We establish some limit theorems for the elephant random walk (ERW), including Berry–Esseen’s bounds, Cramér moderate deviations and local limit theorems. These limit theorems can be regarded as refinements of the central limit theorem for the ERW. Moreover, by these limit theorems, we conclude that the convergence rate of normal approximations and the domain of attraction of normal distribution mainly depend on a memory parameter p which lies between 0 and 3/4.

Background Several factors may affect students going to school toilets, but a few studies have analyzed the reasons for students using toilets. This study aimed to use a structural equation model to understand the factors that impacted children’s toilet behavior. Methods This study was performed in 12 rural nonboarding primary schools (6 schools in the northern and 6 schools in the southern regions of China). All students of the third and sixth grades (761 students) were examined. A questionnaire on students’ toilet behavior was used. The questionnaire included 33 perceptual items based on 5 factors: toilet facilities, cleanliness, hygiene practices, peer relationship, and experience. The questionnaire also covered the frequency of voiding and defecating by themselves. The exploratory factor analysis, confirmatory factor analysis, and pathway analysis were used to analyze the causes of students’ toilet behavior. Results A statistically significant correlation coefficient of 0.300 indicated that cleanliness impacted the toilet frequency of students. The visual experience of the overall cleanliness of the toilet had the most significant impact on students’ toilet behavior (path coefficient, 0.81). Washing facilities and convenient handwashing had the least impact on toilet use (path coefficient, 0.52). Conclusion Cleanliness was the primary consideration for students’ toilet use on campus. The visual experience of the overall cleanliness of toilets had the most significant impact when students used toilets. No pre-survey was conducted to test the reliability and validity of the questionnaire. Using self-reported data might be associated with potential recall errors.

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Let {Z n , n ≥ 0} be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for ln(Z n+n 0 /Z n 0 ) uniformly in n 0 ∈ N, which extend the corresponding results by Grama et al. (Stochastic Process. Appl. 2017) established for n 0 = 0. The extension is interesting in theory, and is motivated by applications. A new method is developed for the proofs; some conditions of Grama et al. ( 2017) are relaxed in our present setting. An example of application is given in constructing confidence intervals to estimate the criticality parameter in terms of ln(Z n+n 0 /Z n 0 ) and n.

We establish some limit theorems for one-dimensional elephant random walk, including Berry-Esseen bounds, Cramér moderate deviations and local limit theorems. These limit theorems can be regarded as refinements of the central limit theorems for the elephant random walk. Moreover, by these limit theorems, we conclude that the domain of attraction of normal distribution mainly depends on a memory parameter p which lies between 0 and 3/4.