Central Limit Theorem for $(t,s)$-sequences, I

Abstract

Let $(X_n{)}_{n\ge 0}$ be a digital $(t,s)$-sequence in base $2$, ${\mathcal{P}}_m=(X_n{)}_{n=0}^{2^m-1}$, and let $D({\mathcal{P}}_m,Y)$ be the local discrepancy of ${\mathcal{P}}_m$. Let $T\oplus Y$ be the digital addition of $T$ and $Y$, and let $${\mathcal{M}}_{s,p}({\mathcal{P}}_m)=({\int }_{[0,1{)}^{2s}}|D({\mathcal{P}}_m\oplus T,Y){|}^p\mathrm{d}T\mathrm{d}Y{)}^{1/p}.$$ In this paper, we prove that $D({\mathcal{P}}_m\oplus T,Y)/{\mathcal{M}}_{s,2}({\mathcal{P}}_m)$ weakly converge to the standard Gaussisian distribution for $m\to \infty$, where $T,Y$ are uniformly distributed random variables in $[0,1{)}^s$. In addition, we prove that \begin{equation} \nonumber \mathcal{M}_{s,p} (\mathcal{P}_m) / \mathcal{M}_{s,2} (\mathcal{P}_m) \to \frac{1}{\sqrt{2π}}\int_{-\infty}^{\infty} |u|^p e^{-u^2/2} \mathrm{d}u \quad {\rm for} \; \; m \to \infty , \;\; p>0. \end{equation}