**
C1. Research Interests **

My research interests concern the theory, statistics and applications of branching stochastic processes and diffusions, sampling sums of dependent variables, random sums, and sums of dependent indicators.

The following could be considered as my basic scientific results:

(a) Proposition of a new scheme of proving limit theorems for sums of a random number of random variables. Demonstration possibilities of this scheme in the case of infinitely divisible limit distributions and sampling sums of dependent variables.

(b) Developing a unified approach to immigration problems in branching processes based on a general limit theory for random superposition of copies of stochastic processes.

(c) Investigation of new functionals of branching processes concerning their genealogical trees, records, exceedances and extreme of family sizes.

These results are included in my Doctor of Physical and Mathematical Sciences thesis submitted in Steklov Mathematical Institute and have been published in the following monographs and articles.

**
C2. Summary of Research**

** **I
have been actively engaged in research since 1977. My research contribution can
be categorized into the following:

1. Sampling and Random Sums.

2. Sums of Dependent Indicators.

3. Immigration Problems in Branching Processes.

4. Functional Limits and Extreme of Branching Processes

5. Estimation Theory for Branching Processes

1. Sampling and Random Sums

It is known that many characteristics of branching stochastic processes can be represented as sums of random number of random variables or random fields. I have suggested a new approach to study such sums based on an idea of their representation as sampling sums of dependent random variables. It turned out that, if we apply this representation to simple random sums, we obtain almost all known and many new results for random sums directly from the representation. Moreover, this representation allows to study branching processes with immigration in cases when reproduction and immigration processes are not independent. It was impossible to treat the latest case by traditional methods of the theory based on generating functions ( see monograph [18], Ch. I, and articles [13], [16] and [17]).

It is known that the sum of n random variables randomly selected from a finite population , , of independent random variables in the equiprobable scheme of sampling without replacement can be represented as

,

where random variables taking values 0 or 1. Here and are independent. Let now vectors and are not independent. It is clear that this scheme (under some assumptions) includes, for example, the case when the decision about including variable into the sum depends on the values of . For such generalized sampling sums I have obtained necessary and sufficient conditions to have a limit distribution, have given an approximation for their distributions and have obtained uniform and non-uniform estimates of the rate of convergence ( see monograph [18 ],Ch. I, papers [2], [5], [9] and [16]).

Necessary and sufficient conditions are obtained for sums of randomly indexed stochastic processes to have a limit distribution in the case when the sequence of indexes and the initial sequence of stochastic processes are not independent. These conditions are extensions of well-known Anskombe condition to the above sums (see monograph [18] , Ch. I, paper [11] ).

Several new estimators are proposed for the population mean, based on the ranked sets of samples with random set size and random number of replications. It is demonstrated that in most cases these estimators are better than existing ones. Conditions for asymptotic normality of those estimators are obtained. (Papers [34], [35]).

2. Sums of Dependent Indicators

Sums of dependent indicators appear in many problems of the probability theory, such as random allocations, queueing theory, branching processes and so on. The study of such sums needs a new techniques in each case of dependence. I have considered two kinds of dependent indicators: the case of multivariate functions of independent random variables and the case of arbitrary dependence. Considering sum of a random number of indicators I have found conditions for a Poisson approximation of the distribution and have given an estimation of the rate of convergence in this approximation. This is in the case of first kind of dependence. In the case of arbitrary dependence I have found approximation of the distribution by a mixture of normal distributions. (Monograph [18], Ch. IV, papers [4], [10].

We must note that models considered in the above two directions are not simple generalizations or extensions of known schemes. They are constructed and investigated in connection with various concrete problems in stochastic processes. Hence they found their non-trivial and interesting applications in the directions which we are going to present below.

3. Immigration Problems

Over the last decade I have developed a unified approach to immigration problems in branching stochastic processes. This approach is based on developing a general limit theory for random superpositions of copies of stochastic processes. Since branching processes maintained by immigration comprise particular cases of this general set-up, it provides a unified approach to a variety of problems that had previously been treated in isolation. Moreover, this approach allows to study new models of the immigration process depending of reproduction, that was impossible to treat by traditional methods based on analysis of probability generating functions. Construction, development and applications of this approach can be seen in my monograph [18], Ch. I and II, and in articles [12], [13], [15] and [22]..

In the case of non-stationary immigration process a perfect description of the asymptotic behavior of the number of particles have been constructed. Some new properties of the limit behavior of the process have been discovered and the causes of these effects are explained. Local limit theorems in the “non-classical” situation (when non-degenerate limit distributions for the number of particles come out by functional non-linear normalization) are also proved. These results which are published in monograph [18], Ch III and in papers [3], [7], [8] and [14 ] have been applied in investigation of various problems in Biology [25 ], [27], in Physics [26], and other areas.

4. Functional Limit Theorems and Extreme of Branching Processes

The main object of investigation in the theory of branching processes is the number of individuals (particles) at a given time. Since this number can be represented as sum of independent processes “shifted” over time, the problem can be reduced to the analysis of equations for probability generating functions. However there are many other characteristics of the process which can not be written as such sums and, therefore, can not be treated by the generating function techniques. Applying result obtained for sums of dependent indicators (see part 2), makes possible to study various new characteristics of branching processes concerning their genealogy. Connections of these characteristics with problems of allocation of particles into cells are also established. (see [18], ch IV and papers [4] and [10]). Extreme problems concerning branching processes, such as maximum of family sizes in the process, records and exceedances of the sequence of branching processes are stated and methods for study of these processes are developed. (Papers [19], [20], [21], [23], [24] and [29]-[31]). One natural approach to study of extremes in branching processes is using limit theorems in the functional form. Using a martingale approach the functional limit theorems are established for critical and nearly critical processes with time-dependent immigration. (Papers [32], [33]).

5. Estimation Theory for Branching Processes

Estimation theory of parameters of branching processes is very important part of statistical theory for stochastic processes. Since properties of branching processes strictly depend on the average number of descendants of one individual, it is important to give an estimator for it based on observed values of the process. However, in the critical and subcritical cases it is difficult, because such processes extinct with probability 1. One of ways to avoid this difficulty is to consider processes with immigration. But, if we consider a stationary immigration, in the critical and subcritical cases the process may be suppressed by immigration. Therefore it is better to consider decreasing or state-dependent (more complicated) immigration component. Considering subcritical processes with state-dependent immigration I have given consistent and asymptotically unbiased estimators for the offspring and immigration averages. It was also shown that these estimators are asymptotically normal. I have proposed consistent and asymptotically unbiased estimators for the expectation and variance of the limit distribution and also for the probability of hitting the state zero by the process. These results were published in [6]. One may observe a rise of interest to this kind problems after more than ten years [28]. It was known that the conditional least squares estimator (CLSE) of the offspring mean in a branching process with stationary immigration is not asymptotically normal. Later it was proved that if the offspring variance tends to zero, it has normal limiting distributions. Note that the condition that the offspring variance tends to there means that in the long ran the process approaches to deterministic process. In recent investigations we proved asymptotic normality of the CLSE when the offspring variance does not tend to zero in non classical model. We were able to show this using functional limit theorems , which are discussed above. (Papers [32], [33])

REFERENCES

[1]* Rahimov, I., On the limit theorems for a sequence of branching processes with non-homogeneous immigration. THEORY OF PROBAB. AND APPL. 1984, N4, (Russian and English trans.).

[2] Rahimov, I., Uniform estimates in the limit
theorems for branching processes with immigration, IZVESTIYA of AS of UzSSR
1984, No4, 24-29** **(Russian).

[3]* Rahimov, I., Critical branching processes with infinite variance and decreasing immigration. THEORY OF PROBAB. AND APPL. 1986, v.31, No1, 98-110 (Russian and English ).

[4]* Rahimov I., Limit theorems for random sums
of dependent indicators and their applications in the theory of branching
processes. THEORY OF PROBAB. AND** **APPL.** **1987, v.32, N2, 317-326**,**
(Russian and English) ** **

[5] Rahimov, I., On approximation of the
distribution of the sum of a random number of random terms, DOKL.of AS of UzSSR,
1987, No1, 5-7** ** (Russian).

[6] Rahimov, I., Statistical estimates for
parameters of a subcritical Galton-Watson processes with a reflecting screen.
“PROBABILITY MODELS AND MATHEM.** **STATIST.” Tashkent, 1987, 76-87
(Russian.).

[7]* Rahimov, I., Local limit theorems for
branching random processes with decreasing immigration.** **THEORY OF PROBAB.
AND APPL. 1988, V.33, No2, 387-392 (Russian and English trans.).

[8] Rahimov, I., A local theorem for Galton-Watson
processes with immigration in the case of uniform limit distribution, **”**SERDICA”
, 1988. V14, No3, 234-244 (Russian).

[9]*** **Rahimov, I., Sirazhdinov S. Kh.,
Approximation of distribution of a sum in a certain scheme for summation of
independent random variables. SOVIET MATH. DOKL**. **Vol.38, 1989, No1,
23-27 (English)

[10]* Rahimov, I., Asymptotic behavior of
families of particles in branching random processes.** **SOVIET MATH.DOKL.,Vol.39,
1989, N2, 322-325 (English).

[11]** **Rahimov, I., Asymptotic behavior of
the sum of randomly indexed processes, In the book “ASYMPTOTIC PROBLEMS OF
PROBABILITY THEORY AND MATH. STATIST.” “Fan”, Tashkent, 1990, 79-92 (Russian).

[12]* Rahimov, I., The General branching processes with immigration depending on reproduction, THEORY OF PROBAB. AND APPL., 1992, V. 37, No 3, 513-525 (Russian and English ).

[13]* Rahimov, I., Sampling sums of dependent
random variables, mixtures of infinitely divisible laws and branching random
processes, DISCRETE MATH. AND APPL**., **1992, V.2, No3, 337-356 (English).

[14]*Rahimov, I., Critical Processes with infinite variance and increasing immigration , MATHEMATICAL NOTES, 1993, V.53, No 6, P.97-107, (Russian and English ).

[15]* Rahimov, I., Branching Processes as a Sums of Dependent Random Variables., BRANCHING PROCESSES, First World Congress, 1993, Varna, Bulgaria. Springer-Verlag, Ser. LNS, Vol. 99, 1995, p. 58-66, Editor Ch. Heyde, (English).

[16]***
**Rahimov, I., ** ** Weak convergence of certain multiple sums of dependent
random variables. Probabilistic Methods in Discrete Mathematics, Progr. Pure
Appl. Discrete Math., 1, VSP, Utrecht, 1993, 376-385 (Russian and English).

[17] Rahimov, I., Mixtures of Infinitely Divisible Laws as Limit Distributions for a Special Form Multiple Sums, UZBEK MATHEMATICAL JOURNAL, 1994, No 4, P. 43-50 (Russian).

[18]** ** Rahimov,
I., Random Sums and Branching Stochastic Processes, SPRINGER VERLAG, LNS, V. 96,
1995. (English).

[19]* Rahimov, I., Record values of a family of
branching processes, Classical and Modern Branching Processes, Springer-Verlag,
Ser.”IMA Volumes in Mathematics** **and its Applications”, Vol. 84, 1996,
Editors K. Athreya and P Jagers, P. 285-295. (English).

[20]* Rahimov, I., Yanev, G., Maximal Number of Direct Offspring in Simple Branching Processes, NONLINEAR ANALYSIS, 1997, V.30, No 4, p.2015-2023. (English),

[21]*** **
Rahimov, I., Kassim, S. Record Values Concerning a Family of Age-Dependent
Processes, THE BULLETIN OF THE MALAYSIAN MATHEMATICAL SOCIETY, 1997, Vol. 20,
No. 2, 57-66. ( English).

[22]* Rahimov, I., Multitype Processes with Reproduction-Dependent Immigration, JOURNAL OF APPLIED PROBABILITY, 1998, V. 35, No. 2, P. 281-292 (English).

[23]* Rahimov, I., Yanev, G., On Maximum Family Size in Branching Processes, JOURNAL OF APPLIED PROBABILITY, (Accepted for publication, will appear in 1999, English).

[24]* Rahimov, I., Hasan, H., Limit Theorems for Exceedances of a Sequence of Branching Processes, THE BULLETIN OF THE MALAYSIAN MATHEMATICAL SOCIETY, (Accepted for publication, will appear in 1999, English).

[25] Berlin, Y. A., Drobnitsky, D., O., et all., On Inherited Fertility in Biological Systems, Biosystems, 1992, V. 26, No. 3, p. 185-192.

[26] Berlin, Y. A., Drobnitsky, D., O., et all., Correlated Fluctuations in Multielement Systems, Physical Review A, 1992, Vol. 45, No. 6, p. 3547-3552.

[27] Lange, K., Fan, R., Z., Branching Process Models for Mutant-Genes in Nonstationary Populations. Theoretical Population Biology, 1997, V. 51, No. 2, P. 118-133.

[28] Jacob C., Peccoud J., Estimation of the
Parameters of Branching Process from Migration Binomial Observations, Advances
in Applied Probability, 1998, V. 30, No.** **4, p. 948-967**.**

[29]***
**Rahimov I. Approximation
of exceedance processes in large populations. Stochastic Models, 2001, (ISI),
V. 17, No 2, P. 147-156.

[30] * Rahimov I. Random Sums of Independent Indicators and Generalized Reduced Processes. Stochastic analysis and applications, (ISI) Vol. 21, No 1, 2003, P. 205-221.

[31]* Rahimov I., Limit Theorems for the size of Subpopulation of Productive Individuals. Stochastic models, (ISI) VOL. 20, 2004, no 3, P. 261-280.

[32]* Rahimov I., Functional Limit Theorems for Critical Processes with Immigration " Advances in Applied Probability, 2007, V. 39, No 4, (ISI)

[33]* Rahimov I. Asymptotic distribution of the CLSE in a critical process with Immigration. Stochastic Processes and Their Applications, 2008, (ISI) (in press ).

[34]* Rahimov I., Muttlak H. A. Estimation of the population mean using random selection in ranked set samples. STATISTICS AND PROBABILITY LETTERS, (ISI) Vol 62, 2003, P. 203-209.

[35]* Rahimov I., Muttlak H. A. Investigating the Estimation of the Population Mean Using Random Ranked Set Samples. NONPARAMETRIC STATISTICS, 2003, V 15, No 3, P. 311-324.

**C3. Works in
Progress**

My current research concerns the following directions.

**a) **
Multivariate
Random Sums.

Let be a random sum of random vectors , where are not necessarily independent of . In the case when are random variables and are stopping times with respect to the filtration generated by , it was shown that the investigation of the random sum could be reduced to study of a simple sum of dependent variables (see Monograph [1] C3). It is interesting to consider such a problem in the case of random vectors. Results, which are going to be obtained in this direction, will be applied in investigation of sampling sums of dependent vectors, multivariate random walks, random fields and multitype general branching processes.

**b)** Immigration-Branching Diffusions

Immigration-Branching Diffusions are models of a population process in which offspring move at random on a bounded space throughout their lives according to diffusions. Each member of the population at the end of its life generates a new population whose members may be located at any point of the space. In addition a random number of new particles may immigrate into the population. Currently the theory of such processes contains many results in the case when diffusion, reproduction and immigration processes are independent. It is important to develop methods of investigation for processes allowing dependence of the above three components. Here I hope that an approach suggested in the article [1], 1992, C4 and in the book [1], C3 will allow to consider such a generalization.

** **c)
Extreme Concerning Branching
Processes

First results on the size of largest family in Galton-Watson processes were obtained in the paper [2], 1997, C4 (see also [3], 1998, C4). Similar problem for Galton-Watson processes allowing immigration have been considered jointly with my student H. Hummieda. In this direction there are many interesting problems, such as to consider more general models of processes or to consider k-th maximal family and other problems.

Another class of problems is to study exceedances and record properties of a sequence of branching processes. First results are obtained in publications [1] 1996, [4] 1997, [2] 1998 C4 and [1], 1996 C6. In particular, investigation of exceedances in this model is a part of Ph.D. program of my student H. Hasan.

** [1]
** Rahimov, I., Random Sums and Branching
Stochastic Processes, **Springer Verlag**, **LNS**, **
V. 96**,
1995. (English).

**[2]
** Badalbaev, I., S., Rahimov, I.,
Non-Homogeneous Flows of Branching Processes, **Tashkent, “FAN”, **1993,
(Russian)

**
C5. Articles Published or Accepted in Refereed
Journals**

** ***Until
1991, because of well-known situation in the former USSR , my scientific results
were published mostly in Russian journals. But some of the articles were
translated and published in an English language version of the journal as well.
Below the star * indicates publications in the international journals which
are either in English or translated into English.*

**
[1]***
Badalbaev I., Rahimov I., Critical branching processes with immigration
decreasing intensity. **THEORY OF PROBAB. AND APPL. , 1978, V.23, No 2,
259-268**, (Russian and English translation).

**
[2]**
Badalbev I,. Rahimov I.,Limit theorems for the critical Galton-Watson processes
with immigration decreasing intensity, **IZVESTIYA of AS of UzSSR, 1978, No2,
9-14,** (Russian).

**
[3] **
Badalbaev, I., Rahimov, I., Bellman-Harris branching processes with
non-homogeneous immigration.**DOKLADI.AS of** **UzSSR,1978, No 9, 3-5**.(Russian)

**
[4]**
Rahimov, I., Some properties of supercritical branching processes with
immigration. In the book **“STOCHASTIC PROCESSES AND MATH. STATIST**. **
“Fan”, UzSSR,****Tashkent, 1978, p.**
**138-143**.(Russian)

**
[5]**
Rahimov, I., On Galton-Watson processes with increasing immigration. **
IZVESTIYA** **of AS of UzSSR**, **1978, N4, 22-31** (Russian).

**
[6]**
Rahimov, I., Some theorems for the continuous-time branching processes with
time-dependent immigration. In the book “**INVESTIGATIONS ON THE PHYSICAL AND
MATHEMATICAL** **SCIENCES”.** **
Tashkent. 1978, 57-63** (Russian).

**1979**

**
[1]**
Ibragimov, R., Rahimov, I., Limit theorems or supercritical multitype Galton-Watson
processes, **IZVESTIYA of** **AS of UzSSR, 1979, No1** ,** 9-15**,
(Russian).

**
[2]**
Badalbev I., Rahimov, I., Critical age-dependent branching processes with
immigration decreasing intensity. In the book “**LIMIT THEOREMS, RANDOM
PROCESSES AND APPL. “Fan”, ****
Tashkent, 1979**,**46-59,**
(Russian).

**
[3]**
Rahimov, I., Supercritical processes with immigration decreasing intensity, In
the book “**LIMIT** **THEOREMS, RANDOM PROCESSES AND APPL**., **“Fan”,
****Tashkent, 1979, 171-179
**(Russian).

**1981**

**
[1]**
Badalbaev I., Rahimov, I., Two limit theorems for Bellman-Harris processes with
immigration, In the book “**LIMIT** **THEOREMS FOR RANDOM PROCESSES AND
STATISTICAL INFERENCES”, “Fan**”, **Tashkent, 1981, 19-29 **(Russian).

**
[2]**
Rahimov, I., On branching processes with increasing immigration, **DOKL. AS of
Uzbek SSR, 1981,** **No1, 3-5 **(Russian).

**
[3]**
Rahimov, I., Juraev, Yu. On transient phenomena in Galton-Watson processes with
immigration decreasing intensity, In the book **“RANDOM PROCESSES AND MATH.
STATIST**.**,****Tashkent**
**Pedagogical Inst. 1981, 76-88**.(Russian).

**[4]**
Rahimov, I., Transient phenomena in branching random processes with
immigration, **IZVESTIYA of** **AS of UzSSR,1981,No5, 30-35 **(Russian).

**
[5]**
Rahimov,I.,Juraev,Yu., Transient phenomena in branching processes with
immigration decreasing intensity. In the **book “LIMIT THEOREMS FOR RANDOM
PROCESSES AND RELATING PROBLEMS”,** **“Fan”, ****
Tashkent, 1981, **
(Russian) .

**1982**

**
[1]**
Rahimov, I., Limit theorems for Bellman-Harris processes with immigration, In
the book **“RANDOM** **PROCESSES AND MATH STAT.“ ****
Tashkent Pedagogical Inst. 1982, 36-45,
**(Russian).

**1983**

**
[1]**
Rahimov, I.,Subcritical processes with non-homogeneous immigration, **IZVESTIYA
of AS of Uz** **SSR, 1983,No3**, **14-19**, (Russian).

**
[2]* **
Badalbaev I., Rahimov, I., Further results on
branching processes with immigration decreasing intensity. **THEORY** **OF
PROBAB.AND APPL., 1983, N4, 775-780,** (Russian and English)

**1984**

**
[1]***
Rahimov, I., On the limit theorems for a sequence of branching processes with
non-homogeneous immigration. **THEORY OF PROBAB. AND APPL. 1984, N4**,
(Russian and English trans.).

**
[2]**
Rahimov, I., Limit theorems for total number of particles in critical Galton-Watson
processes with immigration, In the book “**ASYMPTOTIC PROBLEMS OF PROBABILITY
DISTRIBUTIONS**”, **“Fan”, Tashkent, 1984, 106-119** (Russian).

**
[3]**
Rahimov, I., Uniform estimates in the limit theorems for branching processes
with immigration, **IZVESTIYA of AS of UzSSR 1984, No4**, **24-29 **
(Russian).

**
[4]**
Rahimov, I., A limit theorem for multitype age-dependent branching processes, **
IZVESTIYA of AS of** **UzSSR, 1984, No5, 28-40** (Russian).

**
[5]**
Rahimov, I., Limit theorems for multitype age-dependent branching processes with
immigration, **DOKL. of AS of UzSSR, 1984, No 4**, **3-5** (Russian).

**1985**

**
[1]**
Badalbaev,I. ,Rahimov, I., New limit theorems for multitype branching processes
with immigration decreasing intensity, **IZVESTIYA of AS of UzSSR, 1985, No2**,
**17-22, **(Russian).

**
[2]**
Rahimov, I., Limit distributions for integrals of Bellman-Harris processes with
non-homogeneous immigration, **IZVESTIYA of AS of Uzbek SSR, 1985, No5**, **
20-25** (Russian).

**
[3]**
Rahimov, I., Convergence of a sequence of branching processes with immigration
to the Jirina processes. In the book **“LIMIT THEOREMS FOR PROBABILITY
DISTRIBUTIONS”, ”Fan”, ****Tashkent,
1985, 134-148, **(Russian).

**1986**

**
[1]***
Rahimov, I., Critical branching processes with infinite variance and decreasing
immigration. **THEORY OF PROBAB. AND APPL. 1986, v.31, No1**, **98-110**
(Russian and English ).

**
[2]**
Rahimov, I., Kaverin, S., Class of limit distributions of critical branching
processes with state-dependent immigration, **DOKL. of AS of UzSSR, 1986, No1**,
**4-6, ** (Russian).

**
[3] **
Rahimov, I., Asymptotic behavior of the probability of hitting a fixed state
for Galton-Watson processes with decreasing immigration, **I, IZVESTIYA of AS
of UzSSR, 1986, No2; 33-38** (Russian).

**
[4]**
Rahimov, I., Asymptotic behavior of probability of hitting a fixed state for
Galton-Watson processes with decreasing immigration, **II, IZVESTIYA of AS of
UzSSR, 1986, No3, 38-46** (Russian).

**1987**

**
[1]***
Rahimov I., Limit theorems for random sums of dependent indicators and their
applications in the theory of branching processes. **THEORY OF PROBAB. AND APPL.
1987, v.32, N2**, **317-326,** (Russian and English) ** **

**
[2]**
Rahimov, I., On approximation of the distribution of the sum of a random number
of random terms, **DOKL.of AS of UzSSR,
1987, No1, 5-7 ** (Russian).

**
[3]**
Rahimov, I., Limit theorems for decomposable branching processes with
immigration, **DOKL. of AS of** **UzSSR, 1987, No6**, **5-7**
(Russian).

**
[4]**
Rahimov I., Salahitdinov R., Some generalizations of limit theorems for Galton-Watson
processes with infinite variance and decreasing immigration, **IZVESTIYA of AS
of UzSSR, 1987, No3**, **25-30 **(Russian).

**
[5]**
Rahimov I., New limit theorems for Galton-Watson processes with infinite
variance, **IZVESTIYA of AS of UzSSR, 1987, No4, 29-36 **(Russian).

**
[6]**
Rahimov, I., Kaverin, S., A method for proving limit theorems for branching
processes with state-dependent immigration, In the book **“PROBABILITY MODELS
AND MATH. STATIST”.1987,”Fan”, ****
Tashkent, 61-76 **(Russian).

**
[7]**
Rahimov, I., Statistical estimates for parameters of a subcritical Galton-Watson
processes with a reflecting screen. **“PROBABILITY MODELS AND MATHEM. STATIST.”
****Tashkent, 1987**,
**76-87 (**Russian.).

**
[8]**
Rahimov, I., Two limit theorems for multitupe age-dependent branching processes
with immigration, In the book **“ASYMPTOTIC METHODS IN MATH. STATIST.”,”Fan”,
****Tashkent**,
**1987, 97-108, **(Russian).

**1988**

**
[1]***
Rahimov, I., Sirazhitdinov S. Kh. Approximation of distribution of a sum in a
certain scheme for summation of independent random variables. ** DOKL. ACAD.
NAUK ****USSR, Vol.301, 1988, No1**,
**31-34** ( Russian and English).

**
[2]***
Rahimov, I., Local limit theorems for branching random processes with decreasing
immigration.** THEORY OF PROBAB. AND APPL. 1988, V.33, No2, 387-392**
(Russian and English trans.).

**
[3]**
Rahimov, I., A local theorem for Galton-Watson processes with immigration in
the case of uniform limit distribution, **”SERDICA” , 1988. V14,** **No3,**
**234-244** (Russian).

**
[4]**
Rahimov, I., Asymptotic of non-extinction probability of decomposable branching
processes with immigration, **IZVESTIYA of AS of UzSSR, 1988, No2**, **26-28**
(Russian).

**
[5]**
Rahimov, I., Sirazhdinov, S. Kh., Approximation of the distribution of a sum in
certain a scheme for summation of independent random variables, In the book “**ASYMPTOTIC
METHODS IN THE PROBABILITY THEORY AND MATH**. **STATIST.” “Fan”, **
**Tashkent, 1988**,
**136-151,** (Russian).

**
[6]**
Rahimov, I., Local limit theorem for branching processes with slowly decreasing
immigration, In the book “**ASYMPTOTIC METHODS OF PROBABILITY THEORY AND** **
MATH.STATIST”. ”Fan”, ****Tashkent,
1988, 121-136 **(Russian).

**1989**

**
[1]***
Rahimov, I., Asymptotic behavior of families of particles in branching random
processes. **DOKL**.** AKAD. NAUK ****
USSR, 305, 1989, No3**,
**540-542** (Russian and English).

**
[2]* **
Rahimov, I., Sirazhdinov S. Kh., Approximation of distribution of a sum in a
certain scheme for summation of independent random variables. **SOVIET MATH.
DOKL. Vol.38, 1989, No1**, **23-27** (English)

**
[3]***
Rahimov, I., Asymptotic behavior of families of particles in branching random
processes.** SOVIET** **MATH.DOKL.,Vol.39, 1989, N2**, **322-325**
(English).

**
[4]* **
Rahimov, I., On “A limit theorem for random sums of dependent indicators”,
Letter to the editors, **THEORY OF PROBAB. AND APPL., V 34, No 3, 613 (1989),
**(Russian and English).** **

**
[5]**
Rahimov, I., Kurbanov, S., Branching processes with non-homogeneous migration
and infinite variance, In the book **“FUNCTIONALS OF RANDOM PROCESSES AND
STATIST. INFERENCEES, “Fan”, ****
Tashkent, 1989**, **71-85**
(Russian).

**
[6]**
Rahimov, I., Asymptotic behavior of families of particles in branching random
processes, In the book **“FUNCTIONALS OF RANDOM PROCESSES AND STATIS.
INFERENCES”,”Fan”, ****Tashkent, 1989**,
**58-71,** (Russian).

**
[7]**
Rahimov, I., Branching processes with generalized immigration,** IZVESTIYA of
AS of UzSSR, 1989, No2, 35-40 **(Russian).

**1990**

**
[1] **
Rahimov, I., Asymptotic behavior of the sum of randomly indexed processes, In
the book **“ASYMPTOTIC PROBLEMS OF PROBABILITY THEORY AND MATH. STATIST**.” “**Fan”,
****Tashkent, 1990, 79-92 (**Russian).

**1991**

**
[1]***
Rahimov, I., Sampling sums of dependent variables, mixtures of infinitely
divisible distributions and branching stochastic processes. **DISCRETE
MATHEMATICS, 1991, V. 3, No2, 236-257 **(Russian and English).

**1992**

**
[1]***
Rahimov, I., The General branching processes with immigration depending on
reproduction, **THEORY** **OF PROBAB. AND APPL., 1992, V. 37, No 3**, **
513-525** (Russian and English ).

**
[2]***
Rahimov, I., Sampling sums of dependent random variables, mixtures of infinitely
divisible laws and branching random processes, **DISCRETE MATH. AND APPL.,
1992, V.2, No3**, **337-356** (English).

**
[3**]
Rahimov, I., Critical Bellman-Harris processes with infinite variance and
decreasing immigration, **UZBEK MATHEMATICAL JOURNAL, 1992, No2, 22-31**
(Russian).

**
[4]**
Rahimov, I., Khalilov, V., Abdullev, A., On multitype branching processes with
increasing immigration,** UZBEK** **MATH. JOURNAL, 1992, No 5-6, 63-70,
**(Russian)

**1993**

**
[1]***Rahimov,
I., Critical Processes with infinite variance and increasing immigration , **
MATHEMATICA**L **NOTES, 1993, V.53, No 6, P.97-107**, (Russian and English
).

**1994**

**
[1]**
Rahimov, I., Kurbanov, S., Subcritical Branching Processes with Decreasing
Immigration and Infinite Variance, **UZBEK MATHEMATICAL JOURNAL, 1994, No 1, P.
51-57** (Russian).

**
[2]**
Rahimov, I., Mixtures of Infinitely Divisible Laws as Limit Distributions for a
Special Form Multiple Sums, **UZBEK MATHEMATICAL JOURNAL, 1994, No 4, P. 43-50**
(Russian).

**1996**

**
[1]***
Rahimov, I., Teshabaev, A., Decomposable branching processes with decreasing
immigration, **JOURNAL OF** **APPLIED STATISTICAL SCIENCE, 1996, V.3, No
2/3, P.169-190**, ( English).

**
[2]* **
Rahimov, I., Rosihan, M., Ali,** **On the
Extinction of Families, **The ISOSS Bulletin, 1996, No4, p.7** (English).

**1997**

**
[2]* **
Rahimov, I., Kassim, S. Record Values
Concerning a Family of Age-Dependent Processes, **THE BULLETIN OF** **THE
MALAYSIAN MATHEMATICAL SOCIETY, 1997, Vol. 20, No. 2, 57-66**. ( English).

**
1998**

**
[1]***
Rahimov, I., Multitype Processes with Reproduction-Dependent Immigration, **
JOURNAL OF APPLIED PROBABILITY, 1998, V. 35, No. 2, P. 281-292 **(English) **
(ISI) PDF**

**
[2]***
Rahimov, I., Hasan, H., Exceedance Problems Concerning a family of Branching
Processes, **PAKISTAN JOURNAL OF STATISTICS, 1998, V.14, No. 1, P. 37-47** ,
(English).

**
[3]***
Rahimov, I., Hasan, H., Limit Theorems for Exceedances of a Sequence of
Branching Processes, **BULLETIN OF THE MALAYSIAN MATHEM. SOCIETY,** **V.
21, 1998, p. 37-46, **(English).

**
[4]* **
Rahimov I., Teshabaev A. Some Limit Theorems for Decomposable Branching
Processes, “**Istatistik”, JOURNAL OF TURKISH STAT. ASSOC., 1998, V.1, No 1, p.**
**29-42**.** **(English).

**
1999**

**[1]*
** Rahimov, I., Yanev, G., On Maximum Family
Size in Branching Processes, **JOURNAL OF** **APPLIED PROBABILITY, V.36, No
3, p. 632-643 (ISI),** PDF

[2]* Rahimov, I., Hasan, H. On the number of
productive ancestors in large populations, **Istatistik”, JOURNAL OF TURKISH
STAT. ASSOC., 1999, V.2, No 2, p. 123-133. **(English).

**2000**

**
[**1**]*
**Rahimov, I., A Limit Theorem for Multitype General Branching Processes with
Generalized Immigration, **JOURNAL OF APPLIED STATISTICAL SCIENCES**, **
V.9, No 2, 2000, P.105-122.** ( English).

[2]* Rahimov I., On the Non-Extinction
Probability of Branching Diffusions, **JOURNAL OF APPLIED STATISTICAL SCIENCES**,
**V.9, No 3, 2000, P. 223-236.** ( English).

[3]* Rahimov I., Al-Sabah W. S. Branching
processes with decreasing immigration and tribal emigration. **A. JOURNAL OF
MATH. SC., V.6, No 2, 2000, P. 81-97. PDF**

**2001**

**
[**1**]*
**Rahimov I. Approximation of exceedance processes in large populations. **
STOCHASTIC MODELS, 2001, V. 17, No 2, P. 147-156.
PDF**

[2]* Rahimov I., Ahsanullah M. Records generated
by total progeny of branching stochastic processes. **FAR EAST JOURNAL OF
THEORETICAL STATISTICS, V. 5, No 1, 2001, P. 81-94.**

**
[**3**]*
**Rahimov I., Muttlak H. A. Random ranked set samples. **PAK. JOURNAL OF
STATISTICS, V.17, No 1, 2001, P. 51-66. **

[4]* Rahimov, I., Immigration-Branching
Diffusions ad their extinction, In the book **“APPLIED** **STATISTICAL
SCIENCE V”, ** Ahsanullah M, Kennyon J., Sarkar S. K., Editors, ** Nova
Science Publishers, 2001, P. 97-118. (English).**
PDF

**2002**

[1]* Rahimov I.
Limit Distributions for Generalized Reduced Branching Processes. **
INTERNATIONAL MATHEMATICAL JOURNAL, V.2, No 11,
2002, P. 999-1009.**

[2]* Rahimov I.
Multitype Generalized Reduced Processes. **
JOURNAL OF STATISTICAL THEORY AND APPLICATIONS,
V. 1, No. 3, 2002, P. 149-162.**

**
2003**

[1] * Rahimov I. Random Sums of Independent
Indicators and Generalized Reduced Processes. **
STOCHASTIC ANALYSIS AND APPLICATIONS, Vol. 21,
No 1, 2003, P. 205-221.**

[2]* Rahimov I., Muttlak H. A. Estimation of
the population mean using random selection in ranked set samples. **
STATISTICS AND PROBABILITY LETTERS, Vol 62,
2003, P. 203-209.**

[3]* Rahimov I., Ahsanullah M. Records Related
to Sequence of Branching Stochastic Processes . **
PAKISTAN JOURNAL OF STATISTICS, Vol 19, 2003,
No 1, P. 73-97** .

[4]* Rahimov I.,
Muttlak H. A. Investigating the Estimation of the Population Mean Using Random
Ranked Set Samples. **NONPARAMETRIC
STATISTICS, 2003, V 15, No 3, P. 311-324.**

**2004**

[1]* Rahimov I.,
Malik M. Asymptotic Bahavior of Expected Record Values. **
PAKISTAN JOURNAL OF STATISTICS, Vol 20, 2004, No.
1, P. 129-135 .**

** **[2]*
Rahimov I., Muttlak H. Random
Sums of Random Vectors and Multi Type Families of Productive Individuals. **
INTERNATIONAL JOURNAL OF MATHEMATICS AND MATHEMATICAL SCIENCES** **, Vol.
2004, No 19, P. 975-990.**

[3]* Rahimov I.,
Limit Theorems for the size of Subpopulation of Productive Individuals.
**STOCHASTIC MODELS, VOL. 20, 2004, no 3, P.
261-280.**

[4]* Rahimov I.
On the Number of Large Families in a Branching population. **
Dynamical Systems and Applications-2004,
Proceedings of the Internationl Conference,**
**Antalya, Turkey, July 5-10, 2004, GBS
Publishers & Distributors, P. 582-597. **

**2007**

[1]* Rahimov I.,
Asymptotic Behaviour of a Controlled Branching
Process with Continuous State Space. **
STOCHASTIC ANALYSIS AND APPLICATIONS**,
**(ISI)** 2007, **V. 25, No 2, P. 337-352.
PDF**

[2]* Rahimov I. On a Stochastic
Model for Continuous Mass Branching Population.**
MATHEMATICS AND COMPUTERS IN SIMULATION" (ISI ), 2007, V 76/1-3, P.
171-176. **
PDF

[3]*** **Rahimov I., Kurbanov
S. On the Number of Productive individuals in the Galton-Watson Process with
Immigration. **MATHEMATICS
AND COMPUTERS IN SIMULATION" (ISI ), 2007, V 76/1-3, P. 177-180.**

[4]* Rahimov I., Functional Limit
Theorems for Critical Processes with Immigration "
**ADVANCES IN APPLIED
PROBABILITY, 2007, V. 39, No 4,
P.**
**
1054-1069
(ISI)
PDF**

[5]* Rahimov I., Sabah W.
Controlled branching processes with continuous states. **JOURNAL OF APPLIED
PROBABILITY AND STATISTICS, 2007, V 2, No 2, 123-137**.
PDF

**
2008**

[1]* Rahimov I., Sabah W. Limiting
behavior of a generalized branching process with immigration. **STATISTICS
AND PROBABILITY LETTERS**,
2008, **V. 78/3, 225-230** **
(ISI)**

[2]* Rahimov
I., Chanane B. Branching processes with incubation. **STOCHASTIC MODELS, (ISI)**
**V.
24, No
1 , 2008 , P. 71 - 88 **
.

[3]* Rahimov
I. Asymptotic distribution of the CLSE in a critical process with Immigration.
**STOCHASTIC PROCESSES AND
THEIR APPLICATIONS, (ISI) V 118, 1892-1908. **

[4]* Rahimov I.
Two-type branching processes with sub-exponential life-spans and SIR epidemic
models. **STOCHASTIC ANALYSIS AND
APPLICATIONS**, **2008, V. 26, No 5,
925-940, (ISI) .**

[5]* Rahimov
I., Malik M. Records Generated by a Sequence of Branching Processes with
immigration. **JOURNAL OF APPLIED STATISTICAL SCIENCE**,** 2008, V.16, No
1, P. 27-36.**

[6]*
Rahimov I. Large families and exceedances in branching processes, Invited
chapter for the book **
"RECORDS AND BRANCHING
PROCESSES",
2008, Ch. 5, P. 101-122,
Nova Science Publishers Inc. (USA) . **

[7]* Rahimov I.
Deterministic approximation of a sequence of nearly critical branching
processes. **STOCHASTIC ANALYSIS AND
APPLICATIONS**, **2008, V. 26, No 5,
1013-1024, (ISI) .**

** **

[1]* Rahimov
I. Limit distributions for weighted estimators of the offspring mean in a
branching process. **TEST (ISI), Published online at ****
http://dx.doi.org/10.1007/s11749-008-0124-8
**.

[2]* Rahimov
I. Asymptotically normal estimators for the offspring mean in the branching
process with immigration. **COMMUNICATIONS IN STATISTICS:THEORY AND METHODS,
2009, V. 38, No 1, P. 13-28 (ISI).**

[3] Rahimov I.
Approximation of fluctuations in a sequence of nearly critical
branching processes. **STOCHASTIC MODELS, (ISI) **(accepted).

[4] Rahimov I.,
Omar M. Validity of the bootstrap in the critical process with a non-stationary
immigration. **JOURNAL OF ** **NONPARAMETRIC STATISTICS (ISI) **
(submitted).

[5] Rahimov I.
Bootstrap of the offspring mean in the critical process with a non-stationary
immigration. **STOCHASTIC PROCESSES AND THEIR APPLICATIONS (ISI)**
(submitted)

**
C6. Dissertations**

**
[1]**
Limit Theorems for Branching Processes with Time-Dependent Immigration, **
Tashkent, 1978**,** (PhD Thesis, Russian).**

**
[2]**
Limit Theorems for Branching Processes with Time-Dependent Immigration, **
(Abstract of** **PhD Dissertation), Romanovski Math. Institute, **
**Tashkent, 1978**.
(Russian).

**
[3]**
Summation of a Random Number of Random Terms and Branching Stochastic Processes,
**Steklov Math. Institute, ****
Moscow, 1991, **(Doctor of Phisical and
Mathematical Sciences** **Thesis, Russian).

**
[4]**
Summation of a Random Number of Random Terms and Branching Stochastic Processes
(**Abstract of Doctor of Science Dissertation), Steklov Mathematical Institute
Moscow, 1991**, (Russian).

** **

**
C7. Papers or Abstracts Published in Conferences**

**
1977**

**[1]**
Badalbaev I., Rahimov, I., Some theorems for the continuous-time branching
processes with time-dependent immigration. Proceedings 2nd Vilnius Conference on
**Theory of Probab. and Math.** **Statistics,** Vilnius, 1977, (Russian)

**
1978**

**
[1]**
Rahimov, I., Critical Galton-Watson processes with increasing immigration.**
Proceedings of ****Tashkent**
**Conference of Young Scientists, ****
Tashkent, 1978.**
(Russian.)

**
1981**

**
[1] **
Rahimov, I., Critical processes with non-homogeneous immigration. **IIIth
International ****Vilnius**
**Conference on THEOR. OF PROBAB. AND MATH. STATIST**., Vilnius 1981.

**
1982**

**
[1]***
Rahimov, I., On the branching random processes with immigration in a periodic
environment. Abst.of Comm.**IV USSR-Japan Symposium on Probability Theory and
Mathematical Statistics V.II**, **
Tbilisi, 1982, 2. (**English)

**
1985**

**
[1]**
Rahimov, I., Local limit theorems for Galton-Watson processes with decreasing
immigration. Abstr. Comm. **International Seminar on Mathem. Modelling.and
Stability of Stochastic Models. Varna**, **Bulgaria, 1985**, (Russian).

**
[2**]
Rahimov, I., Asymptotic of local probability of Galton-Watson processes with
decreasing immigration. **IVth International ****
Vilnius Conference on Theory of probab. and
Math. Statist., Vilnius 1985 **
(Russian).

**
1986**

**
[1]* **
Rahimov, I., Sums of dependent indicators and branching stochastic processes.
**1 ^{st} World Congress of**

**
1987**

**
[1]**
Rahimov, I., Local limit theorems for Markov branching processes in a uniform
limit distribution case. **International school “ERGODIC THEORY OF MARKOV
PROCESSES”, Kizil, 1987**, (Russian).

**
1989**

**
[1]***
Rahimov, I., The methods of summation of random variables in theory of branching
processes. **V th** **International ****
Vilnius Conference on THEORY OF PROBAB.AND
MATH.STATIST**.,**Vilnius,
1989,** (English).

**
[2]**
Rahimov, I., Limit theorems for some characteristics of population in a Markov
branching process. **International
School “ERGODIC THEORY OF MARKOV PROCESSES”; Kiev,1989**,
(Russian).

**
1991**

**
[1]***
Rahimov, I., General branching processes with general immigration process. **
VI USSR-JAPAN** **SYMPOSIUM ON PROBABILITY THEORY AND MATHEM. STATISTICS,
Kiev, August**, **1991**, (English).

**
1993**

**
[1]***
Rahimov, I., Branching Processes as a Sums of Dependent Random Variables., **
BRANCHING** **PROCESSES, ****First
World Congress, 1993, Varna, Bulgaria. Springer-Verlag, Ser. LNS**,
**Vol. 99, 1995, p. 58-66, Editor Ch. Heyde**, (English).

**
[2]* **
Rahimov, I., Yildirim, F., Teshabaev, A., ** ** Non Homogeneous Decomposable
Branching Processes., **BRANCHING PROCESSES**, **First World Congress, Varna,
Bulgaria, 1993. Springer-Verlag, Ser. LNS, Vol. 99, 1995, p. 67-76, Editor Ch.
Heyde, (**English).

**
[3]* **
Rahimov, I., ** ** Weak convergence of certain multiple sums of dependent
random variables. **Probabilistic Methods in Discrete Mathematics**, **
Progr. Pure Appl. Discrete Math., 1, VSP, ****
Utrecht, 1993,**
**376-385** (Russian and English).

**
1994**

**
[1]***
Rahimov, I., Branching Processes with Immigration., **Fourth ISOSS Conference
**, **Lahore, Pakistan**,
**1994.** (English)

**
[2]***
Rahimov, I., Decomposable Branching Processes with Decreasing Immigration. **
Fourth** **ISOSS** **Conference , ****
Lahore, Pakistan, 1994**.
(English)

**
[3]**
Rahimov, I., Record values of a family of branching processes, **Classical and
Modern Branching** **Processes**, **IMA Workshop, (by Invitation only)
June , 1994, Springer-Verlag,** **Ser.”IMA Volumes in Mathematics** **and
its Applications”, Vol. 84, 1996**, Editors K. Athreya and P Jagers, P.
285-295. (English).

**
1996**

**
[1]* **
Rahimov, I., Random Sums in the Theory of Branching Processes, Second
International Workshop in **Stochastic Modeling and Experimental Design at St.
Petersburg, 1996, P.305-309**. (Invited Paper, English).

**
[2]***
Rahimov, I., Ahsanullah, M., Record Properties of a Family of Branching
Processes, **Second International Workshop in** **Stochastic Modeling and
Experimental Design at St. Petersburg, 1996, P.300-304.** (English).

**
[3]* **
Rahimov, I., ** ** Summation of a Random Number of Terms and Branching
Stochastic Processes, **Proceedings of Fifth Conference on Statistical
Sciences, Vol. I, 1996, ****Malang,
Indonesia**, (English, Invited Paper).

**
[4]***
Rahimov, I., A Transfer Theorem for Multitype Processes and its Applications,
Proceedings of the Workshop “**Recent Development in Applied Statistics”, **
**Malang, Indonesia, 1996**,
(English, Invited Paper).

**
[5]* **
Rahimov, I., Hasan, H., On the First Exceedance
of a level by Family of Branching Processes, Proceedings of **Fifth Conference
on Statistical Sciences, Vol. II, 1996, Malang, Indonesia** (English).

**
1997**

**
[1]***
Rahimov, I., Yanev, G., Maximal Number of Direct Offspring in Simple Branching
Processes, **Proceedings of The Second World Congress of Nonlinear Analysis,
July 10-17, 1996, Athens, Greece, NONLINEAR ANALYSIS, 1997, V.30, No 4,
p.2015-2023**. (English, Invited Paper).

**[2]***
Rahimov, I., Transfer Limit Theorems for Multitype General Branching
Processes, **51 ^{st} Session of the International Statistical
Institute, Istanbul, August 1997, Bulletin of the International Statistical
Institute, Tome LVII, Book 2, 1997, P. 583-584.**

**[3]*
**Rahimov, I., On a Transfer Theorem
for Multitype General Branching Processes, **ATCM 97** **(The** **Second
Asian Technology Conference in Mathematics), USM, Penang, Malaysia**, 1997,
P.8.

**
[1]***
Rahimov, I., A Transfer Theorem for Multitype Processes and its Applications,
International Workshop (Malang, Indonesia, 1996) on **“APPLIED** **
STATISTICAL SCIENCE II”, ** ** Nova Science Publishers, 1997, P. 37-56.
(English).**

**
2001**

**[1]*
**Rahimov I., Muttlak, H. A. Random Ranked
set Samples. **Proceedings of the ICCS –VII, ****
Lahore, January 2001, P. 145-161. (English).**

**
2004**

**[1]*
**Rahimov I., On the Number of Large Families
in a Branching Population. **Proceedings
of the International Conference "Dynamical Systems and Applications", July
5-10, Antalya, Turkey, P. 582-597**

**
2005**

**[1]*
**Rahimov I., Kurbanov S. On the Number of
Productive individuals in the Galton-Watson Process with Immigration.
Proceedings: **MODELLING 2005, Third IMACS Conference on Mathematical
Modelling. July 4-8, 2005, Pilsen, ****
Szech Republic (accepted).**

**[2]*
**Rahimov
I. On a Stochastic Model for Continuous Mass Branching Population.Proceedings:
** MODELLING 2005, Third IMACS
Conference on Mathematical Modelling. July 4-8, 2005, Pilsen, Szech Republic
(accepted).**

** **

** **

** **