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

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 $${\mathcal{M}}_{s,p}({\mathcal{P}}_m)=({\int }_{[0,1{)}^{2s}}\mathrm{\mid }D({\mathcal{P}}_m\oplus T,Y){\mathrm{\mid }}^p\mathrm{d}T\mathrm{d}Y{)}^{1\mathrm{/}p}\mathrm{.}$$ 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 $m\to \mathrm{\infty }$, 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{2\pi}}\int{-\infty}^{\infty} |u|^p e^{-u^2/2} $\mathrm{d}$u \quad {\rm for} ; ; m \to \infty , ;; p>0. \end{equation}