UDC 519.21

P.S. Knopov
V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine,
Kyiv, Ukraine,  knopov1@yahoo.com

E.J. Kasitskaya
V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine,
Kyiv, Ukraine,  e.kasitskaya@gmail.com

O.S. Samosyonok
V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine,
Kyiv, Ukraine,  samosyonok@gmail.com

ASYMPTOTIC PROPERTIES OF THE EMPIRICAL AVERAGE METHOD

REFERENCES

1. Ermoliev Y., Wets, R.J.-B. (eds). Numerical techniques for stochastic optimization. Hedelberg, Germany: Springer Verlag, 1988. 571 p.
2. 2. Prekopa A. Stochastic programming. Springer, 1995. 600 p.
3. 3. Ruszczynski A., Shapiro D., Dentcheva A. Lectures on stochastic programming: modeling and theory. Philadelphia: SIAM, 2009. 439 p. https://doi.org/10.1137/1.9780898718751.
4. 4. Kall P., Wallace S.W. Stochastic programming. Chichester, UK: John Wiley & Sons, 1994. 307 p.
5. 5. Kankova V. On the convergence rate of empirical estimates in chance constrained stochastic programming. Kybernetika. 1990. Vol. 26, N 6. P. 449–461.
6. Ermoliev Yu.M., Knopov P.S. Method of empirical means in stochastic programming problems. Cybernetics and Systems Analysis. 2006. Vol. 42, N 6. P. 773–785. https://doi.org/10.1007/s10559-006-0118-z.
7. Knopov P.S. Asymptotic properties of some classes of m-estimates. Cybernetics and Systems Analysis. 1997. Vol. 33, N 4. P. 468–481. https://doi.org/10.1007/BF02733103.
8. Knopov P.S., Kasitskaya E.J. Properties of empirical estimates in stochastic optimization and identification problems. Annals of Operations Research, 1995. Vol. 56, N 1. P. 225–239. https://doi.org/10.1007/BF02031709.
9. Knopov P.S., Kasitskaya E.J. Empirical estimates in stochastic optimization and identification. Dordrecht: Kluwer Academic Publishers, 2002. 250 p.
10. Knopov P.S., Kasitskaya E.I. Large deviations of empirical estimates in stochastic programming problems. Cybernetics and Systems Analysis. 2004. Vol. 40, N 4. P. 52–60.
11. Knopov P.S., Kasitskaya E.I. On large deviations of empirical estimates in a stochastic programming problem with time-dependent observations. Cybernetics and Systems Analysis. 2010. Vol. 46, N 5. P. 724–728. https://doi.org/10.1007/s10559-010-9253-7.
12. Deuscel J.-D., Stroock D.W. Large deviations. Boston: Academ. Press, 1989. 310 p.
13. Dembo A., Zeitouni O. Large deviations techniques and applications. New York: Springer-Verlag, 1998. 397 p.
14. Hollander F. Large deviations. Providence, Rhode Island: American Math. Society, 2000. 142 p.
15. Sanov I.N. On probability of large deviations of random variables. Math. Collection. 1957. Vol. 42. P. 11–44.
16. Bryc W. On large deviations for uniformly strong mixing sequences. Stoc. Proc. Appl., 1992. Vol. 41. P. 191–202.
17. Dorogovtsev A.Ya. Theory of Estimation of Parameters of Random Processes. Kyiv: Vyshcha shk., Kyiv, 1982. 192 p.
18. Knopov P.S., Kasitskaya E.J. Large deviations of empirical estimates in the stochastic programming problem for the homogeneous random field with a discrete parameter. Cybernetics and Systems Analysis. 2021. Vol. 57, N 5. P. 704–713. https://doi.org/10.1007/s10559-021-00396-0.
19. Knopov P., Ermolieva T., Kasitskaya E. On the consistency and large deviations of the method of empirical means in stochastic programming problems. In: Theory, Algorithms, and Experiments in Applied Optimization. Springer, 2026. Vol. 226. P. 151–170. https://doi.org/10.1007/978-3-031-91357-0_8.
20. Yurinskyi V.V. On the strengthened law of large numbers for random fields. Mathematical Notes. 1974. Vol. 16, No. 1. P. 141–149.
21. Knopov P.S., Korkhin A.S. Regression analysis under a priory parameter restrictions. Springer, 2012. 234 p.
22. Kaniovski Y.M., King A.J., Wets R.J.-B.: Probabilistic bounds (via large deviations) for the solutions of stochastic programming problems. Ann. Oper. Res. 1995. Vol. 56. P. 189–208. https://doi.org/10.1007/BF0203170723.
23. Dunford N., Schwartz J. Linear operators. P. I: General theory. New York: Interscience, 1957. 896 p.
24. Knopov P.S., Kasitska E.J. Large deviations of empirical estimates in the stochastic programming problem for the homogeneous random field with a discrete parameter. Cybernetics and Systems Analysis. 2021. Vol. 57, N 5. P. 704–713. https://doi.org/10.07/s10559-021-00396-0.
25. Knopov P.S., Pepelyaev V.A. Nonparametric estimate of almost periodic signals. Cybernetics and Systems Analysis. 2007. Vol. 43, N 3, P. 362–367. https://doi.org/10.1007/s10559-007-0057-3.
26. Mikhalevich V.S., Knopov P.S., Golodnikov A.N. Mathematical models and methods of risks assessment in ecologically hazardous industries. Cybernetics and Systems Analysis. 1994. Vol. 30, N 2. P. 259–273. https://doi.org/10.1007/BF02366429.
27. Rychlik I., Rydn J. Probability and risk analysis. An introduction for Engineers. Heidelberg: Springer Berlin, 2006. 286 p. https://doi.org/10.1007/978-3-540-39521-8.
28. Asmussen S., Albrecher H. Ruin probabilities. World Scientific, 2010. 621 p.
29. Marti K., Ermoliev Yu., Makovski M. (eds) Coping with uncertainty: Robust solutions. Heidelberg: Springer Berlin, 2010, 277 p. https://doi.org/10.1007/978-3-642-03735-1.
30. Kirilyuk V.S. Polyhedral coherent risk measure and distributionally robust portfolio optimization. Cybernetics and Systems Analysis. 2023. Vol. 59, N 1. P. 90–100. https://doi.org/10.1007/s10559-023-00545-7.
31. The Global Risks Report 2021 (16th Edition) – World Economic Forum, The Global Risks Report, 2021. https://www3.weforum.org/docs/WEFf.
32. Vovk L.B., Knopov A.P., Pepeljaeva T.V. Some approaches to financial risk assessment. Cybernetics and Systems Analysis. 2010. Vol. 46, N 3. P. 500–505. https://doi.org/10.1007/s10559-010-9225-y.
33. Golodnikov A.N., Knopov P.S., Pepelyaev V.A. Estimation of reliability parameters under incomplete primary information. Theory and Decision. 2004. Vol. 57, N 4. P. 331–344. https://doi.org/10.1007/s11238-005-3217-9.
34. Borodina O.M., Kyryziuk S.V., Fraier O.V., Knopov P.S., Horbachuk V.M. Mathematical modeling of agricultural crop diversification in Ukraine: Scientific approaches and empirical results. Cybernetics and Systems Analysis. 2020. Vol. 56, N 2. P. 213–222. https://doi.org/10.1007/s10559-020-00237-6.
35. Ermolieva T., Havlik P., Frank S., Kail T., Balkovic J., Skalsky R., Ermoliev Yu., Knopov P.S., Borodina O.M., Gorbachuk V.M. A risk-informed decision-making framework for climate change adaptation through robust land use and irrigation planning. Sustainability. 2022. Vol. 14, N 3. 1430. https://doi.org/10.3390/su14031430.
36. Pepelyaev V.A., Golodnikov A.N., Golodnikova N.A. Reviewing climate changes modeling methods. Cybernetics and Systems Analysis. 2023. Vol. 59, N 3. P. 398–406. https://doi.org/10.1007/s10559-023-00574-2.
37. Pepelyaev V.A., Golodnikov A.N. Golodnikova N.A. Modeling the impact of climate change on the crop yield. Cybernetics and Systems Analysis. 2023. Vol. 59, N 6. P. 949–955. https://doi.org/10.1007/s10559-023-00631-w.
38. Pepelyaev V.A., Golodnikov A.N., Golodnikova N.A. Method of optimizing the structure of sowing areas for the adaptation of crop production to climate changes. Cybernetics and Systems Analysis. 2024. Vol. 60, N 3. P. 415–421. https://doi.org/10.1007/s10559-024-00682-7.
39. Pepelyaev V., Golodnikov A., Panteleymonov A. Mathematical tools for modeling the impact of climate change on crop yield. In: The Digital Edge: Transforming Business Systems for Strategic Success. Cham: Springer. 2025. Vol. 604. P. 485–495. ttps://doi.org/10.1007/978-3- 031-95280-7_45.
40. Atoyev K., Ermolieva T., Knopov P. Mathematical modeling of interconnections between ecological, food, and economic dimensions of security. In: Nexus of Sustainability: Understanding of FEWSE Systems І. Studies in Systems, Decision and Control. Cham: Springer, 2024. Vol. 559. P. 33–66. https://doi.org/10.1007/978-3-031-66764-0.
41. Atoyev K.L., Knopov P.S. Mathematical model for assessing the instability in complex systems under pandemics and systemic risks. In Nexus of Sustainability: Understanding of FEWSE Systems IІ. Studies in Systems, Decision and Control. Cham: Springer, 2026. Vol. 627. P. 569-590. https://doi.org/10.1007/978-3-032-03616-2_24.
42. Atoyev K., Ermolieva T., Knopov P. Mathematical modeling of interconnections between ecological, food, and economic dimensions of security. In: Nexus of Sustainability: Understanding of FEWSE Systems І. Studies in Systems, Decision and Control. Cham: Springer, 2024. Vol. 559. P. 33–66. https://doi.org/10.1007/978-3-031-66764-0_2.
43. Atoyev K.L., Knopov P.S. Mathematical model for assessing the instability in complex systems under pandemics and systemic risks. In: Nexus of Sustainability: Understanding of FEWSE Systems IІ. Studies in Systems, Decision and Control. Cham: Springer, 2026. Vol. 627. P. 569–590. https://doi.org/10.1007/978-3-032-03616-2_24.
44. Atoyev K., Ermolieva T., Knopov P. Mathematical modelling of interconnections between ecological, food, and economic dimensions of security. In: Nexus of Sustainability: Understanding of FEWSE Systems І. Studies in Systems, Decision and Control. Cham: Springer, 2024. Vol. 559. P. 33-66. https://doi.org/10.1007/978-3-031-66764-0_2.
45. Atoyev K.L., Knopov P.S. Mathematical modeling of instability in environmental and economic systems. Cybernetics and Systems Analysis. 2025. Vol. 61. P. 943–952. https://doi.org/10.1007/s10559-025-00827-2.
46. Zagorodny A., Bogdanov V., Ermolieva T., Komendantova N. Modeling for managing food-energy-water-social-environmental NEXUS security: Novel systems’ analysis approaches. In: Nexus of Sustainability. Studies in Systems, Decision and Control. Cham: Springer, 2024. Vol. 559. P. 1–33. https://doi.org/10.1007/978-3-031-66764-0_1.
47. Zagorodny A., Bogdanov V., Ermolieva T., Komendantova N., Havlik, P. Integrated solutions to food–energy–water–environmental NEXUS Security modeling and management: robust downscaling and models’ linkage procedures. In: Nexus of Sustainability. Studies in Systems, Decision and Control. Cham: Springer, 2026. Vol. 627. P. 1–37. https://doi.org/10.1007/978-3-032-03616-2_1.
48. Yevtukhova T., Novoseltsev O., Zaporozhets A. Program-targeted approach for improving energy efficiency and sustainability through system-integrated services. In: Nexus of Sustainability. Studies in Systems, Decision and Control. Cham: Springer, 2026. Vol. 627. P. 197–217. https://doi.org/10.1007/978-3032-03616-2_8.
49. Zaporozhets A., Khaustova V., Kyzym M., Trushkina N. Sustainable financing mechanism for energy system development toward a decarbonized economy: conceptual model and management framework. Energies (electronic journal). 2026. Vol. 19, N 2. P. 1–26. https://doi.org/10.3390/en19020422.
50. Ermoliev Y. et al. Robust food–energy–water–environmental security management: Stochastic quasigradient procedure for linkage of distributed optimization models under asymmetric information and uncertainty. Cybernetics and Systems Analysis. 2022. Vol. 58, N 1. P. 45–51. https://doi.org/10.1007/s10559-022-00434-5.
51. Galizia D. Saddle cycles: Solving rational expectations models featuring limit cycles (or chaos) using perturbation method. Quantitative Economics. 2021. Vol. 12, N 3, P. 869–901. https://doi.org/10.3982/QE1491.
52. WHO Coronavirrus (COVID-19) Dashboard. https://covid19.who.int/.
53. Katriel G. Stochastic discrete-time age-of-infection epidemic models. International Journal of Biomathematics. 2013. Vol. 6, N 1. P. 995–1005. https://doi.org/10.1142/S1793524512500660.
54. Bogdanov O. Variants of the stochastic SIR models and vactination strategies. Cybernetics and Systems Analysis. 2023. Vol. 59, N 2. P. 324–330. https://doi.org/10.1007/s10559-023-00566-2.
55. Kermack W., McKendrick A. Contributions to the mathematical theory of epidemics. I. Bulletin of Mathematical Biology. 1991. Vol. 53, N 1–2. P. 57–87. https://doi.org/10.1007/BF02464424.
56. Ishikawa M. Optimal vaccination strategy under saturated treatment using the stochastic SIR model. Transactions of the Institute of Systems, Control and Information Engineers. 2013. Vol. 26, N 11, P. 382–388. https://doi.org/0567scie.26.382.
57. Chowell G., Hyman J.M., Bettencourt L.M.A. Castillo-Chavez models in population biology and epidemiology. Dordrecht: Springer, 2009. 363 p.
58. Statistical Yearbook of Ukraine, 2022. Kyiv: State Statistics Service of Ukraine, 2023. https://stat.gov.ua/en/publications/statistical-yearbook-ukraine-2022.
59. Agriculture of Ukraine, 2022. Kyiv: State Statistics Service of Ukraine, 2023. https://stat.gov.ua/en/publications/agriculture-ukraine-2022.
60. Ermoliev Y. et al. Linking distributed optimization models for food, water, and energy security nexus management. Sustainability, 2022. Vol. 14, N 3. 14 p. https://doi.org/103390.su14031255.
61. Gorbachuk V.M. Dynamics of capital, discount rate and output according to the levels of taxation and budget overbalance. Journal of Automation and Information Sciences. 1999. N 11. P. 118–121. https://doi.org/10.1615/JAutomatInfScien.v31.i11.180.