arXiv subjects

Sixu Liu

Publications and source records attributed to Sixu Liu.

7 recordsLinked to original sources

arXiv2026

Uniform projection designs under the stratified $L_2$-discrepancy

This paper studies a uniform projection criterion for space-filling designs under the stratified $L_2$-discrepancy. The criterion, denoted by $\Phi_{SD}$, is the average squared stratified $L_2$-discrepancy over all two-dimensional projections. For U-type $(n,m,s^p)$ designs, we derive an explicit formula for $\Phi_{SD}$ in terms of row-pairwise weighted hierarchical distances, and we establish sharp lower and upper bounds with equality conditions. We further show that many known optimal constructions attain the lower bound of $\Phi_{SD}$, and that designs attaining the lower bound of the full stratified $L_2$-discrepancy also attain the lower bound of $\Phi_{SD}$. The criterion can be evaluated in $O(n^2m)$ time, with a modest reduction in arithmetic operations compared with direct projection-wise evaluation. Numerical studies illustrate the theoretical results and show that $\Phi_{SD}$ is effective for assessing low-dimensional projection uniformity.

math.ST↗

arXiv2026

Trimmed strong laws and distributional limits for exponentially mixing systems

The Birkhoff Ergodic Theorem establishes pointwise convergence for integrable observables, but for $f\notin L^1$, no normalization yields almost sure convergence. This paper investigates trimmed ergodic sums, where the largest observations are removed, for observables with polynomial tails satisfying $\P(f>t)\,t^{1/\alpha}\to c>0$ as $t\to\infty$, in exponentially mixing dynamical systems. We prove trimmed strong laws of large numbers when $\alpha\geq 1$, extending known results from the i.i.d.\ case. We further establish distributional limit theorems for both lightly and intermediately trimmed sums in the regime $\alpha>1/2$, showing convergence to an explicitly described non-standard law and to a normal distribution, respectively. The proofs rely on approximating trimmed sums by truncated ergodic sums and exploiting the system's exponential mixing properties.

math.DS↗

arXiv2026

Precoding Matrix Indicator in the 5G NR Protocol: A Tutorial on 3GPP Beamforming Codebooks

This paper bridges this critical gap by providing a systematic examination of the beamforming codebook technology, i.e., precoding matrix indicator (PMI), in the 5G NR from theoretical, standardization, and implementation perspectives. We begin by introducing the background of beamforming in multiple-input multiple-output (MIMO) systems and the signaling procedures for codebook-based beamforming in practical 5G systems. Then, we establish the fundamentals of regular codebooks and port-selection codebooks in 3GPP standards. Next, we provide rigorous technical analysis of 3GPP codebook evolution spanning Releases 15-18, with particular focus on: 1) We elucidate the core principles underlying codebook design, 2) provide clear physical interpretations for each symbolic variable in the codebook formulas, summarized in tabular form, and 3) offer intuitive visual illustrations to explain how codebook parameters convey information. These essential pedagogical elements are almost entirely absent in the often-obscure standardization documents. Through mathematical modeling, performance benchmarking, feedback comparisons, and scenario-dependent applicability analysis, we provide researchers and engineers with a unified understanding of beamforming codebooks in real-world systems. Furthermore, we identify future directions and other beamforming scenarios for ongoing research and development efforts. This work serves as both an informative tutorial and a guidance for future research, facilitating more effective collaboration between academia and industry in advancing wireless communication technologies.

cs.IT↗

arXiv2025

Optimal design of experiments with quantitative-sequence factors

A new type of experiment with joint considerations of quantitative and sequence factors is recently drawing much attention in medical science, bio-engineering, and many other disciplines. The input spaces of such experiments are semi-discrete and often very large. Thus, efficient and economical experimental designs are required. Based on the transformations and aggregations of good lattice point sets, we construct a new class of optimal quantitative-sequence (QS) designs that are marginally coupled, pair-balanced, space-filling, and asymptotically orthogonal. The proposed QS designs have a certain flexibility in run and factor sizes and are especially appealing for high-dimensional cases.

stat.ME↗

arXiv2024

Generalized multiple Borel-Cantelli Lemma in dynamics and its applications

Multiple Borel-Cantelli Lemma is a criterion that characterizes the occurrence of multiple rare events on the same time scale. We generalize the multiple Borel-Cantelli Lemma in dynamics established by Dolgopyat, Fayad and Liu [J. Mod. Dyn. 18 (2022) 209--289], broadening its applications to encompass several non-smooth systems with absolute continuous measures. Utilizing this generalization, we derive multiple Logarithm Law for hitting time and recurrence of dispersing billiard maps and piecewise expanding maps under some regular conditions, including tent map, Lorentz-like map and Gauss map.

math.DS↗

arXiv2023

An analogue of Law of Iterated Logarithm for Heavy Tailed Random Variables

We establish functional limit theorems for ergodic sums of observables with power singularities for expanding circle maps. In the regime where the observables have infinite variance, we show that when rescaled by $N^{1/s}(\ln N)^\alpha$, the partial sum process has limit points consisting precisely of increasing piecewise constant functions with finitely many jumps. Our approach combines trimming techniques with a multiple Borel-Cantelli argument. It provides a functional law of the iterated logarithm for heavy-tailed processes where classical almost sure invariance principles do not apply.

math.DS↗

arXiv2021

Multiple Borel Cantelli Lemma in dynamics and MultiLog law for recurrence

A classical Borel Cantelli Lemma gives conditions for deciding whether an infinite number of rare events will almost surely happen. In this article, we propose an extension of Borel Cantelli Lemma to characterize the multiple occurrence of events on the same time scale. Our results imply multiple Logarithm Laws for recurrence and hitting times, as well as Poisson Limit Laws for systems which are exponentially mixing of all orders. The applications include geodesic flows on compact negatively curved manifolds, geodesic excursions, Diophantine approximations and extreme value theory for dynamical systems.

math.DS↗