$(t,s)$ -序列的中心极限定理，I
Central Limit Theorem for $(t,s)$-sequences, I
摘要：
令$ (X_n)_{n \geq 0} $是基$2$中的数字$(t,s)$ -序列，$\mathcal{P}_m =(X_n)_{n=0}^{2^m-1} $，并且令$D(\mathcal{P}_m, Y )$是$\mathcal{P}_m$的局部差异。设$T \oplus Y$是$T$和$Y$的数字加法，令$$\mathcal{M}_{s,p} (\mathcal{P}_m) =\Big( \int_{[0,1)^{2s}} |D(\mathcal{P}_m \oplus T , Y ) |^p \mathrm{d}T \mathrm{d}Y \Big)^{1/p} .$$本文证明了对于$m \rightarrow \infty$，$D(\mathcal{P}_m \oplus T , Y ) / \mathcal{M}_{s,2} (\mathcal{P}_m)$弱收敛于标准高斯分布，其中$T,Y$是$[0,1)^s$中的均匀分布随机变量.此外，我们还证明了\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}
英文摘要：
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{2π}}\int_{-\infty}^{\infty} |u|^p e^{-u^2/2} \mathrm{d}u \quad {\rm for} \; \; m \to \infty , \;\; p>0. \end{equation}
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