Central Limit Theorem for (t,s)-sequences in base 2

Let ${\left(x_n\right)}_{n\ge 0}$ be a digital $(t,s)$-sequence in base 2, $P_m={\left(x_n\right)}_{n=0}^{2^m-1}$, and let $D(P_m,Y)$ be the local discrepancy of $P_m$. Let $T\oplus Y$ be the digital addition of T and Y, and let$M_{s,p}\left(P_m\right)={\left(\underset{{[0,1)}^{2s}}{\int }\right|D(P_m\oplus T,Y){|}^{p}dTdY)}^{1/p}.$ In this paper, we prove that $D(P_m\oplus T,Y)/M_{s,2}\left(P_m\right)$ weakly converges to the standard Gaussian distribution for $m\to \infty$, where $T,Y$ are uniformly distributed random variables in ${[0,1)}^s$. In addition, we prove that$M_{s,p}\left(P_m\right)/M_{s,2}\left(P_m\right)\to \frac{1}{\sqrt{2\pi }}\underset{-\infty }{\overset{\infty }{\int }}|u{|}^p{e}^{-u^2/2}du\text{for}m\to \infty ,p>0.$