| Theory of probability is a branch of mathematics that studies random phenom-ena quantitatively.Its origin is related to the issues of gambling.It has gradually developed into a strict scientific with deeply and widely studied and has a wide range of applications in the natural sciences,technical sciences,management sciences,eco-nomics,finance and other fields.Influenced by physics in the early 20th century,people began to research on stochastic processes.Stochastic process refers to the entire set of random variables whose variation depends on the parameters.There are many kinds of concrete models of stochastic processes.To now,many scholars have conducted in-depth research and many excellent results have been obtained.The Galton-Watson process considered in this paper is a kind of branching pro-cess in stochastic processes,and also a special Markov process.Its original source is a stochastic process model established by Galton and Watson in the 19th century when discussing the inheritance and extinction of British noble surnames.Since then,many scholars have researched and promoted this model.In which the classic model of discrete time and discrete state is often used to discuss population reproduction and particle splitting.Since the 1940s,scholars’research on the branching process has reached a small climax.Kolmogorov and Dmitriev(1947)considered the branching stochastic processes by the method of Markov process.Kolmogorov and Savostyanov(1947)discussed the calculation of final probabilities for branching processes.Yaglom(1947)got certain limit theorems of the branching processes.Harris(1948)further dis-cussed the model of branching processes,and got many results including the asymptotic properties of the moment-generating function,the number of generations of extinction and the estimation of parameters.Moreover,he published the book "The theory of branching processes" in 1963,which introduced the results of early branching processes in great detail,laying a solid foundation for later research on branching processes.At first,recall on the normally discrete Galton-Watson process:(?)where(?) are are two mutually independent i.i.d sequences and array of nonnegative integer-valued random variables.For each k,εk stands for the number of the immigrants of the k-th generation and (?)represent the numbers of the offsprings given birth by the individuals of the(k-1)-th generation.Moreover,the sequence {Nk} represents the size of the population of the k-th generation.In this paper we consider the sub-critical condition.It means the sequence {ζ} satisfying Eζ<1.In chapter 2 and chapter 3,we discuss a class of more general Galton-Watson process.We consider that the random variable {ζ} will change by the increase of generation n.So we use a series of mutually independent variables {ζn} for the generic copies of {ζk-l,j}j.Especially,for any given n∈N+,we assume all the previous offsprings of generation (?)satisfy the same distribution with (?).Then our model shall be written as:Moreover,we not only concentrate on the sub-critical condition limn→∞Eζn<1,but also focus on the nearly unstable case as(?)It means that the limit of Eζn is close to 1 from the negative direction.Then we consider the large and moderate deviation for the total population arising on this model,and give the conclusion corresponding to the different situations and cases.In chapter 2 we consider the following condition:(A1)Suppose the sequence {ζn}satisfying:(?)It’s natural that(?).For our proof,we need some basic assumptions which will be used throughout chapter 2.Let the number of initial generation No≥0 be a deterministic integer.Moreover,(?) exists and less than infinity for any fixed θ;(?)In addition,we defined two functions.One of which is(?)and (?);the other is(?)and (?)Under these assumptions,in chapter 2,we give the large deviations for the total population arising from the Galton-Watson process we considered.Theorem 1.1 In the Galton-Watson process we considered above,under the condition(Al),if(?)and(?)So we have,(?)for any close set F∈R;and(?)for any open set G∈R,where the rate function I1(·)is given as(?)Theorem 1.2 In the Galton-Watson process we considered above,under the condition(A1),if(?)and(?)we have(?)for each close set F∈R;and(?)for each open set G ∈ R,where the rate function I1(·)is given as(?)In chapter 3 we still consider the same model as chapter 2.But we shall change some assumptions.(A2)Let the number of initial generation N0≥0 be a deterministic integer.Moreover,for any n∈N-,(?)In addition,we assume the limit of Eζn and the limit of Var(ζn)both exist.Under these assumptions,in chapter 3,we give the moderate deviations for the total population arising from the Galton-Watson process we considered.Theorem 2.1 Let bn be a sequence of positive numbers satisfying(?)Under the condition(A2),if(?)Then we have,(?)for each close set F∈R;and(?)for each open set G ∈R,where the rate function I1(·)is given as(?)Theorem 2.2 Let bn be a sequence of positive numbers satisfying the condition in Theorem 2.1,under the condition(A2),if(?)We can get the conclusion that the family(?)satisfies the MDP with the speed bn and rate function(?)Theorem 2.3 Let bn be a sequence of positive numbers satisfying the condition in Theorem 2.1,under the condition(A2),if(?)It is natural that(?)So we can get the conclusion that the family (?)satisfies the MDP with the speed bn and rate function(?)In chapter 4,we will consider a multitype Galton-Watson process.Compared with the normal Galton-Watson process,we use Nk to stand for the population of κ-th moment.In this model the total population of k-th moment is affected by not only the(k-1)-th moment,but also the recent p moment.So we use the conception "k-th moment" instead of "k-th generation".Besides,we suppose the offsprings given birth by the individuals of the every generation satisfying the same distribution.Here we use ζ1,ζ2…ζp stand for the offsprings of every individual in its next p moment.And we suppose they are mutually independent with each other.At last,we still use an i.i.d.sequence and array of nonnegative integer-valued random variables (?)to stand for the immigration of k-th moment.Then our model shall be written as:(?)Here we consider this p-type Galton-Watson process.For convenience we suppose every individual in different moment has the same offsprings in next p moments.In this model the sub-critical condition changes to (?).So that the total population will not increase explosively.Due to the value of (?) is nonnegative,the sub-critacal condition means that,for any (?) we have (?).At last,to avoid degeneracy,we assume that P{∈=0}<1 and P{ζi=0}<1 for any(?).We will give the moderate deviations for the total population arising from this model in chapter 4.Theorem 3.1 Let bn be a sequence of positive numbers satisfying,(?)Under the condition introduced above,let the number of initial generation N0,N1,...Np-1 are p deterministic integer,and for any i=1,2,...p,(?)So we have,(?)for each close set F ∈R;and(?)for each open set G∈R,where the rate function Ip(·)is given as(?)... |
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