A study on limit theorems related to stochastic models in pupulation genetics

Let N（t） = N（t，干扰D21干扰D2） be the size of population in generation t。We assume that {N（t，干扰D21干扰D2）}t=0，1，2，...， in a Markov chain of {N干扰D21干扰D2， N干扰D22干扰D2}， with the transition/P（N（t+1，干扰D21干扰D2）≠ N（t，干扰D21干扰D2）|N（t，干扰D21干扰D2） =N干扰D2 j干扰D2） = q干扰D2，P（N（t+1，干扰D21干扰D2） = N（t，干扰D21干扰D2）|N（t，干扰D21干扰D2） =N干扰D2 j干扰D2） = 1 - q干扰D2 j干扰D2，（1 < N干扰D21干扰D2 < N干扰D22干扰D2）。We consider a locus with two alleles A-D21-D2 and A-D22-D2。Let Z（t，干扰D21干扰D2，干扰D22干扰D2） be the number of A干扰D21干扰D2 allele in generation t。For given {N（t，干扰D21干扰D2）}t=0，1，2，...， a Markov chain {Z（t，干扰D21干扰D2）}t=0，1，2...， whose probability law is given by P（Z（t+1，干扰D21干扰D2，干扰D2） = j| Z（Z）（t，干扰D21干扰D2，干扰D22干扰D2） = I） =（イイD72N（t+1，イイD21イエD2）（\）jイエD7）j） （イイD7i（\）2 N（t，干扰D21干扰D2干扰D 7）干扰D1 j干扰D1（1-D7i（/）2 N（t，干扰D21干扰D2）干扰D12 N（t+1+干扰1）-J干扰D1， 0 [less than or equal] I [less than or equal]"N（t）， 0（less than or equal）j（less than or equal）2 N（t+1）， is referred to as a Wright-Fis ... More 她的模型与人口大小的流动有关，在人口遗传学理论中，人口大小的有效概念发挥了作用和重要作用。对于更高的模型，我们可以定义两个有效的尺寸。one is the variance effective size NイイD3（v）（\）eイD3 and the other is the inbreeding effective size NイD3（I）（\）eイD3。在本研究中，下面的属性显示了Wright-Fisher模型与人口大小的流动。variance effective size is the same as the inbreeding effective size NイイD2 = NイD3（v）（/）eイD3 = NイD3（I）（/）eイD3）.2。The explicit form of the effective size N-D2e-D2 is determined.3. effective size N-D2e-D2 is larger （resp. smaller） than the harmonic mean NイイD2HイD2 of NイD21イD2 and NイD22イD2 if the fluctuation of population size is positively （resp. negatively） autocorrelated。The effective size is the same as the harmonic mean （NイイD2eイエD2 = NイイD2 HイエD2） if and only if the fluctuation is uncorrelated. 4。The dependence of qイD21イD2 and qイD22イD2 on the effective size NイD2eイD2 is determined. 5。Asymptotic expression for the effective size NイイD2eイD2 is determined for cases of qイD21イD2， qイD22イD2 ￣ 0 or NイD21イD2， NイD22イエD2 ￣。Less（低）

Let N(t) = N(t,ωィイD21ィエD2) be the size of population in generation t. We assume that {N(t,ωィイD21ィエD2)}t=0,1,2,…, in a Markov chain of {NィイD21ィエD2, NィイD22ィエD2}, with the transition probability/P(N(t+1,ωィイD21ィエD2)≠ N(t, ωィイD21ィエD2)|N(t, ωィイD21ィエD2) =NィイD2jィエD2) = qィイD2jィエD2,P(N(t+1,ωィイD21ィエD2) = N(t, ωィイD21ィエD2)|N(t, ωィイD21ィエD2) =NィイD2jィエD2) = 1 - qィイD2jィエD2,(1 < NィイD21ィエD2 < NィイD22ィエD2). We consider a locus with two alleles AィイD21ィエD2 and AィイD22ィエD2. Let Z(t,ωィイD21ィエD2,ωィイD22ィエD2) be the number of AィイD21ィエD2 allele in generation t. For given {N(t,ωィイD21ィエD2)}t=0,1,2,…, a Markov chain {Z(t,ωィイD21ィエD2)}t=0,1,2,…, whose probability law is given by P(Z(t+1,ωィイD21ィエD2,ωィイD22ィエD2) = j|Z(t,ωィイD21ィエD2,ωィイD22ィエD2) = I) =(ィイD72N(t+1,ωィイD21ィエD2)(/)jィエD7)j) (ィイD7i(/)2N(t,ωィイD21ィエD2ィエD7)ィイD1jィエD1 (1-ィイD7i(/)2N(t,ωィイD21ィエD2)ィエD7)ィイD12N(t+1+ω1)-JィエD1, 0【less than or equal】I【less than or equal】"N(t,ωィイD21ィエD2), 0【less than or equal】j【less than or equal】2N(t+1,ωィイD21ィエD2), is referred to as a Wright-Fis … More her model with fluctuation of population size.In population genetics theory, a concept of effective size of population has played and important role. For the above model, we can define two effective sizes. The one is the variance effective size NィイD3(v)(/)eィエD3 and the other is the inbreeding effective size NィイD3(I)(/)eィエD3. In this study, the following properties are shown for the Wright-Fisher model with fluctuation of population size.1. The variance effective size is the same as the inbreeding effective size NィイD2eィエD2 = NィイD3(v)(/)eィエD3 = NィイD3(I)(/)eィエD3).2. The explicit form of the effective size NィイD2eィエD2 is determined.3. The effective size NィイD2eィエD2 is larger (resp. smaller) than the harmonic mean NィイD2HィエD2 of NィイD21ィエD2 and NィイD22ィエD2 if the fluctuation of population size is positively (resp. negatively) autocorrelated. The effective size is the same as the harmonic mean (NィイD2eィエD2 = NィイD2HィエD2) if and only if the fluctuation is uncorrelated.4. The dependence of qィイD21ィエD2 and qィイD22ィエD2 on the effective size NィイD2eィエD2 is determined.5. Asymptotic expression for the effective size NィイD2eィエD2 is determined for cases of qィイD21ィエD2, qィイD22ィエD2 → 0 or NィイD21ィエD2, NィイD22ィエD2 → ∞. Less