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This chapter compiles various concepts and topics from analysis, combinatorics, and probability theory that can be referenced as needed. These include, among others, the Kolmogorov axiomatic system, key properties of probability measures such as continuity from above and below, and stochastic independence. Other topics include convergence in distribution of random variables, the Lindeberg–Lévy central limit theorem, specific representations of the expectation and variance of nonnegative integer-valued random variables, the Wallis product representation for the number \(\pi \) , Stirling’s formula, generating functions, identities for binomial coefficients, the binomial series, Legendre polynomials, and the Borel-Cantelli lemma.

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Tools from Stochastics, Combinatorics, and Analysis

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  • 发表日期：

    2025年

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  • Norbert Henze

摘要

This chapter compiles various concepts and topics from analysis, combinatorics, and probability theory that can be referenced as needed. These include, among others, the Kolmogorov axiomatic system, key properties of probability measures such as continuity from above and below, and stochastic independence. Other topics include convergence in distribution of random variables, the Lindeberg–Lévy central limit theorem, specific representations of the expectation and variance of nonnegative integer-valued random variables, the Wallis product representation for the number \(\pi \) , Stirling’s formula, generating functions, identities for binomial coefficients, the binomial series, Legendre polynomials, and the Borel-Cantelli lemma.