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1. Non-uniform bounds in the Poisson approximation with applications to informational distances. I.

   Bobkov, S. G.; Chistyakov, G. P.; G\"otze, F. (September 2019, IEEE transactions on information theory)

   We explore asymptotically optimal bounds for deviations of Bernoulli convolutions from the Poisson limit in terms of the Shannon relative entropy and the Pearson chi-squared distance. The results are based on proper non-uniform estimates for densities. This part deals with the so-called non-degenerate case. 

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2. Zero‐inflated count distributions for capture–mark–reencounter data

   https://doi.org/10.1002/ece3.9274  

   Riecke, Thomas V; Gibson, Daniel; Sedinger, James S; Schaub, Michael (September 2022, Ecology and Evolution)

   Abstract The estimation of demographic parameters is a key component of evolutionary demography and conservation biology. Capture–mark–recapture methods have served as a fundamental tool for estimating demographic parameters. The accurate estimation of demographic parameters in capture–mark–recapture studies depends on accurate modeling of the observation process. Classic capture–mark–recapture models typically model the observation process as a Bernoulli or categorical trial with detection probability conditional on a marked individual's availability for detection (e.g., alive, or alive and present in a study area). Alternatives to this approach are underused, but may have great utility in capture–recapture studies. In this paper, we explore a simple concept:in the same way that counts contain more information about abundance than simple detection/non‐detection data, the number of encounters of individuals during observation occasions contains more information about the observation process than detection/non‐detection data for individuals during the same occasion. Rather than using Bernoulli or categorical distributions to estimate detection probability, we demonstrate the application of zero‐inflated Poisson and gamma‐Poisson distributions. The use of count distributions allows for inference on availability for encounter, as well as a wide variety of parameterizations for heterogeneity in the observation process. We demonstrate that this approach can accurately recover demographic and observation parameters in the presence of individual heterogeneity in detection probability and discuss some potential future extensions of this method. 

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3. Harmonic models and Bernoullicity

   https://doi.org/10.1112/S0010437X21007442  

   Hayes, Ben (October 2021, Compositio Mathematica)

   We give many examples of algebraic actions which are factors of Bernoulli shifts. These include certain harmonic models over left-orderable groups of large enough growth, as well as algebraic actions associated to certain lopsided elements in any left-orderable group. For many of our examples, the acting group is amenable so these actions are Bernoulli (and not just a factor of a Bernoulli), but there is no obvious Bernoulli partition. 

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4. Rényi divergences in central limit theorems: Old and new

   https://doi.org/10.1214/25-PS30  

   Bobkov, Sergey G; Götze, Friedrich (January 2025, Probability Surveys)

   We give an overview of various results and methods related to information-theoretic distances of Rényi type in the light of their applications to the central limit theorem (CLT). The first part (Sections 1–9) is devoted to the total variation and the Kullback-Leibler distance (relative entropy). In the second part (Sections 10–15) we discuss general properties of Rényi and Tsall is divergences of order alpha > 1, and then in the third part (Sections 16–21) we turn to the CLT and non-uniform local limit theorems with respect to these strong distances. In the fourth part (Sections 22–31), we discuss recent results on strictly subgaussian distributions and describe necessary and sufficient conditions which ensure the validity of the CLT with respect to the Rényi divergence of infinite order. 

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5. Using Bernoulli maps to accelerate mixing of a random walk on the torus

   https://doi.org/10.1090/qam/1668  

   Iyer, Gautam; Lu, Ethan; Nolen, James (June 2024, Quarterly of Applied Mathematics)

   We study the mixing time of a random walk on the torus, alternated with a Lebesgue measure preserving Bernoulli map. Without the Bernoulli map, the mixing time of the random walk alone is $$O(1/\epsilon^2)$$, where $$\epsilon$$ is the step size. Our main results show that for a class of Bernoulli maps, when the random walk is alternated with the Bernoulli map~$$\varphi$$ the mixing time becomes $$O(\abs{\ln \epsilon})$$. We also study the \emph{dissipation time} of this process, and obtain~$$O(\abs{\ln \epsilon})$$ upper and lower bounds with explicit constants. 

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