A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints

Irina Shevtsova; Mikhail Tselishchev

doi:10.3390/math8040577

journal article Open Access Apr 13, 2020

A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints

Irina Shevtsova Mikhail Tselishchev

Mathematics Vol. 8 No. 4 pp. 577 · MDPI AG

View at Publisher Save 10.3390/math8040577

Abstract

We introduce a generalized stationary renewal distribution (also called the equilibrium transform) for arbitrary distributions with finite nonzero first moment and study its properties. In particular, we prove an optimal moment-type inequality for the Kantorovich distance between a distribution and its equilibrium transform. Using the introduced transform and Stein’s method, we investigate the rate of convergence in the Rényi theorem for the distributions of geometric sums of independent random variables with identical nonzero means and finite second moments without any constraints on their supports. We derive an upper bound for the Kantorovich distance between the normalized geometric random sum and the exponential distribution which has exact order of smallness as the expectation of the geometric number of summands tends to infinity. Moreover, we introduce the so-called asymptotically best constant and present its lower bound yielding the one for the Kantorovich distance under consideration. As a concluding remark, we provide an extension of the obtained estimates of the accuracy of the exponential approximation to non-geometric random sums of independent random variables with non-identical nonzero means.

Topics

No keywords indexed for this article. Browse by subject →

References

35

[1]

Zolotarev "Probability metrics" Theory Probab. Appl. (1984) 10.1137/1128025

[2]

Zolotarev "Ideal metrics in the problems of probability theory and mathematical statistics" Austral. J. Statist (1979) 10.1111/j.1467-842x.1979.tb01139.x

[3]

Zolotarev, V.M. (1997). Modern Theory of Summation of Random Variables, VSP. 10.1515/9783110936537

[4]

Rachev, S. (1991). Probability Metrics and the Stability of Stochastic Models, John Wiley ans Sons.

[5]

Mattner "An optimal Berry–Esseen type theorem for integrals of smooth functions" Lat. Am. J. Probab. Math. Stat. (2019)

[6]

Cambanis "Inequalities for Ek(X,Y) when the marginals are fixed" Z. Wahrsch. Verw. Geb. (1976) 10.1007/bf00532695

[7]

Solovyev "Asymptotic behaviour of the time of first occurrence of a rare event" Engrgy Cybern. (1971)

[8]

Brown "Error Bounds for Exponential Approximations of Geometric Convolutions" Ann. Probab. (1990) 10.1214/aop/1176990750

[9]

Kalashnikov "Metric estimates of the first occurrence time in regenerative processes" Stability Problems for Stochastic Models (1985) 10.1007/bfb0074816

[10]

Kruglov, V.M., and Korolev, V.Y. (1990). Limit Theorems for Random Sums, Moscow State University. (In Russian).

[11]

Sugakova "Estimates in the Rényi theorem for differently distributed terms" Ukr. Math. J. (1995) 10.1007/bf01084909

[12]

Kalashnikov, V.V. (1997). Geometric Sums: Bounds for Rare Events with Applications: Risk Analysis, Reliability, Queueing, Springer. Mathematics and Its Applications. 10.1007/978-94-017-1693-2

[13]

"New rates for exponential approximation and the theorems of Rényi and Yaglom" Ann. Probab. (2011)

[14]

Fundamentals of Stein’s method

Nathan Ross

Probability Surveys 2011 10.1214/11-ps182

[15]

Hung "On the rate of convergence in limit theorems for geometric sums" Southeast Asian J. Sci. (2013)

[16]

Bogachev, V.I. (2018). Weak Convergence of Measures. Am. Math. Soc., 234. 10.1090/surv/234

[17]

Glivenko, V.I. (2007). Stieltjes Integral, URSS. [2nd ed.].

[18]

Bogachev, V.I. (2007). Measure Theory. Vol. I, II., Springer. 10.1007/978-3-540-34514-5

[19]

Folland, G.B. (1999). Real Analysis, John Wiley & Sons. [2nd ed.].

[20]

Feller, W. (1971). An Introduction to the Probability Theory and Its Applications. Vol. II., John Wiley. [2nd ed.].

[21]

Asmussen, W. (2003). Applied Probability and Queues, Springer.

[22]

Lin, X.S. (2006). Integrated Tail Distribution. Encyclopedia of Actuarial Science. Am. Cancer Soc.

[23]

Harkness "Convergence of a sequence of transformations of distribution functions" Pacific J. Math. (1969) 10.2140/pjm.1969.31.403

[24]

Shantaram "On a certain class of limit distributions" Ann. Math. Stat. (1972) 10.1214/aoms/1177690886

[25]

Mulholland "Representation Theorems for Distribution Functions" Proc. Lond. Math. Soc. (1958) 10.1112/plms/s3-8.2.177

[26]

Hoeffding "The extrema of the expected value of a function of independent random variables" Ann. Math. Stat. (1955) 10.1214/aoms/1177728543

[27]

Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Probability Theory, University of California Press.

[28]

Korolev, V., and Zeifman, A. (2020). Bounds for convergence rate in laws of large numbers for mixed Poisson random sums. arXiv. 10.1016/j.spl.2020.108918

[29]

Esseen "A moment inequality with an application to the central limit theorem" Skand. Aktuarietidskr. (1956)

[30]

Kolmogorov "Some recent works in the field of limit theorems of probability theory" Bull. Mosc. Univ. (1953)

[31]

Chistyakov "Asymptotically proper constants in the Lyapunov theorem" J. Math. Sci. (1999) 10.1007/bf02364834

[32]

Chistyakov "A new asymptotic expansion and asymptotically best constants in Lyapunov’s theorem. I" Theory Probab. Appl. (2002) 10.1137/s0040585x97978932

[33]

Chistyakov "A new asymptotic expansion and asymptotically best constants in Lyapunov’s theorem. II" Theory Probab. Appl. (2002) 10.1137/s0040585x97979172

[34]

Chistyakov "A new asymptotic expansion and asymptotically best constants in Lyapunov’s theorem. III" Theory Probab. Appl. (2003) 10.1137/s0040585x97979822

[35]

Shevtsova "On the asymptotically exact constants in the Berry–Esseen–Katz inequality" Theory Probab. Appl. (2011) 10.1137/s0040585x97984772

Metrics

8

Citations

35

References

Details

Published: Apr 13, 2020
Vol/Issue: 8(4)
Pages: 577
License: View

Authors

I

Irina Shevtsova

Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China; Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, GSP-1, 1-52 Leninskiye Gory, Moscow 119991, Russia; Federal Research Center “Informatics and Control” of the Russian Academy of Sciences, Moscow 119333, Russia; Moscow Center for Fundamental and Applied Mathematics, Moscow 119991, Russia

M

Mikhail Tselishchev

Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, GSP-1, 1-52 Leninskiye Gory, Moscow 119991, Russia; Moscow Center for Fundamental and Applied Mathematics, Moscow 119991, Russia

Funding

Russian Science Foundation Award: 18-11-00155

Russian Foundation for Basic Research Award: 19-31-37001

Council on Grants of the President of the Russian Federation Award: MD–189.2019.1

Cite This Article

Irina Shevtsova, Mikhail Tselishchev (2020). A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints. Mathematics, 8(4), 577. https://doi.org/10.3390/math8040577

A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints

You May Also Like