A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints
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...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-04-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/8/4/577 |
| _version_ | 1827718842680344576 |
|---|---|
| author | Irina Shevtsova Mikhail Tselishchev |
| author_facet | Irina Shevtsova Mikhail Tselishchev |
| author_sort | Irina Shevtsova |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-10T20:30:34Z |
| format | Article |
| id | doaj.art-fd601bb4ee4648a4953aef1e9605b016 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T20:30:34Z |
| publishDate | 2020-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-fd601bb4ee4648a4953aef1e9605b0162023-11-19T21:28:34ZengMDPI AGMathematics2227-73902020-04-018457710.3390/math8040577A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support ConstraintsIrina Shevtsova0Mikhail Tselishchev1Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, ChinaDepartment of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, GSP-1, 1-52 Leninskiye Gory, Moscow 119991, RussiaWe 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.https://www.mdpi.com/2227-7390/8/4/577Rényi theoremKantorovich distancezeta-metricsStein’s methodstationary renewal distributionequilibrium transform |
| spellingShingle | Irina Shevtsova Mikhail Tselishchev A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints Mathematics Rényi theorem Kantorovich distance zeta-metrics Stein’s method stationary renewal distribution equilibrium transform |
| title | A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints |
| title_full | A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints |
| title_fullStr | A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints |
| title_full_unstemmed | A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints |
| title_short | A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints |
| title_sort | generalized equilibrium transform with application to error bounds in the renyi theorem with no support constraints |
| topic | Rényi theorem Kantorovich distance zeta-metrics Stein’s method stationary renewal distribution equilibrium transform |
| url | https://www.mdpi.com/2227-7390/8/4/577 |
| work_keys_str_mv | AT irinashevtsova ageneralizedequilibriumtransformwithapplicationtoerrorboundsintherenyitheoremwithnosupportconstraints AT mikhailtselishchev ageneralizedequilibriumtransformwithapplicationtoerrorboundsintherenyitheoremwithnosupportconstraints AT irinashevtsova generalizedequilibriumtransformwithapplicationtoerrorboundsintherenyitheoremwithnosupportconstraints AT mikhailtselishchev generalizedequilibriumtransformwithapplicationtoerrorboundsintherenyitheoremwithnosupportconstraints |