Approximation of the allelic frequency spectrum in general supercritical branching populations
The author considers a branching population model with the following dynamics: starting with a single individual (called the ancestor) whose lifetime is distributed according to an arbitrary probability distribution on \((0,\infty]\), this ancestor, during its lifetime, gives birth to new individuals according to a Poisson process with a constant rate. Each birth event gives a single new individual. Then each child of the ancestor lives and gives birth according to the same mechanism independently of the other individuals in the population. In addition, neutral mutations can occur on individuals and each new mutation confers to its holder a brand new type (i.e., never seen in the population): this is the infinitely many alleles assumption. It is also supposed that every individual inherits the type of its parent. This model leads to a partition of the population by types. The frequency spectrum of the population alive at time t is defined as the sequence \(A(k,t)\), \(k\in\{1,2,\ldots\}\), where, for each \(k\), \(A(k,t)\) is the number of families (i.e., sets of individuals carrying the same type) of size \(k\) in the population. In general the law of the frequency spectrum \(A(k,t)\), \(k\in\{1,2,\ldots\}\), is hardly tractable. Motivated by this, the author introduced two approximations of the frequency spectrum and investigated their errors. The first approximation is \[ (A(k,t))_{k\in\{1,2,\ldots\}} \approx (c_k)_{k\in\{1,2,\ldots\}}{\mathrm{e}}^{\alpha t} \mathcal{E}, \] where \(c_k\), \(k\in\{1,2,\ldots\}\), are some explicitly given deterministic constants, \(\alpha>0\) (the largest root of the Laplace exponent of the corresponding splitting tree), and \(\mathcal{E}\) is a conditionally exponentially distributed random variable given non-extinction. Here \(\alpha>0\) corresponds to the supercritical case, when \(\alpha\) is also called the Malthusian parameter. The second approximation is \[ (A(k,t))_{k\in\{1,2,\ldots\}} \approx (c_k)_{k\in\{1,2,\ldots\}} N_t, \] where \(N_t\) is the number of living individuals in the population at time \(t\). The process \((N_t)_{t\geq 0}\) is known as a binary homogeneous Crump-Mode-Jagers process. For both approximations, the author shows that the appropriately scaled error converges to a Laplace distribution, where the convergence is understood as convergence of processes for the product topology of \({\mathbb{R}}^{\{1,2,\ldots\}}\). The limit law depends on a parameter related to mutation, and this parameter is supposed to be not equal to \(\alpha\).
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