Abstract
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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Notes
- 1.
According to standard conventions, we will write a(z)=o(b(z)) as z→z 0 if b(z)>0 and \(\lim_{z \to z_{0} } \frac{a(z)}{b(z)}=0\), and a(z)=O(b(z)) as z→z 0 if b(z)>0 and \(\limsup_{z \to z_{0} } \frac{|a(z)|}{b(z)} < \infty\).
- 2.
See, e.g., [12], Sect. 2.9.
- 3.
This fact will also easily follow from the properties of characteristic functions dealt with in Chap. 7.
- 4.
An extension of the de Moivre–Laplace theorem to the case of non-identically distributed random variables is contained in the central limit theorem from Sect. 8.4.
References
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1. Wiley, New York (1968)
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Borovkov, A.A. (2013). Sequences of Independent Trials with Two Outcomes. In: Probability Theory. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-5201-9_5
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