论文标题
$（t，s）$ - 序列的中央限制定理，我
Central Limit Theorem for $(t,s)$-sequences, I
论文作者
论文摘要
令$（x_n）_ {n \ geq 0} $为数字$（t，s）$ - 基础$ 2 $，$ \ nathcal {p} _m =（x_n）_ {n = 0}^{2^{2^m-1} $，以及n = 0}^{2^m-1} $ $ \ MATHCAL {P} _M $。令$ t \ oplus y $为$ t $和$ y $的数字添加，然后让$$ \ mathcal {m} _ {s，p}（\ Mathcal {p} _m）= \ big（\ big（\ int _ {[0,1） \ Mathrm {d} t \ Mathrm {d} y \ big）^{1 / p}。$$在本文中，我们证明$ d（\ nathcal {p} _m \ oplus t，y） / \ mathcal {m} {m}对于$ m \ rightarrow \ infty $，其中$ t，y $是$ [0,1）^s $均匀分布的随机变量。此外，我们证明\ begin {qore} \ nonumber \ mathcal {m} _ {s，p}（\ Mathcal {p} _m） / \ Mathcal {m} _ {m} _ {s，2}（\ Mathcal {p} _m）\ \ frac {1} {\ sqrt {2π}} \ int _ { - \ infty}^{\ infty} | \; m \ to \ infty，\; \; p> 0。 \ end {equation}
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}
