Abstract
We present a brief survey of the main results obtained by V. S. Korolyuk in probability theory and mathematical statistics.
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REFERENCES
B. V. Gnedenko and V. S. Korolyuk, “Some remarks on the theory of domains of attraction of stable distributions,” Dopov. Akad. Nauk Ukr. RSR, No. 4, 275–278 (1950).
B. V. Gnedenko and V. S. Korolyuk, “On the maximum deviation of two empirical distributions,” Dokl. Akad. Nauk SSSR, 80, No.4, 525–528 (1951).
V. S. Korolyuk, On the Goodness-of-Fit Tests of Kolmogorov and Smirnov [in Russian], Author's Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Kiev (1954).
V. S. Korolyuk, “Asymptotic expansions for the goodness-of-fit tests of Kolmogorov and Smirnov,” Dokl. Akad. Nauk SSSR, 95, No.3, 443–446 (1954).
V. S. Korolyuk, “Asymptotic expansions for the goodness-of-fit tests of Kolmogorov and Smirnov,” Izv. Akad. Nauk SSSR, Ser. Mat., 19, No.2, 103–124 (1955).
V. S. Korolyuk, “Asymptotic expansions for the distribution of maximum deviations in the Bernoulli scheme,” Dokl. Akad. Nauk SSSR, 108, No.2, 183–186 (1956).
V. S. Korolyuk, “Asymptotic expansions of certain functionals of sums of random variables,” in: Proceedings of the Third Mathematical Congress (Moscow, June-July 1956) [in Russian], Vol. 2, Academy of Sciences of the USSR, Moscow (1956), p. 47.
V. S. Korolyuk, “Asymptotic analysis of distributions of maximum deviations in the Bernoulli scheme,” Teor. Ver. Primen., 4, Issue 4, 369–397 (1959).
V. S. Korolyuk, “Asymptotic analysis of the probability of concentration of random walks with lattice distribution of transition probabilities in a one-dimensional scheme,” Dopov. Akad. Nauk Ukr. RSR, No. 7, 702–707 (1959).
V. S. Korolyuk, “On the asymptotics of certain functionals in a two-dimensional scheme of walks,” Dokl. Akad. Nauk SSSR, 135, No.3, 520–523 (1960).
V. S. Korolyuk, “On the asymptotics of distributions in the scheme of walks with lattice distribution of transition probabilities,” Teor. Ver. Primen., 5, Issue 2, 255–256 (1960).
V. S. Korolyuk, “On the asymptotics of certain functionals in a lattice scheme of walks,” Teor. Ver. Primen., 6, Issue 3, 334–341 (1961).
V. S. Korolyuk, “Asymptotic analysis of distributions of maximum deviations in a lattice scheme of walks,” Teor. Ver. Primen., 7, Issue 4, 393–409 (1962).
V. S. Korolyuk, “On the asymptotics of distributions of maximum deviations,” Dokl. Akad. Nauk SSSR, 142, No.3, 522–525 (1962).
V. S. Korolyuk and D. V. Gusak, “On the asymptotics of distributions of maximum deviations in Poisson processes,” Ukr. Mat. Zh., 14, No.2, 138–144 (1962).
V. S. Korolyuk, Asymptotic Analysis in Boundary-Value Problems for Random Walks [in Russian], Author's Abstract of the Doctoral-Degree Thesis (Physics and Mathematics), Kiev (1963).
V. S. Korolyuk, “Asymptotics of certain functionals in boundary-value problems for two-dimensional random walks,” in: Limit Theorems in Probability Theory [in Russian], Academy of Sciences of Uzbek SSR, Tashkent (1963), pp. 43–48.
D. V. Gusak, On the Asymptotics of the Distribution of the First Exit Time of a Process with Independent Increments [in Russian], Author's Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Kiev (1964).
D. V. Gusak and V. S. Korolyuk, “On the time of the first passage through a given level for processes with independent increments,” Teor. Ver. Primen., 13, No.3, 471–478 (1967).
D. V. Gusak and V. S. Korolyuk, “On the joint distribution of a process with stationary increments and its maximum,” Teor. Ver. Primen., 14, No.3, 421–430 (1969).
D. V. Gusak and V. S. Korolyuk, “Distribution of functionals of homogeneous processes with independent increments,” Teor. Ver. Mat. Statist., 1, 55–73 (1970).
N. S. Bratiichuk and V. S. Korolyuk, “Resolvent of a homogeneous process with independent increments and cut-off on a semiaxis,” Teor. Ver. Primen., 30, No.2, 368–372 (1985).
N. S. Bratiichuk and B. Pirliev, “On the value of a jump over a level for a random walk on the superposition of two renewal processes,” Ukr. Mat. Zh., 35, No.6, 689–695 (1985).
V. S. Korolyuk, N. S. Bratiichuk, and B. Pirdjanov, Boundary-Value Problems for Random Walks [in Russian], Ylym, Ashkhabad (1987).
V. S. Korolyuk, Boundary-Value Problems for Complex Poisson Processes [in Russian], Naukova Dumka, Kiev (1975).
V. S. Korolyuk and Yu. V. Borovskikh, Analytic Problems in the Asymptotics of Probability Distributions [in Russian], Naukova Dumka, Kiev (1981).
V. S. Korolyuk, V. N. Suprun, and V. M. Shurenkov, “Potential method in boundary-value problems for processes with independent increments,” Teor. Ver. Primen., 21, No.2, 253–259 (1974).
V. N. Suprun and V. M. Shurenkov, “On the resolvent of a process with independent increments that is terminated at the moment of hitting the negative semiaxis,” in: Investigations in the Theory of Random Processes [in Russian], Institute of Mathematics, Ukrainian Academy of Science (1975), pp. 170–174.
V. N. Suprun, “Ruin problem and the resolvent of a cut-off process with independent increments,” Ukr. Mat. Zh., 28, No.1, 53–61 (1976).
N. S. Bratiichuk and D. V. Gusak, Boundary-Value Problems for Processes with Independent Increments [in Russian], Naukova Dumka, Kiev (1990).
V. S. Korolyuk and V. M. Shurenkov, “Potential method in boundary-value problems for random walks on a Markov chain,” Ukr. Mat. Zh., 29, No.4, 464–471 (1977).
N. S. Bratiichuk, Investigation of Boundary-Value Problems for Random Processes [in Russian], Author's Abstract of the Doctoral-Degree Thesis (Physics and Mathematics), Kiev (1990).
V. S. Korolyuk and A. F. Turbin, “Potential method in problems of queuing theory,” in: Proceedings of the Third All-Union Conference on Queuing Theory (Pushchino-na-Oke) [in Russian], Vol. 1, Moscow University, Moscow (1976), pp. 119–127.
T. K. Butko and V. S. Korolyuk, “Potential method in the investigation of the queuing system GI/M1/∞,” in: Analytic Methods in Problems of Probability Theory [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences (1984), pp. 16–27.
V. S. Korolyuk and B. Pirliev, “A random walk on a semiaxis on the superposition of two renewal processes,” Ukr. Mat. Zh., 36, No.4, 433–436 (1984).
B. Pirdjanov, “A random walk generated by two renewal processes,” Ukr. Mat. Zh., 30, No.5, 680–685 (1978).
M. S. Bratiychuk, “Limit theorems for some characteristics of system GI/GI/1,” in: Exploring Stochastic Laws, VSP, Netherlands (1995), pp. 77–90.
M. S. Bratiychuk and B. Pirdjanov, “On a new approach in studying the busy period of system GI/G/1,” in: Proceedings of the Sixth USSR-Japan Symposium on Probability Theory and Mathematical Statistics (Kiev, August 1991), World Scientific, Singapore (1992).
V. S. Korolyuk, “Semi-Markov random walk,” in: Proceedings of the Second International Symposium “Markov Models: Theory and Applications,” (1998), pp. 1–14.
V. S. Korolyuk, “Sojourn time of a semi-Markov process in a fixed set of states,” Ukr. Mat. Zh., 17, No.3, 123–128 (1965).
I. I. Ezhov and V. S. Korolyuk, “Semi-Markov processes and their applications,” Kibernetika, No. 5, 58–65 (1967).
A. A. Tomusyak, “Determination of the ergodic distribution of Markov and semi-Markov processes,” Kibernetika, No. 1, 67–71 (1969).
V. S. Korolyuk and A. A. Tomusyak, “Description of the operation of reserve systems by using semi-Markov processes,” Kibernetika, No. 5, 55–59 (1965).
S. N. Simonova, “On a multilinear system with losses and incoming semi-Markov flow of demands,” Kibernetika, No. 6, 48–53 (1967).
V. Ya. Valakh and V. S. Korolyuk, “Stochastic automata with random response time and their operation in random media,” in: Proceedings of the Fourth All-Union Meeting on Automatic Control (Technical Cybernetics) [in Russian], Tbilisi (1968), pp. 102–103.
V. S. Korolyuk, L. I. Polishchuk, and A. A. Tomusyak, “On one limit theorem for semi-Markov processes,” Kibernetika, No. 4, 144–145 (1969).
V. S. Korolyuk, “On the asymptotic behavior of the sojourn time of a process in a subset of states,” Ukr. Mat. Zh., 21, No.6, 842–845 (1969).
V. S. Korolyuk and A. F. Turbin, “On the asymptotic behavior of the sojourn time of a semi-Markov process in a reducible subset of states,” Teor. Ver. Mat. Statist., 2, 133–143 (1970).
V. S. Korolyuk and A. F. Turbin, “On one method for the proof of limit theorems for certain functionals of semi-Markov processes,” Ukr. Mat. Zh., 24, No.2, 234–239 (1972).
V. S. Korolyuk, I. P. Penev, and A. F. Turbin, “Asymptotic expansion for the distribution of the absorption time of a Markov chain,” Kibernetika, No. 4, 133–135 (1973).
V. S. Korolyuk and A. F. Turbin, “Lumping of complex systems,” in: Mathematization of Knowledge and Scientific and Technological Progress [in Russian], Naukova Dumka, Kiev (1975), pp. 46–65.
V. S. Korolyuk and A. F. Turbin, Semi-Markov Processes and Their Applications [in Russian], Naukova Dumka, Kiev (1976).
V. S. Korolyuk and A. F. Turbin, Phase Lumping of Complex Systems. A Textbook [in Russian], Vyshcha Shkola, Kiev (1978).
V. S. Korolyuk and A. F. Turbin, Mathematical Foundations of the State Lumping of Large Systems [in Russian], Naukova Dumka, Kiev (1978).
V. S. Korolyuk, “Evolution of systems in a semi-Markov random medium,” Kibernetika, No. 5, 106–109 (1987).
V. S. Korolyuk and A. V. Svishchuk, “Phase lumping of inhomogeneous semi-Markov random evolutions,” Ukr. Mat. Zh., 41, No.2, 163–170 (1989).
V. S. Korolyuk and A. V. Svishchuk, Limit Theorems for Semi-Markov Evolutions in the Scheme of Asymptotic Phase Lumping [in Russian], Preprint No. 16, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1984).
V. S. Korolyuk, A. V. Svishchuk, and V. V. Korolyuk, “A central limit theorem in the scheme of phase lumping for semi-Markov random evolutions,” Ukr. Mat. Zh., 39, No.3, 314–319 (1987).
V. S. Korolyuk and A. V. Swishchuk, Semi-Markov Random Evolutions, Kluwer, Dordrecht (1995).
V. S. Korolyuk and A. V. Svishchuk, Semi-Markov Random Evolutions [in Russian], Naukova Dumka, Kiev (1992).
V. S. Korolyuk and A. V. Swishchuk, Evolutions of Systems in Random Media, CRC Press, Florida (1995).
V. S. Korolyuk and A. V. Svishchuk, Evolution Stochastic Systems in Random Media [in Russian], Naukova Dumka, Kiev (1995).
V. S. Korolyuk, “Phase averaging of inhomogeneous semi-Markov random evolutions,” Ukr. Mat. Zh., 41, No.2, 163–170 (1989).
V. S. Korolyuk and A. V. Svishchuk, “A central limit theorem for inhomogeneous semi-Markov random evolutions,” Ukr. Mat. Zh., 41, No.8, 1064–1070 (1989).
A. V. Svishchuk, “Representation of multiplicative operator functionals of semi-Markov processes,” Dokl. Akad. Nauk Ukr. SSR, No. 10, 27–28 (1988).
A. V. Swishchuk, Random Evolutions and Their Applications, Kluwer, Dordrecht (1997).
V. S. Korolyuk, A. F. Turbin, and A. V. Svishchuk, “Limit theorems for Markov random evolutions,” in: Abstracts of the Sixth USSR-Japan Symposium on Probability Theory and Mathematical Statistics (Tbilisi, August 1982), Vol. 2, Tbilisi (1982), pp. 39–40.
V. S. Korolyuk and A. V. Svishchuk, “Phase lumping of semi-Markov random evolutions,” in: Abstracts of the International Conference “Stochastic Optimization” (Kiev, September 1984) [in Russian], Kiev (1984), pp. 127–129.
V. S. Korolyuk and A. V. Svishchuk, “Phase merging of semi-Markov random evolutions and storage theory,” in: Abstracts of the Fourth Vilnius Conference on Probability Theory and Mathematical Statistics, Vol. 2, Mosklas, Vilnius (1985), pp. 62–64.
V. S. Korolyuk and A. V. Svishchuk, “Semi-Markov random evolutions. Phase aggregations and applications,” in: Proceedings of the Fifth International Summer School on Probability Theory and Mathematical Statistics (Varna, 1985), Sofia (1988), pp. 116–131.
V. S. Korolyuk, Stochastic Models of Systems [in Russian], Naukova Dumka, Kiev (1989).
V. S. Korolyuk, Stochastic Models of Systems [in Ukrainian], Lybid', Kiev (1993).
A. V. Svishchuk, “Weak convergence of semi-Markov random evolutions in the scheme of averaging (martingale approach),” Ukr Mat. Zh., 41, No.12, 1680–1686 (1989).
A. V. Svishchuk, “A martingale approach to semi-Markov random evolutions,” in: Proceedings of the Scientific Conference of Young Scientists (Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, November 1986) [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1987).
A. V. Svishchuk, “Estimates for the rate of convergence in limit theorems for semi-Markov evolutions,” in: Stochastic Systems and Their Applications [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1990), pp. 86–92.
V. S. Korolyuk and A. V. Svishchuk, “A central limit theorem for semi-Markov random evolutions,” Ukr. Mat. Zh., 38, No.3, 330–334 (1986).
V. S. Korolyuk, “CLT for semi-Markov random evolutions,” Comp. Math. Appl., 19, No.1, 83–88 (1990).
V. S. Korolyuk and A. V. Svishchuk, “Weak convergence of semi-Markov random evolutions in the scheme of averaging,” Teor. Ver. Primen., 42, 53–64 (1990).
V. S. Korolyuk and A. V. Svishchuk, “Limit representation of semi-Markov random evolutions in the scheme of series,” Ukr. Mat. Zh., 41, No.11, 1476–1482 (1989).
V. S. Korolyuk and A. V. Svishchuk, “Weak convergence of semi-Markov random evolutions (martingale approach),” Probab. Theory Math. Statist., 1, 1–9 (1990).
A. V. Svishchuk, “Solution of a martingale problem for semi-Markov random evolutions,” in: Asymptotic and Applied Problems in the Theory of Random Evolutions [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1990), pp. 98–107.
V. S. Korolyuk and A. V. Svishchuk, Applied Problems in the Theory of Random Evolutions [in Russian], Znanie, Kiev (1990).
A. V. Svishchuk, Approximation and Ruin Problem for Semi-Markov Risk Processes, Preprint No. 20, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1996).
V. S. Korolyuk, “Stability of autonomous dynamical systems with rapid Markov switchings,” Ukr. Mat. Zh., 43, No.2, 1176–1181 (1991).
V. S. Korolyuk, Averaging and Stability of Dynamical Systems with Rapid Markov Switchings, Preprint, University of Umea, Umea (1991).
V. S. Korolyuk, “Stability of stochastic systems in the diffusion-approximation scheme,” Ukr. Mat. Zh., 50, No.1, 36–47 (1998).
A. V. Svishchuk, “Stability of semi-Markov evolution systems in schemes of averaging and diffusion approximation,” in: Asymptotic Analysis of Random Evolutions [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1994), pp. 255–269.
A. V. Svishchuk and A. G. Burdeinyi, “Stability of semi-Markov evolution systems and its application in financial mathematics,” Ukr. Mat. Zh., 48, No.10, 1386–1401 (1996).
A. V. Svishchuk and S. Ya. Goncharova, “Stability of semi-Markov risk processes in schemes of averaging and diffusion approximation,” Ukr. Mat. Zh., 51, No.7, 972–979 (1999).
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Gusak, D.V., Bratiichuk, N.S. & Svishchuk, A.V. On the Creative Contribution of V. S. Korolyuk to the Development of Probability Theory. Ukrainian Mathematical Journal 52, 1161–1178 (2000). https://doi.org/10.1023/A:1010388617778
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DOI: https://doi.org/10.1023/A:1010388617778