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

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-dimensional Hammersley’s point set. Let \(D(\bar{\textbf{x}},(\mathcal {H}_{s+1,N})_{n=0}^{N-1} )\) $D(\overline{\mathbf{x}},{({\mathcal{H}}_{s+1,N})}_{n=0}^{N-1})$ be the local discrepancy of \((\mathcal {H}_{s+1,N})_{n=0}^{N-1}\) ${({\mathcal{H}}_{s+1,N})}_{n=0}^{N-1}$ , and let \(D_{s,p} ( (\mathcal {H}_{s+1,N})_{n=0}^{N-1})\) $D_{s,p}({({\mathcal{H}}_{s+1,N})}_{n=0}^{N-1})$ be the \(L_p\) $L_p$ discrepancy of \((\mathcal {H}_{s+1,N})_{n=0}^{N-1}\) ${({\mathcal{H}}_{s+1,N})}_{n=0}^{N-1}$ . In this Part II of our paper, we prove the Central Limit Theorem for Hammersley’s net: \(\begin{aligned} D(\bar{\textbf{x}},\mathcal {H}_{s+1,N} ) /D_{s+1,2}(\mathcal {H}_{s+1,N}) \overset{w}{\rightarrow }\ \mathcal {N}(0,1), \end{aligned}\) $$\begin{array}{c}\hfill D(\overline{\mathbf{x}},{\mathcal{H}}_{s+1,N})/D_{s+1,2}({\mathcal{H}}_{s+1,N})\stackrel{w}{\to }\mathcal{N}(0,1),\end{array}$$ where \(\bar{\textbf{x}}\) $\overline{\mathbf{x}}$ is a uniformly distributed random variable in \([0,1]^{s+1}\) ${[0,1]}^{s+1}$ and we claim that the lower bound of \(L_p\) $L_p$ discrepancy for p>0 is of the same order as the upper bound: \(\begin{aligned} \frac{ D_{s+1,p}( \mathcal {H}_{s+1,N} )}{ D_{s+1,2}( \mathcal {H}_{s+1,N} )} \overset{N \rightarrow \infty }{\longrightarrow }\ \kappa _p^{1/p}, \quad \kappa _p= \frac{1}{\sqrt{2 \pi }}\int _{-\infty }^{\infty } |u|^p e^{-u^2/2} d u. \end{aligned}\) $$\begin{array}{c}\hfill \frac{D_{s+1,p}({\mathcal{H}}_{s+1,N})}{D_{s+1,2}({\mathcal{H}}_{s+1,N})}\stackrel{N\to \infty }{⟶}{\kappa }_p^{1/p},{\kappa }_p=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{|u|}^p{e}^{-u^2/2}du.\end{array}$$