In 1943 the Institute of Mathematics of the Academy of Science of the Republic of Uzbekistan was established under which the future of mathematical statistics and theory of possibilities (future director V.I. Romanovsky) was opened. The laboratory has made a significant contribution to the development of mathematics, special accounting theory and mathematical statistics in Uzbekistan, to the growth of Uzbekistan's prestige in the world, to the training of qualified personnel not only for the republic, but also abroad. A great mathematician of the XX century academician V.I. Romanovsky (1879-1954) takes a great part in the field of mathematics. He is a high-level mathematician and one of the founders of the great pedagogical period and the School of Mathematics of Uzbekistan.

The laboratory staff has obtained many significant results in the theory of probability and mathematical statistics, the theory of random processes. Limit theorems and estimates for the sum of independent random variables, limit theorems for Markov chains, limit theorems for random branching processes, statistical methods of product quality control, public service theory, asymptotic problems of statistical estimation theory, stationary random great scientific results have been achieved in quantities, ergodic theory, stochastic dynamic systems theory, random fields in statistical physics, quadratic stochastic processes, and other areas. In the theory of Markov chains academicians V.I. Romanovsky, T.A. Sarimsakov, S.H. Sirojiddinov and Sh.K. Farmonov, in the theory of branching processes S.V. Nagaev, A.V. Nagayev, Sh.K. Farmonov, R.E. Muhammedkhanova, I.S. Badalbaev and I. Rakhimov, discrete problems of probability theory Sh.A. Mirakhmedov in the field of application, M.A. Mirzakhmedov, Sh.A. Hoshimov and Sh.A. Mirakhmedov in the field of parametric and non-parametric statistics, academician T.A. Azlarov in flat estimates in limit theorems for public service systems, in functional spaces, in particular Banax O.Sh.Sharipov in limit theorems for the sum of weakly connected elements in space, A.A. Abdushukurov in the asymptotics of statistics based on censored observations, in the theory of stationary random quantities, ergodic theory and stochastic dynamic systems V.G. Vinokurov and N.N. Ganikhodjayev. N.N. Ganikhodjayev and their students in the theory of random fields and quadratic stochastic processes in statistical physics have achieved high scientific results, which are recognized by experts.

From 1944 to 2021 in the department of probability theory and mathematical statistics academicians V.I. Romanovskiy, T.A. Sarimsoqov, S.H. Sirojiddinov, T.A. Azlarov, Sh.Q. Farmonov, professorlar S.V. Nagayev, M.K. Kamolov, M.I. Eydelnat, T.L. Malevich, V.G. Vinokurov, A.V. Nagayev, M.M. Mamatov, B. Abdalimov, I.S. Badalbayev, I. Raximov, A. Startsev, Sh. Xoshimov, Sh.A. Miraxmedov, Sh. Shoraxmetov, M.U. Gafurov, N.N. Ganixodjayev and many other well-known scientists have worked and continue to work. The first graduate student of the institute was S.H. Sirojiddinov, a student of V.I. Romanovsky. Later he served as a Vice President of the Academy of Science for many years (1970-1983). He made a significant contribution to the development of mathematics in Uzbekistan, especially to the probability theory and mathematical statistics.

Department of Probability Theory and Mathematical Statistics was lead by V.I. Romanovsky (1949), by S.X. Sirojiddinov (1953-1954), by M.I. Eydelnant (1954-1960), by R.X. Diveyev (1960-1967), by V.G. Vinokurov (1967-1972), by T.A. Azlarov (1972-1976) and by Sh.K.Farmonov (2001-2018). In 1971 the Department of Probability Theory and the Laboratory of Statistics of the Institute were merged to form the Department of Probability Theory and Mathematical Statistics (Head of the Department T.A. Azlarov). In 1976 on the basis of the Department of Probability Theory and Mathematical Statistics were created 2 sections – “Probability Theory” (head of the department Sh.K. Formanov) and “Mathematical Statistics” (head of the department T.A. Azlarov).

 The Probability Theory Department was lead 1976-1990 by Sh.K. Formanov, 1991-1995 by I.S. Badalbaev, 1995-2001 by Sh.K. Formanov and the Mathematical Statistics Department was lead 1976-1980 by T.A.Azlarov, 1980-1983 by S.X.Sirojiddinov, 1983-1988 by M.U.Gafurov, 1988-1994 by A.V.Nagaev, 1994-2001 by Sh.Hashimov. In 2001 the 2 departments - Probability Theory and Mathematical Statistics departments  were merged, and the new laboratory was headed by Sh.K. Formanov from 2001 to 2018.

 In 2020 the department was renamed as a Stochastic Analysis. Today the laboratory is headed by Y.M.Khusanbaev.

The staff of the laboratory conducts research in the following areas:

- Classical and non-classical variants of the central limit theorem for random sums;

- Limit theorems for random quantities and sequences of vectors;

- Mathematical methods of statistical acceptance control of the finished product;

- Random locations (probability methods of discrete mathematics);

- The theory of asymptotic effectiveness of statistical symptoms;

- Spacings-statistics, their application;

- Limit theorems and their applications for weakly bound random variables that take value in finite and infinite-dimensional spaces;

- Limit theorems for weakly bound random fields that receive values in finite and infinite-dimensional spaces, weak connection conditions for random fields;

- Limit theorems and footsteps for U-statistics consisting of observations;

- Evaluation of the parameters of autoregression processes, autoregression processes with weakly linked innovation, autoregression processes in Banach spaces;

- Time series modeling;

- Analysis of functional observations;

- Inequalities for moments and probabilities;

- Construction of the distribution function (copula function) of a random vector with a given marginal distribution;

- Asymptotics of the sum of conditionally independent random variables;

- Limit theorems for Galton-Watson branching processes;

- Functional limit theorems for immigrant branching random processes;

- Analysis of branching random processes in a changing environment;

- Monte Carlo method and its applications, mathematical models of risk decision-making;

- Limit Gibbs measurement theory for grid models of statistical physics;

- Nonlinear dynamic systems and their applications;

- Quadratic stochastic operators and processes;

- Asymptotic analysis of extreme ordered statistics;

- Limit theorems for convex coating functionals;

- The theory of point processes.

1.     A generalized model of random placement of particles in boxes (Generalized Urna Model) is introduced. . The results were applied to a multinomial, finite set of nonlinear selection and the Polya-Egenberger models.

2.     An algorithm for constructing an asymptotic distribution for a selective aggregate distribution function constructed by the non-return method from a finite set is presented. In the two-step random sampling scheme of finite complex analysis, an asymptotic distribution for student-selected sample mean values is established in stratification and clustering designs.

3.     In the grouping method of observations, a class of compatibility traits based on statistics, such as the sum of group-frequency functions, was studied. In comparing symptoms belonging to this class, the concept of Second Order Effectiveness based on the asymptotic power of symptoms, as well as the concept of Intermediate Effectiveness, which is considered between the two “extreme cases” Pitman and Bahadur asymptotic effects approach. In terms of these effects, the problem of optimal symptoms is solved.

4.     The unknown parameter of the distribution includes the Generalized Spacings Estimators class, which is based on high-order spacings-statistics and minimizes the line distance in various convex functions. To do this all kinds of limit theorems for the approximation of the sum of the functions of flat-spacing statistics by the normal distribution for the visible statistics have been proved.

5.     Probable representations in the form of mathematical expectations of random variables were obtained for nonlinear problem solutions; random branching processes were constructed in accordance with the probabilistic assumptions, their modeling formulas were determined, the constructed processes were completed with a single probability, and the number of generation dice was proved to be less than one; fixed values in the process trajectory, shifted value in low-branched process trajectories are constructed, the variance of the constructed values is limited, the number of branches is defined and modeled in limited process trajectories; the invariance of the values and the limitation of the variances have been proved by the use of marking moments and martingale theories.

6.     For boundary value problems in systems random processes are constructed according to probabilistic appearances, modeling formulas for process trajectories are defined, and a modeling algorithm is given; the variance of the solutions of the problems in the trajectories of random processes is constructed with limited displacement and low displacement estimates.

7.     In Lebeg space, a series of continuously growing dimensional fractions have been shown to be unique in the accuracy of isomorphism.

8.     The problem of the existence of a limit Gibbs distribution for several competing interactive models of statistical physics with a finite set of cases is solved, the problem of having a limit Gibbs distribution of more than one is solved.

9.     The concept of quadratic stochastic processes was introduced and analytical methods for such processes were developed. Delayed argument differential and integro-differential equations describing such processes are derived.

10.  In the theory of nonlinear dynamic systems the quadratic stochastic operators have been constructed and studied that describe a mathematical model of the transfer of blood group and rhesus factor from parents to children. The ergodic problem is solved for a wide class of quadratic stochastic operators. A method for constructing the quadratic stochastic operators with a set of genotypes that are numerical and continuous has been proposed, and the problem of ergodicity for the separation of such operators has been solved.

11. Functional limit theorems have been proved for branching random processes involving the flow of random quantities associated with immigration.

12. For the moments of branching random processes the values are set from above using single particle moments and immigration moments.

In 1973 S.H.Sirojiddinov, T.A.Azlarov, Sh.K.Formanov and M.M.Mamatov were awarded with the Beruni State Prize of Uzbekistan for their research in the field of “Limit theorems of probability theory and their applications”.

In 1977 young prospect M.U.Gafurov, Sh.A.Mirakhmedov, Sh.A.Hoshimov and Sh.A.Ismatullayev were awarded with the Prize of Uzbekistan in the field of science and technology for young scientists.

The laboratory actively cooperates with many foreign institutes and universities, including:

- V.A.Steklov Institute of Mathematics (Moscow, Russia);

- V.A.Steklov Petersburg branch of the Steklov Institute of Mathematics;

- S.L.Sobolev Institute of Mathematics (Novosibirsk, Russia);

- M.V.Lomonosov Moscow State University (Moscow, Russia);

- University of Bochum (Bochum, Germany)

- Kiev National University named after Taras Shevchenko (Kiev, Ukraine);

- University of Copenhagen (Copenhagen, Denmark);

- Stanford University (California, USA);

- Putra University (Kuala Lumpur, Malaysia);

- University of Magdeburg (Magdeburg, Germany);

- University of Debrecen (Debrecen, Hungary);

- Institute of Mathematics of the Ukrainian Academy of Science (Kiev, Ukraine);

- Russian Humanities University (Moscow, Russia).