Russian Mathematical Surveys, 2023, Volume 78, Issue 3, Pages 573–583 DOI: https://doi.org/10.4213/rm10101e (Mi rm10101)

DOI: https://doi.org/10.4213/rm10101e

Russian version:
Uspekhi Matematicheskikh Nauk, 2023, Volume 78, Issue 3(471), Pages 183–195
DOI: https://doi.org/10.4213/rm10101

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On 15 July 2022 Academician of the Russian Academu of Sciences, prominent mathematician, and one of the leading world experts in probablity theory and mathematical statistics Ildar Abdullovich Ibragimov observed his 90th birthday.

He was born in Leningrad in 1932. At that time his father Abdulla Shakirovich worked in the Department of Mechanics of Materials of the Forest Technical Academy (which he had himself graduated from, having enrolled there just after the end of the Civil War). His mother Bella Gil’manovna, a graduate of Kazan’ University, worked as a doctor her whole life.

In 1937 the father was imprisoned after a false anonymous denunciation. He did not confess guilt and was freed half a year later for lack of corpus delicti. After I. V. Stalin’s death he was fully rehabilitated. However, Bella Gilmanovna and their son were exiled for three years to Muromtsevo village in Omsk Oblast, where Ildar went to the first grade of a primary school. Then they moved to Staraya Russa, where the father was at that time and where WW2 caught them. The father was mobilized to the army, while the mother and son did not manage to flee and spent a long period of time under Nazi occupation. After the war the father found an engineering position at a factory in Verkhnyaya Tavda in Sverdlovsk Oblast, and the family moved there.

After graduating from the secondary school Ibragimov wanted to continue education at the Faculty of Mathematics and Mechanics of Leningrad University, but he was not admitted there because of his father’s arrest record and his own life under occupation. Actually, Ibragimov believed at that time that he was just not good enough. However, neither was he admitted to Leningrad Electotechnical Institute, and there a young guy from the selection committee was frank enough to explain to him the actual reason. Still, Ildar managed to enroll in that year – to the Forest Engineering Academy.

In the academy Ibragimov’s extraordinary capabilities drew the attention of professor N. V. Lipin, and thanks to his help and the assistance of A. D. Alexandrov, the rector of Leningrad University, he was transferred to the Faculty of Mathematics and Mechanics of the university. There, as a second-grade student, Ibragimov won a student competition. After that Yu. V. Linnik, the chairman of the jury of this competition, invited the talented student to his home, where, after a long conversation, he proposed Ibragimov to work in conjunction. This is how Ibragimov became involved in probability theory with Linnik as his scientific advisor.

Ibragimov established his first serious research result during his fourth year at the university, by finding a simple necessary and sufficient condition for a unimodal distribution function which ensures that its convolutions with all other unimodal distributions are unimodal too [14].

In the early 1960s Ibragimov became involved in the topic of limit theorems for sums of weakly dependent random variables. In particular, he considered stationary random sequences satisfying the strong mixing condition proposed by M. Rosenblatt [91]. He also introduced a new notion of uniformly strong mixing, which is now called mixing in the sense of Ibragimov. He showed that, under certain conditions of weak dependence, normalized partial sums of random variables can converge only to stable laws, and he also proved a number of new limit theorems (see [16]). At about the same time Ibraginov [18] and P. Billingsley [3] proved independently a central limit theorem for stationary random sequences of variables with finite variance whose partial sums are a martingale. Subsequently, in [27] he established several versions of a central limit theorem for stationary sequences whose maximal correlation coefficient (introduced by A. N. Kolmogorov and Yu. A. Rozanov [83]) tends to zero.

The part of probability theory linked with limit theorems for sums of weakly dependent random variables was covered in the famous monograph [66] authored by Ibragimov and Linnik. Ibragimov also conjectured there (see [66], Chap. XX) that the central limit theorem holds for stationary sequences satisfying the uniformly strong mixing condition, provided that the partial sums have finite variances which tend to infinity with the number of terms. This beautiful conjecture is still open. For their investigations of limit theorems Ibragimov, Linnik, Yu. V. Prokhorov, and Rozanov were awarded the Lenin Prize in science (the highest prize in the USSR) in 1970.

The property of mixing is closely connected with various regularity conditions for stationary processes. Ibragimov made an enormous contribution to finding all types of regularity criteria for stationary Gaussian processes in terms of their spectral measures. In conjunction with Solev [76], [77], he found necessary and sufficient conditions for the regularity, in the sense of Kolmogorov, of a stationary Gaussian sequence in terms of its spectral density. He also obtained necessary [19] and sufficient [25] conditions in these terms for its strong mixing. In [24] he found the general properties of the spectral densities of completely regular multidimensional stationary processes with discrete time. Questions concerning the regularity of stationary processes were thoroughly treated in the remarkable book [74] by Ibragimov and Rozanov.

Using the methods developed in [19], in investigations of the strong mixing conditions Ibragimov managed to find necessary and sufficient conditions [23] in ‘Szegő’s strong limit theorem’, which put an end to a series of papers by a number of prominent mathematicians, such as M. Kac [82], G. Baxter [1], and others, who were gradually relaxing rather restrictive assumptions made by Szegő himself in [95].

Ibragimov made a great contribution to the analysis of estimates for the rate of convergence in the central limit theorem. For instance, it was shown in [71] that, apart from a Lindeberg-type expression, such estimates must include some moment characteristics, and in [21] he presented necessary and sufficient conditions for this rate of convergence to be power-like. Furthermore, in [22] he found an existence criterion for Chebyshev–Cramér (Chebyshev–Edgeworth) expansions, and in the quite recent join paper with E. L. Presman and Sh. K. Formanov [73] they presented conditions for the validity of the central limit theorem, which are equivalent to Lindeberg’s and Rotar’s ones and in which the moment functions can be replaced by more general functions.

From the early 1960s on, Ibragimov also became engaged in questions of statistical estimates and prediction. His first results in this area related to estimates for the spectral functions of stationary sequences. For instance, in [15] and [17] he showed that for Gaussian stationary sequences the empiric spectral distribution function is unbiased, consistent, and asymptotically normal (also see [20]). In [15] he presented necessary and sufficient conditions in order that the linear prognosis error decay exponentially or power-like. Jointly with V. N. Solev [75], he also considered a similar problem for sequences with spectral density of a certain special form. In the recent paper [46] with Z. A. Kabluchko and M. A. Lifshits they considered the linear prediction problem in the case of some or other constraints imposed on the predictor.

An important stage in Ibragimov’s research was his extremely fruitful collaboration with R. Z. Khas’minskii. In conjunction they developed a very powerful — now classical – machinery of asymptotic estimation. The works written during the first period of their collaboration, in the 1970s, and underlying the world-famous monograph [57], were mainly devoted to parametric problems of estimation. For instance, in [51] (also see [57], Chap. 1, § 5.2) they examined the consistency of Bayes estimators and maximum likelihood estimators so that the convergence rate of risk functions was expressed in terms of the Hellinger distance. An asymptotic lower bound for the risk functions was established in [13]. Moreover, in [57], Chap. 1, § 10, they proved limit theorems for such estimators. For regular families of experiments, in [47], [48], and [50] they showed that such estimates are asymptotically efficient. Ibragimov and Khas’minskii also examined cases without regularity, when singularities of densities improve the accuracy of estimation significantly. For instance, in [49] they considered densities with discontinuities of the first kind, and in [52] and [55] unbounded densities.

The asymptotic normality of maximum likelihood estimation from observations of a smooth signal in Gaussian white nose, which had been established by V. A. Kotel’nikov [84] on the physical level, was rigorously proved in [53] for a one-dimensional parameter, and in [57], Chap. 3, § 5, for a multidimensional one. The case of a discontinuous signal was treated in [54].

Since the late 1970s, Ibragimov and Khas’minskii were actively engaged in questions of non-parametric estimation, when the parameter to be estimated does not belong to a finite-dimensional set any longer. The central idea was to use Kolmogorov’s approximation theory to find an approximation to the parameter in question in a finite-dimensional (or even finite) set, and then to use the classical machinery of parametric estimation. In the framework of this approach (apparently initiated by N. N. Chentsov [7], [8]) Ibragimov and Khas’minskii [58]–[60] considered the problem of the minimax estimation of the density of an unknown distribution in $\mathbb R^n$. They found upper and lower estimates for the risk functions; they also showed that for a wide class of parameter sets the optimal estimators, from the standpoint of the convergence of the risk function, are kernel estimations with the de la Vallée Poussin kernel.

Ibragimov and Khas’minskii also made an invaluable contribution to the study of the problem, mentioned above, of estimating a signal in white noise and functionals of this signal, in the case when the signal belongs to an infinite-dimensional set. In this case the possibility of efficient asymptotic estimation depends on the relation between the smoothness of the functional and the possibility of a nice approximation of the parameter set by finite-dimensional linear spaces. The corresponding theory was developed in [56] and [70]. Subsequently, A. Nemirovski [89] showed that the estimators in [70] were asymptotically efficient. In should also be mentioned that in his recent paper [44] Ibragimov indicated classes of parameter sets that are poorly approximated by finite-dimensional linear spaces, but possess nice estimates of sufficiently smooth functionals.

At the end of the 1990s, during the late stage of their collaboration, Ibragimov and Khas’minskii [9], [61]–[63] (also see [36]) considered estimation problems for the functional coefficients of stochastic partial differential equations. They were the first to propose a setup in which the level of noise decays to zero.

At the same time Ibragimov became interested in statistical problems where the parameter estimated is an entire analytic function. For instance, in this setting he looked [34] at problems of estimation for the density of a probability distribution, for the regression function, and for a function observed in white noise, while in [38] he considered the problem of estimation for the spectral density of a Gaussian stationary process.

Also, in [35] and [37] Ibragimov stated and considered the problem of estimation of an analytic distribution density on the basis of censored data, when only elements occurring in a prescribed finite interval are observed. In this way he obtained a statistical analogue of the uniqueness theorem for analytic functions. He answered the question of how far away from a finite interval we can obtain some information about the behaviour of an analytic function in the case when we only partly measure it inside this interval. Subsequently, he also considered the multidimensional case [39] and also stated and solved a similar problem for the estimation of the analytic spectral density of a Gaussian stationary process [40] and the analytic intensity density of a one-dimensional [45] or a multidimensional [43] Poisson point process.

We can separately note another area in probability theory that attracted Ibragimov’s interest in the early 1970s: the behaviour of the zeros of polynomials with independent identically distributed random coefficients. Jointly with N. P. Maslova, they succeeded in putting a full stop, in a certain sense, to a series of outstanding papers by A. Bloch and G. Pólya [4], J. E. Littlewood and A. C. Offord [85]–[87], M. Kac [80], [81], P. Erdős and Offord [11], by showing that for a quite wide class of distributions the average number of real zeros of a random polynomial increases logarithmically with the degree of the polynomial [67]–[69]. Complex zeros were also considered: with D. N. Zaporozhets [78] they showed that, in order that the zeros concentrate asymptotically near the unit circle with probability one, it is necessary and sufficient that the distribution of the coefficients have a finite logarithmic moment. Jointly with O. Zeitouni [79] Ibragimov indicated explicitly the scaling limit for a given concentration in the case when the distribution of the coefficients lies in the attraction domain of a stable law. In addition, in [78] the authors showed that the arguments of zeros are always uniformly distributed. The multidimensional case was considered in [72] and [96].

Ibragimov was interested in regularity questions for trajectories of stochastic processes. Using various embedding theorems for classes of smooth functions and ideas from approximation theory and the theory of smooth extension of functions, in [26] and [28] he established several easily verifiable conditions for such properties of trajectories as continuity, smoothness, the Hölder property, the boundedness of variation, and some others; also see a theorem in [57] (Appendix 1, Theorem 19) on the properties of realizations of random fields, which generalizes a theorem of Kolmogorov. In [31] and [32] he considered such questions for multivariate random fields.

In the 1980s, apart from other problems, Ibragimov considered functionals of random walks. In [29] and [30] he investigated their limiting behaviour for quite a wide class of distributions of the increments and under very broad assumptions about the functionals themselves. These and other deep results became the basis of their joint monorgaph [5] with A. N. Borodin.

Ibragimov contributed to the development of almost sure limit theorems, which establish weak convergence with probability one of empirical measures. This type of convergence was originally considered independently by G. A. Brosamler [6] and P. Schatte [92], who established an almost sure version of the central limit theorem. In [33] Ibragimov generalized their result to the attraction domain of an arbitrary stable law. Together with M. A. Lifshits [65], he found a number of other generalizations: in particular, to multidimensional (or even arbitrary separable metric) spaces and to weakly dependent random variables, established the principle of almost sure invariance, and in [64] they proposed a stronger version of almost sure convergence by considering a class of unbounded test functions.

Ibragimov also showed interest to such an area as the characterization of a normal distribution by the independence of linear functionals of a random vector. This topic originated from S. N. Bernstein’s classical paper [2] on the characterization of the normal distribution in terms of the independence of the sum and difference of two independent copies of random variables, which was generalized to several dimensions by G. Darmois [10] and V. P. Skitovich [93], [94]. Subsequently, L. V. Mamai [88] and B. Ramachandran [90] treated the infinite-dimensional case. S. G. Ghurye and I. Olkin [12] and A. A. Zinger [97] considered a matrix analogue of this problem. Ibragimov managed to relax the assumptions imposed in known characterizations, both for the vector form [42] and the matrix one [41].

Ibragimov’s life path is a vivid example of selfless service to science. The scope of his research and the fundamentality of results are astounding. It is already 50 years that he is the head of the Leningrad/St Petersburg school in probability theory, mathematical statistics, and stochastic processes, which has always occubied one of the leader positions in the world. He supervises the St Petersburg seminar on probability theory and mathematical statistic, with its creative but demanding atmosphere, thanks to which younger researchers get used to high scientific standards from the very first days.

In 1990 Ibragimov was elected a corresponding member of the Academy of Sciences of the USSR, and in 1997 an acting member of the Russian Academy of Sciences. Since 2000 to 2006 he was the director of the St. Petersburg Department of the Steklov Mathematical Institute, and in 1997–2005 he was the head of the celebrated Department of Probability Theory of the Faculty of Mathematics and Mechanics at St. Petersburg State University, which had been founded by Linnik. Ibragimov is the father of a well-known scientific school. His former student include such well-known experts as T. Arak, A. N. Borodin, M. I. Gordin, Yu. A. Davydov, A. Yu. Zaitsev, Ya. Yu. Nikitin, V. N. Solev, A. N. Tikhomirov, B. S. Tsirelson, and others.

Ibragimov has enormous erudition not only in mathematics, but also in literature and history. He has alwas been a first-rate athlete, who participated in many extremal trips. With his wife Emiliya Leont’evna he raised a son and a daughter. His characeristic features are humility, cheerfulness, kindness, and high ethic standards.

From our hearts we congratulate Ildar Abdullovich Ibragimov on his birthday and wish him many happy years of scientific research and communication with his family, friends, and colleagues.

Bibliography
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17.	I. A. Ibragimov, “On estimation of the spectral function of a stationary Gaussian process”, Theory Probab. Appl., 8:4 (1963), 366–401
18.	I. A. Ibragimov, “A central limit theorem for a class of dependent random variables”, Theory Probab. Appl., 8:1 (1963), 83–89
19.	I. A. Ibragimov, “On the spectrum of stationary Gaussian sequences satisfying the strong mixing condition. I. Necessary conditions”, Theory Probab. Appl., 10:1 (1965), 85–106
20.	I. A. Ibragimov, “A class of estimates for the spectral function of a stationary sequence”, Theory Probab. Appl., 10:1 (1965), 123–126
21.	I. A. Ibragimov, “On the accuracy of Gaussian approximation to the distribution functions of sums of independent random variables”, Theory Probab. Appl., 11:4 (1966), 559–579
22.	I. A. Ibragimov, “On the Chebyshev–Cramér asymptotic expansions”, Theory Probab. Appl., 12:3 (1967), 455–469
23.	I. A. Ibragimov, “On a theorem of G. Szegö”, Math. Notes, 3:6 (1968), 442–448
24.	I. A. Ibragimov, “Completely regular multidimensional stationary processes with discrete time”, Proc. Steklov Inst. Math., 111 (1970), 269–301
25.	I. A. Ibragimov, “On the spectrum”, Theory Probab. Appl., 15:1 (1970), 23–36
26.	I. A. Ibragimov, “Properties of sample functions for stochastic processes and embedding theorems”, Theory Probab. Appl., 18:3 (1974), 442–453
27.	I. A. Ibragimov, “A note on the central limit theorems for dependent random variables”, Theory Probab. Appl., 20:1 (1975), 135–141
28.	I. A. Ibragimov, “On smoothness conditions for trajectories of random functions”, Theory Probab. Appl., 28:2 (1984), 240–262
29.	I. A. Ibragimov, “Some limit theorems for functions of a random walk”, Soviet Math. Dokl., 29:3 (1984), 627–630
30.	I. A. Ibragimov, “Théorèmes limites pour les marches aléatoires”, École d'été de probabilités de Saint-Flour XIII – 1983, Lecture Notes in Math., 1117, Springer, Berlin, 1985, 199–297
31.	I. A. Ibragimov, “Conditions for Gaussian homogeneous fields to belong to classes”, J. Math. Sci. (N. Y.), 68:4 (1994), 484–497
32.	I. A. Ibragimov, “Remarks on variations of random fields”, J. Math. Sci. (N. Y.), 75:5 (1995), 1931–1934
33.	I. A. Ibragimov, “On almost-everywhere variants of limit theorems”, Dokl. Math., 54:2 (1996), 703–705
34.	I. Ibragimov, “On estimation of analytic functions”, Studia Sci. Math. Hungar., 34:1-3 (1998), 191–210
35.	I. A. Ibragimov, “On the extrapolation of entire functions observed in a Gaussian white noise”, Ukrainian Math. J., 52:9 (2000), 1383–1395
36.	I. A. Ibragimov, “An estimation problem for quasilinear stochastic partial differential equations”, Problems Inform. Transmission, 39:1 (2003), 51–77
37.	I. A. Ibragimov, “Estimation of analytic densities based on censored data”, J. Math. Sci. (N. Y.), 133:3 (2006), 1290–1297
38.	I. Ibragimov, “Estimation of analytic spectral density of Gaussian stationary processes”, Parametric and semiparametric models with applications to reliability, survival analysis, and quality of life, Stat. Ind. Technol., Birkhäuser Boston, Boston, MA, 2004, 419–443
39.	I. A. Ibragimov, “On censored sample estimation of a multivariate analytic probability density”, Theory Probab. Appl., 51:1 (2007), 142–154
40.	I. A. Ibragimov, “On the estimation of an analytic spectral density outside of the observation band”, Topics in stochastic analysis and nonparametric estimation, Papers presented at the conference “Asympototic analysis in stochastic processes, nonparametric estimation, and related problems”, dedicated to professor R. Z. Khasminskii on the occasion on the 75th birthday (Detroit, MI 2006), IMA Vol. Math. Appl., 145, Springer, New York, 2008, 85–103
41.	I. A. Ibragimov, “On the Ghurye–Olkin–Zinger theorem”, J. Math. Sci. (N. Y.), 199:2 (2014), 174–183
42.	I. A. Ibragimov, “On the Skitovich–Darmois–Ramachandran theorem”, Theory Probab. Appl., 57:3 (2013), 368–374
43.	I. A. Ibragimov, “On estimation of the intensity density function of a Poisson random field outside the observation region”, J. Math. Sci. (N. Y.), 214:4 (2016), 484–492
44.	I. A. Ibragimov, “On estimation of functions of a parameter observed in Gaussian noise”, J. Math. Sci. (N. Y.), 238:4 (2019), 463–470
45.	I. A. Ibragimov, “An estimation problem for the intensity density of Poisson processes”, J. Math. Sci. (N. Y.), 251:1 (2020), 88–95
46.	I. Ibragimov, Z. Kabluchko, and M. Lifshits, “Some extensions of linear approximation and prediction problems for stationary processes”, Stochastic Process. Appl., 129:8 (2019), 2758–2782
47.	I. A. Ibragimov and R. Z. Khas'minskii, “Asymptotic behavior of generalized Bayes estimates”, Soviet Math. Dokl., 11 (1970), 1181–1185
48.	I. A. Ibragimov and R. Z. Khas'minskii, “Asymptotic behavior of statistical estimators in the smooth case. I. Study of the likelihood ratio”, Theory Probab. Appl., 17:3 (1973), 445–462
49.	I. A. Ibragimov and R. Z. Has'minskii (Khas'minskii), “Asymptotic behavior of statistical estimates for samples with a discontinuous density”, Math. USSR-Sb., 16:4 (1972), 573–606
50.	I. A. Ibragimov and R. Z. Khas'minskii, “Asymptotic behavior of some statistical estimators. II. Limit theorems for the a posteriori density and Bayes' estimators”, Theory Probab. Appl., 18:1 (1973), 76–91
51.	I. A. Ibragimov and R. Z. Khas'minskii, “On moments of generalized Bayessian estimators and maximum likelihood estimators”, Theory Probab. Appl., 18:3 (1974), 508–520
52.	I. A. Ibragimov and R. Z. Khas'minskii, “Asymptotic behavior of statistical estimators of the location parameter for samples with continuous density with singularities”, J. Soviet Math., 9 (1978), 50–72
53.	I. A. Ibragimov and R. Z. Has'minskii (Khas'minskii), “Estimation of a signal parameter in Gaussian white Noise”, Problems Inform. Transmission, 10:1 (1974), 31–46
54.	I. A. Ibragimov and R. Z. Has'minskii (Khas'minskii), “Parameter estimation for a discontinuous signal in white Gaussian nsoise”, Problems Inform. Transmission, 11:3 (1975), 203–212
55.	I. A. Ibragimov and R. Z. Khas'minskii, “Asymptotic behavior of statistical estimates of the shift parameter for samples with unbounded density”, J. Soviet Math., 16:2 (1981), 1035–1041
56.	I. A. Ibragimov and R. Z. Khas'minskii, “A problem of statistical estimation in Gaussian white noise”, Soviet Math. Dokl., 18 (1978), 1351–1354
57.	I. A. Ibragimov and R. Z. Has'minskii (Khas'minskii), Statistical estimation. Asymptotic theory, Appl. Math., 16, Springer-Verlag, New York–Berlin, 1981, vii+403 pp.
58.	I. A. Ibragimov and R. Z. Khas'minskii, “Asymptotic bounds on the quality of the nonparametric regression estimation in $\mathscr L_p$”, J. Soviet Math., 24:5 (1984), 540–550
59.	I. A. Ibragimov and R. Z. Khas'minskii, “Estimation of distribution density”, J. Soviet Math., 21:1 (1983), 40–57
60.	I. A. Ibragimov and R. Z. Khas'minskii, “More on the estimation of distribution densities”, J. Soviet Math., 25:3 (1984), 1155–1165
61.	I. A. Ibragimov and R. Z. Khas'minskii, “Estimation problems for coefficients of stochastic partial differential equations. Part I”, Theory Probab. Appl., 43:3 (1999), 370–387
62.	I. A. Ibragimov and R. Z. Khas'minskii, “Estimation problems for coefficients of stochastic partial differential equations. Part II”, Theory Probab. Appl., 44:3 (2000), 469–494
63.	I. A. Ibragimov and R. Z. Khas'minskii, “Estimation problems for coefficients of stochastic partial differential equations. Part III”, Theory Probab. Appl., 45:2 (2001), 210–232
64.	I. Ibragimov and M. Lifshits, “On the convergence of generalized moments in almost sure central limit theorem”, Stat. Probab. Lett., 40:4 (1998), 343–351
65.	I. A. Ibragimov and M. A. Lifshits, “On almost sure limit theorems”, Theory Probab. Appl., 44:2 (2000), 254–272
66.	I. A. Ibragimov and Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971, 443 pp.
67.	I. A. Ibragimov, “On the expected number of real zeros of random polynomials. I. Coefficients with zero means”, Theory Probab. Appl., 16:2 (1971), 228–248
68.	I. A. Ibragimov and N. B. Maslova, “On the expected number of real zeros of random polynomials. II. Coefficients with non-zero means”, Theory Probab. Appl., 16:3 (1971), 485–493
69.	I. A. Ibragimov and N. B. Maslova, “Average number of real roots of random polynomials”, Soviet Math. Dokl., 12 (1971), 1004–1008
70.	I. A. Ibragimov, A. S. Nemirovskii, and R. Z. Has'minskii (Khas'minskii), “Some problems on nonparametric estimation in Gaussian white noise”, Theory Probab. Appl., 31:3 (1987), 391–406
71.	I. A. Ibragimov and L. V. Osipov, “On an estimate of the remainder in Lindeberg's theorem”, Theory Probab. Appl., 11:1 (1966), 125–128
72.	I. A. Ibragimov and S. S. Podkorytov, “Random real algebraic surfaces”, Dokl. Math., 52:1 (1995), 101–103
73.	I. A. Ibragimov, E. L. Presman, and Sh. K. Formanov, “On modifications of the Lindeberg and Rotar' conditions in the central limit theorem”, Theory Probab. Appl., 65:4 (2021), 648–651
74.	I. A. Ibragimov and Y. A. Rozanov, Gaussian random processes, Appl. Math. (N. Y.), 9, Springer-Verlag, New York–Berlin, 1978, x+275 pp.
75.	I. A. Ibragimov and V. N. Solev, “The asymptotic behavior of the prediction error of a stationary sequence with a spectral density of special type”, Theory Probab. Appl., 13:4 (1968), 703–707
76.	I. A. Ibragimov and V. N. Solev, “A condition for regularity of a Gaussian stationary process”, Soviet Math. Dokl., 10 (1969), 371–375
77.	I. A. Ibragimov and V. N. Solev, “A condition for the regularity of a stationary Gaussian sequence”, Semin. Math., 12, V. A. Steklov Math. Inst., Leningrad, 1971, 54–60
78.	I. Ibragimov and D. Zaporozhets, “On distribution of zeros of random polynomials in complex plane”, Prokhorov and contemporary probability theory, In honor of Yu. V. Prokhorov on the occasion of his 80th birthday, Springer Proc. Math. Stat., 33, Springer, Heidelberg, 2013, 303–323
79.	I. Ibragimov and O. Zeitouni, “On roots of random polynomials”, Trans. Amer. Math. Soc., 349:6 (1997), 2427–2441
80.	M. Kac, “On the average number of real roots of a random algebraic equation”, Bull. Amer. Math. Soc., 49 (1943), 314–320
81.	M. Kac, “On the average number of real roots of a random algebraic equation (II)”, Proc. London Math. Soc. (2), 50 (1948), 390–408
82.	M. Kac, “Toeplitz matrices, translation kernels and a related problem in probability theory”, Duke Math. J., 21:3 (1954), 501–509
83.	A. N. Kolmogorov and Yu. A. Rozanov, “On strong mixing conditions for stationary Gaussian processes”, Theory Probab. Appl., 5:2 (1960), 204–208
84.	V. A. Kotel'nikov, Theory of potential noise immunity, Gosenergoizdat, Moscow–Leningrad, 1956, 153 pp. (Russian)
85.	J. E. Littlewood and A. C. Offord, “On the number of real roots of a random algebraic equation”, J. London Math. Soc., 13:4 (1938), 288–295
86.	J. E. Littlewood and A. C. Offord, “On the number of real roots of a random algebraic equation. II”, Proc. Cambridge Philos. Soc., 35:2 (1939), 133–148
87.	J. E. Littlewood and A. C. Offord, “On the number of real roots of a random algebraic equation (III)”, Mat. Sb., 12(54):3 (1943), 277–286
88.	L. V. Mamai, “On the theory of characteristic functions”, Sel. Transl. Math. Stat. Probab., 4 (1963), 153–170
89.	A. Nemirovski, “Topics in non-parametric statistics”, Lectures on probability theory and statistics, École d'été de probabilités de Saint-Flour XXVIII – 1998 (Saint-Flour 1998), Lecture Notes in Math., 1738, Springer, Berlin, 2000, 85–277
90.	B. Ramachandran, Advanced theory of characteristic functions, Statist. Publ. Soc., Calcutta, 1967, vii+208 pp.
91.	M. Rosenblatt, “A central limit theorem and a strong mixing condition”, Proc. Nat. Acad. Sci. U.S.A., 42 (1956), 43–47
92.	P. Schatte, “On strong versions of the central limit theorem”, Math. Nachr., 137 (1988), 249–256
93.	V. P. Skitovich, “One property of normal distribution”, Dokl. Akad. Nauk SSSR, 89:2 (1953), 217–219 (Russian)
94.	V. P. Skitovič, “Linear combinations of independent random variables and the normal distribution law”, Sel. Transl. Math. Stat. Probab., 2 (1962), 211–228
95.	G. Szegő, “On certain Hermitian forms associated with the Fourier series of a positive function”, Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.], 1952, Tome Suppl. (1952), 228–238
96.	D. N. Zaporozhets and I. A. Ibragimov, “On random surface area”, J. Math. Sci. (N. Y.), 176:2 (2011), 190–202
97.	A. A. Zinger, “On a characterization of the multi-dimensional normal law by the independence of linear statistics”, Theory Probab. Appl., 24:2 (1979), 388–392

Citation in format AMSBIB

\Bibitem{BorBulVer23}
\by A.~A.~Borovkov, Al.~V.~Bulinski, A.~M.~Vershik, D.~Zaporozhets, A.~S.~Holevo, A.~N.~Shiryaev
\paper Ildar Abdullovich Ibragimov (on his ninetieth birthday)
\jour Russian Math. Surveys
\yr 2023
\vol 78
\issue 3
\pages 573--583
\mathnet{http://mi.mathnet.ru//eng/rm10101}
\crossref{https://doi.org/10.4213/rm10101e}
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