Sequences of Independent Trials with Two Outcomes
The weak and strong laws of large numbers are established for the Bernoulli scheme in Sect. 5.1. Then the local limit theorem on approximation of the binomial probabilities is proved in Sect. 5.2 using Stirling’s formula (covering both the normal approximation zone and the large deviations zone). The same section also contains a refinement of that result, including a bound for the relative error of the approximation, and an extension of the local limit theorem to polynomial distributions. This is followed by the derivation of the de Moivre–Laplace theorem and its refinements in Sect. 5.3. In Sect. 5.4, the coupling method is used to prove the Poisson theorem for sums of non-identically distributed independent random indicators, together with sharp approximation error bounds for the total variation distance. The chapter ends with derivation of large deviation inequalities for the Bernoulli scheme in Sect. 5.5.
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