Abstract
The subject of this chapter is the application of the theory of probability metrics to limit theorems arising from summing independent and identically distributed (i.i.d.) random variables (RVs).
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Notes
- 1.
- 2.
See, for example, Dunford and Schwartz [1988, Theorem IV.8.3.5].
- 3.
See, for example, Erdös and Spencer [1974, p. 17].
- 4.
- 5.
- 6.
- 7.
See Lemma 3.3.1, (3.4.18), and (3.3.13) in Chap. 3.
- 8.
See Billingsley [1999].
- 9.
- 10.
- 11.
See Kalashnikov and Rachev [1988, Sect. 3, Theorem 10.1].
- 12.
See Corollary 5.5.1 and Theorem 6.2.1.
- 13.
See Theorem 6.4.1 or Theorem 8.3.1 with \(c(x,y) = \vert x - y\vert \).
- 14.
See, for example, Rachev and Rüschendorf [1992] for an application of ideal metrics in the multivariate CLT, Rachev and Rüschendorf [1994] for an application of the Kantorovich metric, Maejima and Rachev [1996] for rates of convergence in operator-stable limit theorems, and Klebanov et al. [1999] for rates of convergence in prelimit theorems.
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Rachev, S.T., Klebanov, L.B., Stoyanov, S.V., Fabozzi, F.J. (2013). Ideal Metrics with Respect to Summation Scheme for i.i.d. Random Variables. In: The Methods of Distances in the Theory of Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4869-3_15
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