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Mat. Sb. (N.S.), 1976, Volume 101(143), Number 3(11), Pages 416–454 (Mi msb2908)  

This article is cited in 36 scientific papers (total in 36 papers)

Metric distances in spaces of random variables and their distributions

V. M. Zolotarev


Abstract: In this paper the concept of a metric in the space of random variables defined on a probability space is introduced. The principle of three stages in the study of approximation problems is formulated, in particular problems of approximating distributions.
Various facts connected with the use of metrics in these three stages are presented and proved. In the second part of the paper a series of results is introduced which are related to stability problems in characterizing distributions and to problems of estimating the remainder terms in limiting approximations of distributions of sums of independent random variables.
Both the account of properties of metrics and the application of these facts in the second part of the paper are presented under the assumption that the random variables take values in a general space (metric, Banach or Hilbert space).
Bibliography: 11 titles.

Full text: PDF file (3791 kB)
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English version:
Mathematics of the USSR-Sbornik, 1976, 30:3, 373–401

Bibliographic databases:

UDC: 519.2
MSC: Primary 60E05, 60F05; Secondary 54E35
Received: 19.04.1976

Citation: V. M. Zolotarev, “Metric distances in spaces of random variables and their distributions”, Mat. Sb. (N.S.), 101(143):3(11) (1976), 416–454; Math. USSR-Sb., 30:3 (1976), 373–401

Citation in format AMSBIB
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\by V.~M.~Zolotarev
\paper Metric distances in spaces of random variables and their distributions
\jour Mat. Sb. (N.S.)
\yr 1976
\vol 101(143)
\issue 3(11)
\pages 416--454
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\zmath{https://zbmath.org/?q=an:0367.60003}
\transl
\jour Math. USSR-Sb.
\yr 1976
\vol 30
\issue 3
\pages 373--401
\crossref{https://doi.org/10.1070/SM1976v030n03ABEH002280}
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  • http://mi.mathnet.ru/eng/msb/v143/i3/p416

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. M. Zolotarev, “IDEAL METRICS IN THE PROBLEMS OF PROBABILITY THEORY AND MATHEMATICAL STATISTICS”, Aust N Z J Stat, 21:3 (1979), 193  crossref  mathscinet  zmath
    2. Senatov V., “Several Estimates of the Speed of Convergence in the Multidimensional Central Limit-Theorem”, 254, no. 4, 1980, 809–812  mathscinet  isi
    3. Bentkus V. Rachkauskas A., “Estimates of Speed of Convergence in the Central Limit-Theorem in the Prokhorov Metric”, 264, no. 3, 1982, 521–524  mathscinet  zmath  isi
    4. Senatov V., “On a Relation Between Levy - Prohorov Metrics and Ideal Metrics”, 982, 1983, 213–216  mathscinet  zmath  isi
    5. Siganov I., “Several Remarks on Applications of One Approach to Studies of Characterization Problems of Polyas”, 982, 1983, 227–237  mathscinet  isi
    6. V. V. Senatov, “On Estimating the Rate of Convergence in the Central Limit Theorem Over a System of Balls in $R^k $”, Theory Probab Appl, 28:2 (1984), 463  mathnet  crossref  mathscinet  zmath  isi
    7. “Seminars VI and VII on Problems of Continuity and Stability of Stochastic Models”, Theory Probab Appl, 28:4 (1984), 841  mathnet  crossref
    8. L. B. Klebanov, R. V. Yanushkyavichyus, “$\varepsilon $-Independence of the Sample Mean and Tubular Statistics”, Theory Probab Appl, 28:1 (1984), 165  mathnet  crossref  mathscinet  zmath  isi
    9. Kakosian A., Klebanov L., Ianushkiavichius R., “Probabilistic Applications of the Analog of an Kolmogorov,a.N. Inequality”, 279, no. 3, 1984, 535–538  mathscinet  isi
    10. Rachev S., “A Dudley Problem”, 275, no. 1, 1984, 28–31  mathscinet  zmath  isi
    11. V. V. Senatov, “An Estimate of the Lévy–Prokhorov Metric”, Theory Probab Appl, 29:1 (1985), 109  mathnet  crossref  mathscinet  zmath  isi
    12. V. Yu. Bentkus, A. Rachkauskas, “Estimates of the Distance between Sums of Independent Random Elements in Banach Spaces”, Theory Probab Appl, 29:1 (1985), 50  mathnet  crossref  mathscinet  zmath  isi
    13. S. T. Rachev, “The Monge–Kantorovich Mass Transference Problem and Its Stochastic Applications”, Theory Probab Appl, 29:4 (1985), 647  mathnet  crossref  mathscinet  zmath  isi
    14. “Summary of Reports Presented at Sessions of the Probability and Statistics Seminar at the Leningrad Section at the V. A. Steklov Mathematical Institute of the USSR Academy of Sciences, February–December 1983”, Theory Probab Appl, 29:4 (1985), 836  mathnet  crossref  isi
    15. Zubkov A., “On Compound Probabilistic Metrics with Given Minimal Metrics”, 1155, 1985, 443–447  mathscinet  zmath  isi
    16. V. V. Senatov, “On the Dependency of Estimates of the Convergence Rate in the Central Limit Theorem on the Covariance Operator of the Summands”, Theory Probab Appl, 30:2 (1986), 380  mathnet  crossref  mathscinet  zmath  isi
    17. I. B. Klebanov, R. V. Yanushkyavichyus, “Estimation of Stability in S. N. Bernshtein’s Theorem”, Theory Probab Appl, 30:2 (1986), 383  mathnet  crossref  mathscinet  zmath  isi
    18. S. Yu. Vsekhsvyatskii, V. V. Kalashnikov, “Estimates of the Moments of Occurrence of Rare Events in Regenerative Processes”, Theory Probab Appl, 30:3 (1986), 618  mathnet  crossref  mathscinet  isi
    19. Koicheva M., “The Method of Metric Distances in the Problem of Estimation of the Deviation From the Exponential-Distribution”, Lect. Notes Math., 1233 (1986), 32–35  crossref  mathscinet  isi
    20. A. M. Zubkov, “On Compound Probability Metrics with Fixed Minimal Ones”, Theory Probab Appl, 31:1 (1987), 135  mathnet  crossref  mathscinet  zmath  isi
    21. A. Yu. Zaitsev, “Estimates of the Lévy–Prokhorov Distance in the Multivariate Central Limit Theorem for Random Variables with Finite Exponential Moments”, Theory Probab Appl, 31:2 (1987), 203  crossref  mathscinet  isi
    22. J. Beirlant, S.T. Rachev, “The problem of stability in insurance mathematics”, Insurance: Mathematics and Economics, 6:3 (1987), 179  crossref  mathscinet  zmath
    23. Zaitsev A., “On the Gaussian Approximation of Convolutions Under Multidimensional Analogs of Bernstein,S.N. Inequality Conditions”, Probab. Theory Relat. Field, 74:4 (1987), 535–566  crossref  mathscinet  isi
    24. Gurvits L., Zakharin A., “Randomization of Random-Variables and their Distributions”, 23, no. 1, 1987, 136–145  crossref  mathscinet  zmath  isi
    25. Paul Deheuvels, Alan Karr, Dietmar Pfeifer, Robert Serfling, “Poisson approximations in selected metrics by coupling and semigroup methods with applications”, Journal of Statistical Planning and Inference, 20:1 (1988), 1  crossref  mathscinet
    26. S. T. Rachev, “The problem of stability in queueing theory”, Queueing Syst, 4:4 (1989), 287  crossref  mathscinet  zmath
    27. P. Deheuvels, D. Pfeifer, Madan L. Puri, “A new semigroup technique in Poisson approximation”, Semigroup Forum, 38:1 (1989), 189  crossref  mathscinet  zmath  isi
    28. Heribert Kirschfink, “The Generalized Trotter Operator and Weak Convergence Of Dependent Random Variables in Different Probability Metrics”, Results. Math, 15:3-4 (1989), 294  crossref  mathscinet  zmath
    29. S. T. Rachev, J. E. Yukich, “Rates of convergence of α-stable random motions”, J Theoret Probab, 4:2 (1991), 333  crossref  mathscinet  zmath
    30. S. T. Rachev, L. Rüschendorf, “Rate of Convergence for Sums and Maxima and Doubly Ideal Metrics”, Theory Probab Appl, 37:2 (1993), 222  mathnet  crossref  mathscinet  isi
    31. Olga Fiedler, Werner Römisch, “Stability in multistage stochastic programming”, Ann Oper Res, 56:1 (1995), 79  crossref  mathscinet  zmath  isi
    32. Emmanuel Rio, “Distances minimales et distances idéales”, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 326:9 (1998), 1127  crossref  mathscinet  zmath
    33. Paul B Slater, J Phys A Math Gen, 32:47 (1999), 8231  crossref  mathscinet  zmath
    34. E Gordienko, “Estimates of stability of geometric convolutions”, Applied Mathematics Letters, 12:5 (1999), 103  crossref  mathscinet  zmath  elib
    35. Tardella L., “A Note on Estimating the Diameter of a Truncated Moment Class”, Stat. Probab. Lett., 54:2 (2001), 115–124  crossref  mathscinet  zmath  isi
    36. V. I. Bogachev, A. V. Kolesnikov, “The Monge–Kantorovich problem: achievements, connections, and perspectives”, Russian Math. Surveys, 67:5 (2012), 785–890  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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