ON THE UPPER BOUND OF THE L\(_p\) DISCREPANCY OF HALTON’S SEQUENCE AND THE CENTRAL LIMIT THEOREM FOR HAMMERSLEY’S NET

Let \((H_s(n))_{n \ge 1}\) ${(H_s(n))}_{n\ge 1}$ be an \(s-\) $s-$ dimensional Halton’s sequence, and let \(\mathcal {H}_{s+1,N}=(H_s(n),n/N)_{n=0}^{N-1}\) ${\mathcal{H}}_{s+1,N}={(H_s(n),n/N)}_{n=0}^{N-1}$ be the \(s+1-\) $s+1-$ dimensional Hammersley’s point set. Let \(D({\textbf {x}},(H_n)_{n=0}^{N-1})\) $D(\mathbf{x},{(H_n)}_{n=0}^{N-1})$ be the local discrepancy of \((H_n)_{n=0}^{N-1}\) ${(H_n)}_{n=0}^{N-1}$ , and let \(D_{s,p} ( (H_n)_{n=0}^{N-1})\) $D_{s,p}({(H_n)}_{n=0}^{N-1})$ be the \(L_p\) $L_p$ discrepancy of \((H_n)_{n=0}^{N-1}\) ${(H_n)}_{n=0}^{N-1}$ . It is known that \(\limsup _{N \rightarrow \infty } (\log N)^{-s/2} D_{s,p} ( H_{s}(N) )_{n=0}^{N-1} >0\) ${lim\; sup}_{N\to \infty }{(logN)}^{-s/2}{D}_{s,p}{(H_s(N))}_{n=0}^{N-1}>0$ . In this paper, we prove that \(\begin{aligned} D_{s,p} (( H_{s}(N) )_{n=0}^{N-1}) =O( \log ^{s/2} N) \quad \mathrm{for} \; \; N \rightarrow \infty . \end{aligned}\) $$\begin{array}{c}\hfill D_{s,p}({(H_s(N))}_{n=0}^{N-1})=O({log}^{s/2}N)\mathrm{for}N\to \infty .\end{array}$$ That is, we found the smallest possible order of magnitude of \(L_p\) $L_p$ discrepancy of Halton’s sequence. Then, we prove the Central Limit Theorem for Hammersley’s net: \(\begin{aligned} D(\bar{{\textbf {x}}},\mathcal {H}_{s+1,N} ) /D_{s+1,2}(\mathcal {H}_{s+1,N}) {\mathop {\rightarrow }\limits ^{w}} \mathcal {N}(0,1), \end{aligned}\) $$\begin{array}{c}\hfill D(\overline{\mathbf{x}},{\mathcal{H}}_{s+1,N})/D_{s+1,2}({\mathcal{H}}_{s+1,N})\stackrel{w}{\to }\mathcal{N}(0,1),\end{array}$$ where \(\bar{{\textbf {x}}}\) $\overline{\mathbf{x}}$ is a uniformly distributed random variable in \([0,1]^{s+1}\) ${[0,1]}^{s+1}$ . The main tool is the theorem on \(\mathbbm {p}\) $\mathbb{p}$ -adic logarithmic forms.