Titlebook: Probability Theory; A Comprehensive Cour Achim Klenke Textbook 20081st edition Springer-Verlag London 2008 Probability theory.Random variab

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Characteristic Functions and the Central Limit Theorem,The main goal of this chapter is the central limit theorem (CLT) for sums of independent random variables (Theorem 15.37) and for independent arrays of random variables (Lindeberg-Feller theorem, Theorem 15.43). For the latter, we prove only that one of the two implications (Lindeberg’s theorem) that is of interest in the applications.

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Convergence of Markov Chains,We consider a Markov chain . with invariant distribution π and investigate conditions under which the distribution of . converges to π for . → ∞. Essentially it is necessary and sufficient that the state space of the chain cannot be decomposed into subspaces

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Achim KlenkeComprehensive and modern introduction to the most important fields of probability theory.Unique selection of topics, including many not usually found in introductory texts.Includes supplementary mater

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0172-5939 dition, plenty of figures, computer simulations, biographic details of key mathematicians, and a wealth of examples support and enliven the presentation. .978-1-84800-048-3Series ISSN 0172-5939 Series E-ISSN 2191-6675

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Textbook 20081st editioner to display the power of the abstract concepts in the world of probability theory. In addition, plenty of figures, computer simulations, biographic details of key mathematicians, and a wealth of examples support and enliven the presentation. .

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es an imbalance classification strategy with heterophily-aware GNNs to effectively address the class imbalance problem while significantly reducing training time. Our experiments on real-world graphs demonstrate our model’s superiority in classification performance and efficiency for node classifica