The asymptotic behavior of rarely visited edges of the simple random walk

In this paper, we study the asymptotic behavior of the number of rarely visited edges (i.e., edges that visited only once) of a simple symmetric random walk on $\mathbb{Z}$. Let $\alpha(n)$ be the number of rarely visited edges up to time $n$. First, we evaluate $\mathbb{E}(\alpha(n))$, show that $n\to \mathbb{E}(\alpha(n))$ is non-decreasing in $n$ and that $\lim\limits_{n\to+\infty}\mathbb{E}(\alpha(n))=2$. Then we study the asymptotic behavior of $\mathbb{P} (\alpha(n)>a(\log n)^2)$ for any $a>0$ and use it to show that there exists a constant $C\in(0,+\infty)$ such that $\limsup\limits_{n\to+\infty}\frac{\alpha(n)}{(\log n)^2}=C$ almost surely

Favorite Downcrossing Sites of One-Dimensional Simple Random Walk

For a one-dimensional simple symmetric random walk $(S_n)$, a site $x$ is called a favorite downcrossing site at time $n$ if its downcrossing local time at time $n$ achieves the maximum among all sites. In this paper, we study the cardinality of the favorite downcrossing site set, and will show that with probability 1 there are only finitely many times at which there are at least four favorite downcrossing sites and three favorite downcrossing sites occurs infinitely often. Some related open questions will be introduced.Comment: 17 pages. arXiv admin note: substantial text overlap with arXiv:2111.0068

Three Favorite Edges Occurs Infinitely Often for One-Dimensional Simple Random Walk

For a one-dimensional simple symmetric random walk $(S_n)$, an edge $x$ (between points $x-1$ and $x$) is called a favorite edge at time $n$ if its local time at $n$ achieves the maximum among all edges. In this paper, we show that with probability 1 three favorite edges occurs infinitely often. Our work is inspired by T\'{o}th and Werner [Combin. Probab. Comput. {\bf 6} (1997) 359-369], and Ding and Shen [Ann. Probab. {\bf 46} (2018) 2545-2561], disproves a conjecture mentioned in Remark 1 on page 368 of T\'{o}th and Werner [Combin. Probab. Comput. {\bf 6} (1997) 359-369].Comment: 23 page

Insulin resistance in Chinese patients with type 2 diabetes is associated with C-reactive protein independent of abdominal obesity

<p>Abstract</p> <p>Background</p> <p>There is debate as to whether the association between C-reactive protein (CRP) and insulin resistance is independent of body fatness, particularly central obesity. Therefore, the association among CRP, insulin resistance and obesity was analyzed in Chinese patients with type 2 diabetes.</p> <p>Methods</p> <p>The study included 520 Chinese patients diagnosed with type 2 diabetes with CRP levels not exceeding 10 mg/L. The degree of insulin resistance was determined with the homeostasis model assessment of insulin resistance (HOMA-IR). The CRP levels were categorized into quartiles from the lowest to the highest concentrations (Q1-Q4).</p> <p>Results</p> <p>Body mass index (BMI) and waist circumference (WC) were both higher in Q4, Q3 and Q2 than those in Q1. HOMA-IR was higher in Q2, Q3 and Q4 than that in Q1 (Q1 vs Q4, P < 0.001; Q1 vs Q3, P < 0.001; Q1 vs Q2, P = 0.028). Log CRP was significantly correlated with log HOMA-IR (correlation coefficient: 0.230, P < 0.001) and BMI (correlation coefficient: 0.305, P < 0.001) and WC (correlation coefficient: 0.240, P < 0.001) by Spearman correlation analysis. Multiple linear regression analysis adjusting for age, gender and components of metabolic syndrome, log CRP was also independently associated with log HOMA-IR (β coefficient, 0.168; P < 0.001) and WC (β coefficient, 0.131; P = 0.006).</p> <p>Conclusion</p> <p>These findings showed that insulin resistance was associated with CRP levels independent of abdominal obesity in Chinese patients with type 2 diabetes, suggesting that abdominal obesity could only partly explain the link between subclinical inflammation and insulin resistance.</p

Prevalence of non-alcoholic fatty liver disease and its relation to hypoadiponectinaemia in the middle-aged and elderly Chinese population

Introduction: Hypoadiponectinaemia is an important risk factor for non-alcoholic fatty liver disease (NAFLD). However, little is known about its role in the Chinese population. This study sought to assess the prevalence of NAFLD and its association with hypoadiponectinaemia in middle-aged and elderly Chinese. Material and methods: We conducted a population-based cross-sectional study in an urban Shanghai sample of 2201 participants age 50 years to 83 years (973 men, 1228 women). Hepatic ultrasonographic examination was performed for all participants. Serum adiponectin concentrations were measured by ELISA methods. Results: The prevalence of NAFLD was 19.8% (16.0% in men, 22.8% in women). Serum adiponectin levels were significantly higher in female than in male subjects (p < 0.001). Serum adiponectin levels were significantly lower in NAFLD subjects than those in control subjects (p < 0.001). The prevalence of NAFLD progressively increased with declining adiponectin levels (p(for) (trend) < 0.001). The participants in the lowest adiponectin quartile had a significantly increased risk for acquiring NAFLD (OR = 2.31, 95% CI 1.72-3.15) after adjustment for potential confounders. Conclusions: Population-based screening suggests that NAFLD is highly prevalent in middle-aged and elderly people in Shanghai, particularly among women. Serum adiponectin level is negatively associated with NAFLD independently of potential cofounders, indicating that hypoadiponectinaemia may contribute to the development of NAFLD

Association of lipid profiles and the ratios with arterial stiffness in middle-aged and elderly Chinese

BACKGROUND: Serum lipids and the ratios are known to be associated with the cardiovascular diseases (CVD). However, the associations of serum lipids and the ratios related to arterial stiffness are unclear. We sought to compare the strength of these serum lipids and the ratios with arterial stiffness assessing by brachial-ankle pulse wave velocity (baPWV) in the middle-aged and elderly Chinese subjects. METHODS: A total number of 1133 Chinese aged from 50 to 90 years old were recruited from Shanghai downtown district. The serum lipids, baPWV and major cardiovascular risk factors of the participants were measured. RESULTS: Participants with high baPWV exhibited higher levels of non-HDL-c, TC/HDL-c, TG/HDL-c, LDL-c/HDL-c, and non-HDL-c/HDL-c, while HDL-c worked in the opposite direction (all P<0.05). In addition, TC, TG, LDL-c, non-HDL-c, TC/HDL-c, TG/HDL-c, LDL-c/HDL-c, and non-HDL-c/HDL-c had a positive relationship with the baPWV value, while HDL-c was on the contrary (all P <0.05). Finally, individuals with high non-HDL-c/HDL-c (OR 1.71, 95% CI 1.06-2.55, P = 0.013) and low HDL-c (OR 0.57, 95% CI 0.35-0.96, P = 0.024) were seem to be at high risk of arterial stiffness. CONCLUSIONS: As a risk indicator, non-HDL-c/HDL-c, which could be readily obtained from routine serum lipids, was significantly associated with baPWV. Non-HDL-c/HDL-c was superior to traditional lipid variables for estimating arterial stiffness in the middle-aged and elderly Chinese population

Investigation of Type 2 Diabetes Risk Alleles Support CDKN2A/B, CDKAL1, and TCF7L2 As Susceptibility Genes in a Han Chinese Cohort

Background: Recent genome-wide association studies (GWASs) have reported several genetic variants to be reproducibly associated with type 2 diabetes. Additional variants have also been detected from a metaanalysis of three GWASs, performed in populations of European ancestry. In the present study, we evaluated the influence of 17 genetic variants from 15 candidate loci, identified in type 2 diabetes GWASs and the metaanalysis, in a Han Chinese cohort. Methodology/Principal Findings: Selected type 2 diabetes-associated genetic variants were genotyped in 1,165 type 2 diabetic patients and 1,136 normoglycemic control individuals of Southern Han Chinese ancestry. The OR for risk of developing type 2 diabetes was calculated using a logistic regression model adjusted for age, sex, and BMI. Genotype-phenotype associations were tested using a multivariate linear regression model. Genetic variants in CDKN2A/B, CDKAL1, TCF7L2, TCF2, MC4R, and PPARG showed a nominal association with type 2 diabetes (P <= 0.05), of whom the three first would stand correction for multiple testing: CDKN2A/B rs10811661, OR: 1.26 (1.12-1.43) P = 1.8* 10(-4); CDKAL1 rs10946398, OR: 1.23 (1.09-1.39); P = 7.1* 10(-4), and TCF7L2 rs7903146, OR: 1.61 (1.19-2.18) P = 2.3* 10(-3). Only nominal phenotype associations were observed, notably for rs8050136 in FTO and fasting plasma glucose (P = 0.002), postprandial plasma glucose (P = 0.002), and fasting C-peptide levels (P = 0.006) in the diabetic patients, and with BMI in controls (P = 0.033). Conclusions/Significance: We have identified significant association between variants in CDKN2A/B, CDKAL1 and TCF7L2, and type 2 diabetes in a Han Chinese cohort, indicating these genes as strong candidates conferring susceptibility to type 2 diabetes across different ethnicities

Signaling Role of Prokineticin 2 on the Estrous Cycle of Female Mice

The possible signaling role of prokineticin 2 (PK2) and its receptor, prokineticin receptor 2 (PKR2), on female reproduction was investigated. First, the expression of PKR2 and its co-localization with estrogen receptor (ERα) in the hypothalamus was examined. Sexually dimorphic expression of PKR2 in the preoptic area of the hypothalamus was observed. Compared to the male mice, there was more widespread PKR2 expression in the preoptic area of the hypothalamus in the female mice. The likely co-expression of PKR2 and ERα in the preoptic area of the hypothalamus was observed. The estrous cycles in female PK2-null, and PKR2-null heterozygous mice, as well as in PK2-null and PKR2-null compound heterozygous mice were examined. Loss of one copy of PK2 or PKR2 gene caused elongated and irregular estrous cycle in the female mice. The alterations in the estrous cycle were more pronounced in PK2-null and PKR2-null compound heterozygous mice. Consistent with these observations, administration of a small molecule PK2 receptor antagonist led to temporary blocking of estrous cycle at the proestrous phase in female mice. The administration of PKR2 antagonist was found to blunt the circulating LH levels. Taken together, these studies indicate PK2 signaling is required for the maintenance of normal female estrous cycles