  Let $ (X_n)_{n \geq 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 $$\mathcal{M}_{s,p} (\mathcal{P}_m) =\Big(
\int_{[0,1)^{2s}} |D(\mathcal{P}_m \oplus T , Y ) |^p \mathrm{d}T \mathrm{d}Y
\Big)^{1/p} .$$ 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 $m \rightarrow \infty$, 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}
