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2013 | OriginalPaper | Buchkapitel

19. Ideal Metric with Respect to Maxima Scheme of i.i.d. Random Elements

verfasst von : Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi

Erschienen in: The Methods of Distances in the Theory of Probability and Statistics

Verlag: Springer New York

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Abstract

In this chapter, we discuss the problem of estimating the rate of convergence in limit theorems arising from the maxima scheme of independent and identically distributed (i.i.d.) random elements.

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Vorheriges Kapitel How Close Are the Individual and Collective Models in Risk Theory?

Nächstes Kapitel Ideal Metrics and Stability of Characterizations of Probability Distributions

See, for example, Resnick [1987a] and references therein.

See Maejima and Rachev [1997] for a discussion of the convergence rates in the multivariate max-stable limit theorem.

See (15.3.12) and (15.3.13).

See Example 4.2.3 in Chap. 4.

See the definition of \(\boldsymbol \rho _{r}\) provided in (19.2.4).

See Resnick [1987a].

See the definition in (19.2.4) and Theorems 19.2.1 and 19.2.2.

See Sect. 2.7 and Remark 2.7.2 in Chap. 6.

See Definition 19.2.1.

See Definition 3.3.2 in Chap. 3.

See Example 3.4.1 in Chap. 3 with \(d(x,y) =\| x - y\|_{r}\).

See Corollary 5.3.2 in Chap. 6 and (3.3.12) in Chap. 3.

See Corollary 7.4.2 and (7.5.15) in Chap. 6.

See Lemma 8.3.1 and Corollary 8.3.1 in Chap. 6.

See, for example, Resnick [1987a].

See (19.3.6) and Sect. 7.2 in Chap. 6.

See Example 4.4.3 and (4.4.41) in Chap. 6.

See (4.4.42) and (4.4.43) in Chap. 6 and Remark 7.2.3 in Chap. 6.

See Remark 7.2.3 in Chap. 6.

For additional details, see Kakosyan et al. [1988, Lemmas 2.4.1 and 2.4.2 and Theorem 2.4.1].

See Remark 19.2.1.

See Lemmas 19.2.2 and 19.2.3 for similar results.

The same is true for the summation scheme; see (16.3.7) in Chap. 66.

See (19.3.3) and (19.3.4).

See (19.3.4).

See, for example, Sazonov [1981, pp. 69–70].

See also Example 3.4.1 in Chap. 3.

See, for example, Theorem 7.2.2 in Chap. 6.

See Corollary 7.4.2 in Chap. 6.

The Kantorovich metric referred to by Zolotarev in the quote is (19.4.8) in this chapter.

See Rachev and Rüschendorf [1992] for applications of double ideal metrics in estimating convergence rates.

See Definition 15.3.1.

See Hoffman-Jorgensen and Pisier [1976].

See (8.3.7) and (8.3.21) in Chap. 6.

See (15.3.1) in Chap. 65.

See Zolotarev [1976, Theorem 5].

See Zolotarev [1978, Theorem 4].

See Zolotarev [1978, p. 272].

See Remark 7.2.3 in Chap. 6.

See Chow and Teicher [1978, p. 396].

It is identical to \({\mathcal{L}}^{p}\) for one-point measures μ.

See Shiryayev [1984, p. 469] and Chow and Teicher [1978, p. 367].

See (19.4.1) and Example 3.4.1 in Chap. 3.

See (3.4.12) and (19.4.31).

See (4.4.64) in Chap. 6.

See Chow and Teicher [1978, p. 247].

Balkema AA, de Haan L (1990) A convergence rate in extreme-value theory. J. Applied Probability 27, 577–585MATHCrossRef

Bentkus VY, Rackauskas A (1985) Estimates of the distance between sums of independent random elements in banach spaces. Theor Prob Appl 29:50–65MATHCrossRef

Chow YS, Teicher H (1978) Probability theory: independence, interchangeability, martingales. Springer, New YorkMATH

Cohen JP (1982) Convergence rate for the ultimate and penultimate approximation in extreme-value theory. Adv Appl Prob 14:833–854MATHCrossRef

de Haan L (1984) A spectral representation for max-stable processes. Ann Prob 12:1197–1204

Gine E, Leon JR (1980) On the central limit theorem in Hilbert space. Stochastica 4:43–71MathSciNetMATH

Hoffman-Jorgensen J (1977) Probability in banach spaces. Ecole d ’Ete de Probabilités de Saint-Flow VI. In: Lecture Notes in Mathematics, vol 598, pp 1–186. Springer, Berlin

Hoffman-Jorgensen J, Pisier G (1976) The law of large numbers and the central limit theorem in Banach spaces. Ann Prob 4:587–599CrossRef

Kakosyan AB, Klebanov L, Rachev ST (1988) Quantitative criteria for convergence of probability measures. Ayastan Press, Erevan (in Russian)

Maejima M, Rachev S (1997) Rates-of-convergence in the multivariate max-stable limit theorem. Stat Probab Lett 32:115–123MathSciNetMATHCrossRef

Omey E, Rachev S (1991) Rates of convergence in multivariate extreme value theory. J Multivar Anal 37:36–50MathSciNetCrossRef

Pisier G, Zinn J (1977) On the limit theorems for random variables with values in the spaces L ^p. Z Wahrsch Verw Geb 41:289–305MathSciNetCrossRef

Rachev S, Rüschendorf L (1992) Rate of convergence for sums and maxima and doubly ideal metrics. Theor Probab Appl 37(2):276–289

Resnick SI (1987a) Extreme values: regular variation and point processes. Springer, New YorkMATH

Resnick SI (1987b) Uniform rates of convergence to extreme-value distributions. In: Srivastava J (ed) Probability and statistics: essays in honor of Franklin A. Graybill. North-Holland, Amsterdam

Sazonov VV (1981) Normal approximation – some recent advances. In: Lecture Notes in Mathematics, vol 879. Springer, New York

Senatov VV (1981) Some lower estimates for the rate of convergence in the central limit theorem in Hilbert space. Sov Math Dokl 23:188–192MATH

Shiryayev AN (1984) Probability. Springer, New YorkMATH

Smith RL (1982) Uniform rates of convergence in extreme-value theory. Adv Appl Prob 14:600–622MATHCrossRef

Woyczynski WA (1980) On Marcinkiewicz-Zygmund laws of large numbers in Banach spaces and related rates of convergence. Prob Math Stat 1:117–131MathSciNetMATH

Zolotarev VM (1976) Metric distances in spaces of random variables and their distributions. Math. USSR sb. 30:373–401CrossRef

Zolotarev VM (1978) On pseudomoments. Theor Prob Appl 23:269–278MathSciNetCrossRef

Zolotarev VM (1983) Probability metrics. Theory Prob Appl 28:278–302CrossRef

Titel: Ideal Metric with Respect to Maxima Scheme of i.i.d. Random Elements
verfasst von: Svetlozar T. Rachev
Lev B. Klebanov
Stoyan V. Stoyanov
Frank J. Fabozzi
Verlag: Springer New York
Buch: The Methods of Distances in the Theory of Probability and Statistics
Print ISBN: 978-1-4614-4868-6

Electronic ISBN: 978-1-4614-4869-3

Copyright-Jahr: 2013
DOI: https://doi.org/10.1007/978-1-4614-4869-3_19

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