P_n)$ values are a powerful tool to compare the quality of different $(t,m,s)$-nets, and to enhance our understanding of how scrambling can improve the quality of deterministic point sets.