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

Let N(T)=N(t，ω-i D21-D 2)be the size of population in generation t.We assume that{N(t，ω-I D 21-D 2)}t=0，1，2，…，in a马尔可夫chain of{N I D21 I D 2，N I D 22 I D 2}，with the transition probability/P(N(t+1，ωI D 21 I D 2)≠N(t，ωI D 21 I D 2)|N(t，ωI D 21 I D 2)=N I D 2 j I D 2)=q I D 2 j I D 2，P(N(t+1，ωI D 21 I D 2)=N(t，ωI D 21 D 2)|N(t，t，t，ωI D21 I D 2)=N I D 2 j I D 2)=1-q I D 2 j I D 2 2，(1&lt；N I D 21 I D 2&lt；N I D 22 I D 2)。We consider a locus with two alleles A I D 21 D 2 and A I D 22 D 2.Let Z(t，ω-i D21-D 2，ω-I D 22-D 2)be the number of A I D 21 D 2 allele in generation t.For given{N(t，ωI D 21 I D 2)}t=0，1，2，…，a马尔可夫chain{Z(t，ωI D21 I D 2)}t=0，1，2，…，whose probability law is given by P(Z(t+1，ω-I D21-I D2，ω-I D22-D 2)=j|Z(t，ω-I D21-D 2-D2)=I)=(I)=(D-72N(t+1，ω-I D 21-D 2)(/)j I D 7)j)(I D 7 i(/)2N(t，ωI D 21 I D 2 D 7)I D 1 j D 1(1-I D 7 i(/)2N(t，i D 21 D 2 D 7)I D 1(1-I D 7i(/)2N(t，t，ωD 21 D 2)I D 7)I D 12 N(t+1+ω1)-J D 1，0[less than or equal]I[less than or equal]N(t，t，ωI D 21 i D 2)，0[less than or equal]j[less than or equal]2N(t+1，ωI D 21 I D 2)，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 I D3(V)(/)e i D3 and the other is the inbreeding effective size N I D3(I)(/)e e D 3.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 I D 2 e I D 2=N I D 3(V)(/)e I D 3=N I D 3(I)(/)e I D 3).2.The explicit form of the effective size N I D 2e D 2 is determined.3.The effective size N I D2e I D2is larger(resp.Smaller)than the harmonic mean N I D 2 H D 2 of N I D 21 I D 2 and N I D 22 I D 2 if the fluctuation of population size is positively(resp.Negatively)autocorrelated。The effective size is the same as the harmonic mean(N I D 2 e I D 2=N I D 2 H D 2)if and only if the fluctuation is uncorrelated.4.The dependence of q I D21个D 2 and q个D 22个D 2 on the effective size N个D 2 e个D 2 is determined.5.Asymptotic expression for the effective size N I D2e I D2is determined for cases of q I D21ⅡD2，qⅡD22ⅡD2→0or NⅡD21ⅡD2，NⅡD22ⅡD2→∞。Less：Less

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