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Levy’s continuity theorem and central limit theorem
This chapter explores the use of characteristic functions in probability theory. The characteristic function of a random variable contains all the...
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Faster Convergence in the Free Central Limit Theorem
We show that there is a faster convergence in the free central limit theorem for measures of bounded support when we have vanishing free cumulants....
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Richter’s local limit theorem, its refinement, and related results*
We give a detailed exposition of the proof of Richter’s local limit theorem in a refined form and establish the stability of the remainder term in...
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Classical Limit and Ehrenfest’s Theorem Versus Non-relativistic Limit of Noncommutative Dirac Equation in the Presence of Minimal Uncertainty in Momentum
In this article, we discuss the classic limit and investigate the Ehrenfest’s theorem of the Dirac equation in the context of minimal uncertainty in...
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The Local Limit Theorem in the Reducible Case
We prove the local limit theorem for $${\mathbb {P}}(S_{N}-z_{N}\in...
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Examples of Autonomous Differential Systems with Contrasting Combinations of Lyapunov, Perron, and Upper-Limit Stability Measures
AbstractNew characteristics of differential systems are studied, which meaningfully develop the concepts of Lyapunov, Perron, and upper limit...
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A SIR epidemic model on a refining spatial grid II-central limit theorem
A stochastic SIR epidemic model taking into account the heterogeneity of the spatial environment is constructed. The deterministic model is given by...
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Rates of Convergence in the Central Limit Theorem for the Elephant Random Walk with Random Step Sizes
In this paper, we consider a generalization of the elephant random walk model. Compared to the usual elephant random walk, an interesting feature of...
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Some Aspects of the Central Limit Theorem and Related Topics
Very often the observation of natural phenomena leads to an average trend with fluctuations around it. One of the most well known example is the...
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Variance Growth, Center-Tightness, and the Central Limit Theorem
We analyze the variance of $$S_N=f_1(X_1,X_2)+\cdots...
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Martingale Central Limit Theorem
The martingale central limit theorem provides convergence of suitably centered and scaled sums of martingale difference sequences having finite...
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Local Central Limit Theorem for Long-Range Two-Body Potentials at Sufficiently High Temperatures
Dobrushin and Tirozzi (Commun Math Phys 54(2):173–192, 1977) showed that, for a Gibbs measure with the finite-range potential, the Local Central...
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Functional Central Limit Theorem and Strong Law of Large Numbers for Stochastic Gradient Langevin Dynamics
We study the mixing properties of an important optimization algorithm of machine learning: the stochastic gradient Langevin dynamics (SGLD) with a...
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Upper bound limit analysis for 3D slope stability analysis based on rigid block structure
In this paper, an upper bound limit analysis for 3D slope stability analysis based on rigid block structure (SLU-3D) is proposed. Specifically, a...
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Three-dimensional collapse mechanism and failure probability analysis of deep-buried tunnel based on the upper bound theorem of limit analysis and Hoek-Brown failure criterion
In this work, the three-dimensional collapse mechanism of deep-buried tunnel roof is established primarily based on the Hoek-Brown failure criterion...
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Central Limit Theorems
In the previous chapter, we have studied some random variables which come up naturally when a stationary Poisson hyperplane process is observed...
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Application of the upper bound theorem for metal forming processes considering an arbitrary isotropic pressure-independent yield criterion with no strength differential effect
The present paper concerns the application of the upper bound theorem based on the work function to analyze metal forming processes. An advantage of...
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Examples of Differential Systems with Contrasting Combinations of Lyapunov, Perron, and Upper-Limit Properties
AbstractA number of examples of systems of differential equations are given that have, in a sense, opposite properties of stability or instability of...
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A Hardy–Hilbert-type integral inequality involving two multiple upper-limit functions
By means of the weight functions, the idea of introducing parameters and the technique of real analysis, a new Hardy–Hilbert-type integral inequality...