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- W3117268004 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{2pi}}int_{-infty}^{infty} |u|^p e^{-u^2/2} mathrm{d}u quad {rm for} ; ; m to infty , ;; p>0. end{equation}" @default.
- W3117268004 created "2021-01-05" @default.
- W3117268004 creator A5000741106 @default.
- W3117268004 date "2020-12-27" @default.
- W3117268004 modified "2023-09-27" @default.
- W3117268004 title "Central Limit Theorem for $(t,s)$-sequences, I" @default.
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- W3117268004 hasPublicationYear "2020" @default.
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