Mathematics & Physics 2015, 8(4), 394–405 УДК 519.218.2 On Long-time Behaviors of States of Galton-Watson Branching Processes Allowing Immigration Azam A.Imomov∗ State Testing Center under the Cabinet of MRU Karshi State University Kuchabag, 17, Karshi city, 180100 Uzbekistan Received 13.06.2015, received in revised form 01.08.2015, accepted 02.09.2015 We observe the discrete-time Branching Process allowing Immigration. <...> Limit properties of transition functions and their convergence to invariant measures are investigated. <...> In the critical situation a speed of this convergence is defined. <...> Keywords: branching process, immigration, transition functions, invariant measures, ratio limit property, rate of convergence. <...> The state sequence {Xn} is a homogeneous Markov chain with state space on N0 and can be expressed recursively as Xn−1 Xn = ∑ k=1 ξnk +ηn, for n ∈ N, where independent and identically distributed (i.i.d.) random variables ξnk denote the offspring number of k-th individual in the (n−1)-th generation, and i.i.d. variables ηn are not depend on ξnk interpreted as number of immigrants-individuals at the moment n. <...> Each individual reproduces independently of each other and according to the offspring law pk := P{ξ11 = k}. <...> Throughout the paper we assume p0 > 0 and ∑ hj = 1. j∈N0 functions We denote S ⊆ N0 to be the state space of the chain {Xn}. <...> All rights reserved c – 394 – P(i) n (s) := EisXn =∑ j∈S p(n) ij sj Azam A.Imomov On Long-time Behaviors of States of Galton-Watson Branching Processes . is probability generating function (PGF). <...> Process {Xn} is classified as sub-critical, critical and supercritical if A < 1, A = 1 and A > 1 accordingly. <...> The above described evolution process of individuals was considered first by Heathcote [3] in 1965. <...> Further long-term properties of states and a problem of existence and uniqueness of invariant measures of GWPI were investigated in papers of Seneta [8, 10, 11], Pakes [4–7] and by many other authors. <...> Therein some moment conditions for PGF F(s) and G(s) was required to be satisfied. <...> In aforementioned works <...>