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• Author or Editor: Zhan Shi
Clear All Modify Search  # Favourite sites of simple random walk

Periodica Mathematica Hungarica
Authors: Zhan Shi and Bálint Tóth
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# An Lp-view of the Bahadur-Kiefer Theorem

Periodica Mathematica Hungarica
Authors: Miklós Csörgő and Zhan Shi

Summary Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\alpha_n$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\beta_n$ \end{document} be respectively the uniform empirical and quantile processes, and define \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $R_n = \alpha_n + \beta_n$ \end{document}, which usually is referred to as the Bahadur--Kiefer process. The well-known Bahadur-Kiefer theorem confirms the following remarkable equivalence: \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\|R_n\| /\sqrt{\| \alpha_n \| }\, \sim \, n^{-1/4} (\log n)^{1/2}$ \end{document} almost surely, as \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $n$ \end{document} goes to infinity, where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\| f\| =\sup_{0\le t\le 1} |f(t)|$ \end{document} is the \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $L^\infty$ \end{document}-norm. We prove that \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\|R_n\|_2 /\sqrt{\| \alpha_n \|_1}\, \sim \, n^{-1/4}$ \end{document} almost surely, where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\| \, \cdot \, \|_p$ \end{document} is the \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $L^p$ \end{document}-norm. It is interesting to note that there is no longer any logarithmic term in the normalizing function. More generally, we show that \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $n^{1/4} \|R_n\|_p /\sqrt{\| \alpha_n \|_{(p/2)}}$ \end{document} converges almost surely to a finite positive constant whose value is explicitly known.

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# Weak convergence for the minimal position in a branching random walk: A simple proof

Periodica Mathematica Hungarica
Authors: Elie Aïdékon and Zhan Shi

## Summary

Consider the boundary case in a one-dimensional super-critical branching random walk. It is known that upon the survival of the system, the minimal position after n steps behaves in probability like

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac{3} {2}$$ \end{document}
log n when n → ∞. We give a simple and self-contained proof of this result, based exclusively on elementary properties of sums of i.i.d. real-valued random variables.

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# Differential diagnosis of Moniezia benedeni and M. expansa (Anoplocephalidae) by PCR using markers in small ribosomal DNA (18S rDNA)

Acta Veterinaria Hungarica
Authors: HongBin Yan, XinWen Bo, Youyu Liu, Zhongzi Lou, XingWei Ni, WanGui Shi, Fang Zhan, HongKean Ooi and WanZhong Jia

Moniezia benedeni and M. expansa are common ruminant tapeworms of worldwide distribution, causing gastrointestinal disorders and even death in sheep and goats. In this study, a polymerase chain reaction- (PCR-) based approach for precise species identification was developed and also applied to faecal DNA diagnosis of the tapeworm infection. Since nuclear ribosomal DNA (rDNA) appears to be a useful target for species and/or strain markers, the 18S regions of the rDNA of M. benedeni and M. expansa were amplified and sequenced. The lengths and GC contents of the regions sequenced were 2476–2487 bp and 51.9–52.1% for M. benedeni and 2473 bp and 51.9–52.0% for M. expansa, respectively. Alignment and comparison of the 18S sequences of the two species showed 92.5–93.3% homology. No matches for the 18S regions of M. benedeni and M. expansa were found with other species by BLAST search, which made the 18S sequences appropriate markers for the design of distinctive primers for the two Moniezia species. Our 18S-based PCR could detect as low levels as 0.5 pg genomic DNA or the DNA extracted from 0.2 g faecal sample collected from the rectum of an M. expansa-infected goat. The results indicate that this PCR approach is a reliable alternative for the differential diagnosis of Moniezia species in faecal samples.

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# Synthesis and preliminary evaluation of a novel 125I-labeled T140 analog for quantitation of CXCR4 expression

Journal of Radioanalytical and Nuclear Chemistry
Authors: Yanjiang Han, Duanzhi Yin, Mingqiang Zheng, Wei Zhou, ZhenHong Lee, Lan Zhan, Yufei Ma, Mingxing Wu, Lingli Shi, Ni Wang, Jianbo Lee, Cheng Wang, Zheng Lee and Yongxian Wang

## Abstract

The aim of this study was to develop a radiopharmaceutical for the imaging of CXCR4-expressing tumors in vivo. For 125I-labeling, 125I-SIB was synthesized and conjugated with the ε-NH2 group of Ac-TZ14011, a specific CXCR4 antagonist. The specific radioactivity of the product was 5 GBq/μmol and the radiochemical purity (RCP) was 96% (n = 3). After 6 h, the RCP of the product in PBS was 93%. The MCF-7 cell uptake of Ac-TZ14011 was rapid and high. Primary biodistribution studies indicated that 125I-IB-Ac-TZ14011 was mainly excreted via the kidney, and further evaluation in mice with induced tumors was necessary.

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