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

TL;DR

This paper establishes a central limit theorem for the local discrepancy of digital $(t,s)$-sequences, showing that normalized discrepancy measures converge in distribution to a Gaussian as the sequence length increases.

Contribution

It proves the weak convergence of the normalized local discrepancy to a Gaussian distribution and characterizes the asymptotic behavior of discrepancy moments for digital $(t,s)$-sequences.

Findings

Normalized discrepancy converges to Gaussian distribution.

Discrepancy moments stabilize to a constant involving the Gaussian integral.

Results hold for sequences as length tends to infinity.

Abstract

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}