ASYMPTOTIC PROPERTIES AND LIMITING DISTRIBUTIONS OF BRANCHING PROCESSES

SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 10 OCTOBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 158 ASYMPTOTIC PROPERTIES AND LIMITING DISTRIBUTIONS OF BRANCHING PROCESSES D. Aroev1, Kh.U. Abdusamatova2 PhD, Associate Professor, Dean of the Faculty of Exact Sciences and Engineering, Kokand State University1 Doctoral Student, Kokand State University2 https://doi.org/10.5281/zenodo.17553665 Abstract. This article examines issues related to probability theory and the theory of branching processes. The main focus is on probability functions, generating functions, and their asymptotic properties. The application of Kolmogorov’s theorem is also demonstrated, and the limiting properties of distributions are investigated. The results obtained are of great significance for the advanced study of mathematical statistics and the theoretical foundations of branching processes. Keywords: branching process, generating function, probability theory, Kolmogorov theorem, distribution, asymptotic properties. Introduction. The theory of branching processes occupies one of the central places in modern probability and statistical science. It emerged from the need to mathematically describe reproduction processes, population evolution, and the stochastic dynamics of systems in which individual elements generate new elements according to probabilistic laws. The idea of branching processes was first formulated in the works of F. Galton and G. Watson in the 19th century in the context of studying the extinction of family names. Later, significant contributions to the development of this field were made by such prominent scientists as A. N. Kolmogorov, T. E. Harris, K. B. Athreya, B. A. Sevastyanov, and many others who laid the fundamental foundations of modern theory. Branching processes have found wide application in various fields of science and technology. In biology, they are used to model the growth of cell cultures, population genetics, and the spread of epidemics; in physics — to study nuclear reactions and particle diffusion processes; in cybernetics and computer science — to analyze algorithms and network structures; and in demography and ecology — to forecast population sizes. Branching processes are of particular importance in the theory of random structures, where they serve as a natural tool for analyzing the asymptotic behavior of random variables and distributions. One of the key problems in the study of branching processes is the investigation of their asymptotic properties and limiting distributions. In particular, an important direction is the analysis of the probability of process extinction, the conditions for the existence of non-degenerate limits, and the application of limit theorems such as Kolmogorov’s theorem. Generating functions play a crucial role in this regard, as they allow for a compact description of the probabilistic characteristics of the process and enable the derivation of rigorous mathematical results. This article is devoted to the analysis of the asymptotic properties of branching processes defined using generating functions. The paper examines the conditions for the existence of limiting distributions, studies the probabilities of extinction and survival of the process, and presents proofs of key inequalities and formulas. Moreover, the results obtained deepen the understanding of the SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 10 OCTOBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 159 limiting behavior of random processes, which is of interest not only for theoretical probability but also for applied research in related fields. Let there be       0 1, , 0Z Z t t    (b.p.) generated by a generating function       0 1, , 0Z Z t t    Then the generating function of the given process     , , 1 Zt F t x Ex x has the form of a fractional-linear function. Let us represent the generating function     , , 1 Zt F t x Ex x in the form       2 11 2 b f x a x x    (1) where     ' '' 1 , 1a f b f are constants. Using the equality     ' '' 1 , 1a f b f , we can write   1 2 0 2 2 2 0 2 1 2 2 , 2 ,a p p p p p p p b p p a b           (2) It is well known that another important characteristic of a branching process is the probability of extinction   0 min 1,qx , where 0 x is the root of the equation   0fx (3) Besides the trivial root 1x , equation (3) has another root 01x when 0a , and it is expressed by the formula 0 2 1a xb  Hence, we have the following cases: 1) 0 2 0, 1 1 a ax b     ; 2) 0 2 0, 1 1 a ax b     , then the root is unique 0 2 0, 1 1 a ax b     ;     02 02 1, 0 , 2 1 , 0 при a p p qaпри a p p b         , then 0a . Now let us prove the validity of the following inequalities 21 a b or 2 b a (4) for   '10fa , for any generating function   fx that satisfies   '' 1fb . Indeed, we have       '12 '' 23 1 2 ... ..., 1 2 6 ... 1 ... n n a f p p np b f p p n n p              where 10, 0, 2 n p p n   . Therefore,   1 22 1, 22 nn nn nn bp a p np        SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 10 OCTOBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 160 Consequently,     2 3 2 1 23 2 2 3 ... ... 2 11 3 ... ... 222 n nn nn n np p p np ap abn n n n bp p p p                    It is obvious that   21 112 2 n n np nn n p   when 5n . Hence,   22 1 1 22 nn nn nn np p      The latter confirms the validity of relations (4). Now let us return to the continuous-time branching process generated by the generating function   fx , defined by the given formula. From the above theorem, it follows that the probability of the continuation (survival) of the process       0 111 2 at at e Q t P t be a     (5) If 0a , i.e., the process is subcritical, then from (1.9.5) it follows that   0 , 12 at at ee Q t t bx a      (6) where 012 b xa  is the root of equation   0fx . Relation (1.9.6) corresponds to the classical Kolmogorov theorem, according to which       1, at Q t Ke o t t   (7) where K is a positive constant depending on the form of the generating function   fx . By comparing the limiting relations (6) and (7), we arrive at the conclusion that in the case when the generating function   fx is defined by formula (6), the constant K in (6) is equal to 1 12 Kb a   Let 0a be   fx the generating function defined by formula (1.7.6). In this case, the extinction probability of the critical process is given by         0 22 1 1 1 , 2 Q t P t o t bt bt         (8) and the constant ( C ) in the corresponding Kolmogorov theorem is 2 Kb  . Next, let us note the following. If 0a , then from the asymptotic formulas (6) and (8) it follows that     01, 0, , 1,2,... k P t P t t k    (9) SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 10 OCTOBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 161 for any 1k . Consequently, the limiting distribution for the sequence of distributions         , 0,1,2,... k P t P Z t k k   (10) is a degenerate distribution ( F_0 ). As the following theorem shows, in order for the sequence of distributions (9) to have a nondegenerate limiting distribution, it is necessary to consider conditional distributions.         */ 0 , 1,2,... k P t P Z t k Z t k    . Event             1 0 \ 0 k Z t Z t k Z t         means the non-extinction of the process     ,0Z t t   at time t . The probability of this event                 0 1 0 1 0 1 k P Z t P Z t k P Z t P t Q t             From the last equality and the asymptotic relation (6), the limiting statements (9) follow. Theorem 1. Let     ,0Z t t   be a subcritical branching process with continuous time and a generating function   fx defined by the given formula. Then, for any 1k , the limits exist.       * lim / 0 k tP Z t k Z t p     (11) Generating function   ** 1 ,1 k k k F x p x x     has following view     * 00 1 x Fx x x x  where 0 x is the root of the equation   0fx not equal to one 01x . Proof. First, let us present the explicit formulas for the distribution       , 0,1,2,... k P t P Z t k k   The formulas presented in B. A. Sevastyanov’s book [ ] (Ch. I, §8, p. 44) are inaccurate. They lack the multiplicative expression   exp at . It has been shown above that the probability generating function (p.g.f.) of the continuous-time branching process     ,0Z t t   , defined by the p.g.f.     ,0Z t t   has the form         1 , 1 , 0 1 1 1 2 at at ex F t x a bex a        The last equality can be transformed into the formula        1 ,1 11 at t ex F t x x     (12) when   1, 21 at t t t be a        . SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 10 OCTOBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 162 Using the expansion   0 1 1 k k x x       , equality (12) can be rewritten as         1 ,1 11 at at k k tt ee F t x x x          . From this equality, it follows that for 0a and 1k             1 02 1 2 1, 1 1 1 1 11 22 2 k at at at k at at at be ee a P t P t bb b ee e aa a                   (13) Now, let us calculate the probability generating function (p.g.f.)   *,F t x of the conditional probabilities         */ 0 , 1 k P t P Z t k Z t k    Next, we have                       *1 1 , ,0 , , / 0 1 n k kk k P t x F t x F t R t x F t x P Z t k Z t x Q t Q t Q t              (14) when         , 1 , , ,0R t x F t x Q t R t   . It has been proven above that               0 1 , 1 , 1 1 1 1 1 1 22 at at at at ex e F t x Q t P t bb e x e aa            It is evident that the following asymptotic formulas hold: for 0,at     0 12 at at ee Qt bx a   (15)         1 , 1 , 11 2 at ex R t x F t x bx a      (16) Then using formula 00 1 , 1 22 bb xx aa     from (15) and (16) we get for 1x       0 0 11 2 ,1 1 lim , 1 1 122 t b xa R t x x x bb Q t x x x aa                . (17) From the last formula (17), we finally have       * 00 1 1 11 1 1 x xx Fx x x x x x             (18) SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 10 OCTOBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 163 t is evident that the generating function   * Fx is a probability generating function (p.g.f.)   *11F . Theorem 1 is proved. Remark 1. The stated Theorem 1 can also be proved in a different way, using the theorem on the continuous correspondence between the set of generating functions and the set of distributions (see [Sev.] Theorem 2, Ch.I, §3, p.21). According to this theorem, in order that for each fixed 1n   ** lim nn tp t p   it is sufficient that for any 1x     ** lim , tF t x F x   At the same time, in Theorem 3 of the book [Sev.] (Ch. I, §3, p. 23), an estimate is provided. So we have                     11 , 1 , 1 11 2 1 ,0 1 1 1 1 1 22 111 2 at at at at at t at be R t x F t x e x xx a bb Q t F t e x e x e aa x be a                 when     1 2 11 2 at tat be a be a     Consequently, the probability generating function (p.g.f.) of the distribution     *,1 k P t k            *,1 1 , 1 1 11 t k k tt R t x x x F t x P t x Q t x x             (20) Next                 0 00 1 211 11 1 1 1 1 1 222 1 1 1 11 1 at tat at at at at at be a bbb e e O e aaa Oe x x O e x O e                       (*) Here, a basic asymptotic relation is used   11 , 0 1O x x x    Taking the above into account, the chain of equalities (*) has     0 00 1 1 1 , , at at tt x O e O e t xx          (21) SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 10 OCTOBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 164 we place 0 0 1x x    ,     ** lim , t F x F t x   . Then from the equality (1.9.20) and (1.9.21) it follows that             ** 11 11 sup , sup 11 tat t xx t xx F t x F x O O e xx            (22) From the last relation, the following theorem follows. Theorem 2. Let   ,0 t Zt be a subcritical branching process with generating function   fx     '10af . Then the following estimates hold:       ** 1 sup , at xF t x F x O e   , where     * 00 1 x Fx x x x  , 0 x root equation   0fx , which is more than 1 . From this, Theorem 2, in view of (19), follows: Theorem 3. Let   ,0 t Zt be a subcritical branching stochastic process. Then the local limit theorem holds:     ** sup , at nn nP t p O e t   where the probability distribution   *,1 n pn is generated by the probability generating function     * 100 1 n n n x F x p x x x x      has equality place     * 100 1 n n n x F x p x x x x      Hence, the limiting distribution   *,1 n pn forms a geometric distribution *1 00 ,1 n n P b n    with parameters   0 0 0 0 0 00 1 1 , , , x bb xx    . Conclusion. The conducted research allows us to draw several important conclusions regarding the behavior of branching processes and their asymptotic properties. It has been shown that generating functions serve as an effective tool for describing the dynamics of the process and identifying its limiting characteristics. The established relationships and proven theorems make it possible to rigorously describe the probability of extinction, the conditions for the existence of non-degenerate limiting distributions, as well as the specifics of asymptotic behavior in critical and subcritical cases. Of particular importance are the results related to the application of Kolmogorov’s theorem, which provides a deep understanding of the limiting nature of distributions in branching processes. It has been demonstrated that in several cases, limiting distributions may degenerate, and only by considering conditional probabilities can a non-degenerate limit exist. The obtained estimates and limit assertions broaden the analytical capabilities and allow the theory of branching processes to be applied to more complex practical problems. Thus, the study of the asymptotic properties and limiting distributions of branching processes not only holds fundamental importance for the further development of probability theory but also opens new perspectives for practical applications in biology, physics, demography, cybernetics, and other scientific fields. SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 10 OCTOBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 165 REFERENCES 1. Sevastyanov, B. A. Branching Processes. Moscow: Nauka, 1971. 2. Kolmogorov, A. N., & Fomin, S. V. Elements of the Theory of Functions and Functional Analysis. Moscow: Nauka, 1981. 3. Gnedenko, B. V. Course in Probability Theory. Moscow: Nauka, 1988. 4. Athreya, K. B., & Ney, P. Branching Processes. Springer, 1972. 5. Harris, T. E. The Theory of Branching Processes. Springer, 1963. 6. Kendall, D. G. Stochastic Processes. Moscow: Mir, 1972. 7. Feller, W. An Introduction to Probability Theory and Its Applications. Moscow: Mir, 1984. 8. 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