The Central Limit Theorem

Abstract

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 sn=∑1⩽k⩽rnXn,k{s_n} = \sum\nolimits_{1 \leqslant k \leqslant {r_n}} {{X_{n,k}}}. The Lindeberg condition is sufficient for Sn
/sn
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.