Summary
16.1 We study some properties of the convergence in law of random variables.
16.2 Let \({\left( {{X_{n,k}}} \right)_{\begin{array}{*{20}{c}} {n \geqslant 1} \\ {1 \leqslant k \leqslant {r_n}} \\ \end{array}}}\) be a triangular array of independent, centered, and square-integrable random variables. For every n ≥ 1, write \(s_n = \left[ {\sum {_{1 \le k \le r_n } {\rm{ }}Var\left( {X_{n,k} } \right)} } \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}\) and \({s_n} = \sum\nolimits_{1 \leqslant k \leqslant {r_n}} {{X_{n,k}}}\). The Lindeberg condition is sufficient for S n /s n to converge in law to the normal law (Theorem 16.2.1). Note, incidentally, that this condition is also necessary in the most usual cases (as a consequence of a Feller’s theorem, which is not proved here).
16.3 We prove the central limit theorem (Theorem 16.3.1), as well as some refinements of this theorem.
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© 1996 Springer-Verlag New York, Inc.
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Simonnet, M. (1996). The Central Limit Theorem. In: Measures and Probabilities. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4012-9_16
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DOI: https://doi.org/10.1007/978-1-4612-4012-9_16
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