Approximation of the Number of Descendants in Branching Processes

Abstract

We discuss approximations of the relative limit densities of descendants in Galton–Watson processes that follow from the Karlin–McGregor near-constancy phenomena. These approximations are based on the fast exponentially decaying Fourier coefficients of Karlin–McGregor functions and the binomial coefficients. The approximations are sufficiently simple and show good agreement between approximate and exact values, which is demonstrated by several numerical examples.

Generalized Schröder-type functional equations for Galton–Watson processes in random environments

“…Thus, H is a contraction mapping if h > 0 is small enough. Hence, there is a unique fixed point Φ -the solution of (7). Now, we have…”

“…and, hence, |P r (z)| < h, see (14). Thus, if Φ(z) is analytic for |z| < h < 1 then it is also analytic for |z| < Rh by (7). Hence, we can increase the domain of analyticity, since R > 1 for h < 1.…”

“…For small x ⩽ h, the inequality is obvious for any α < 0. Using the same arguments as above, we can increase the interval in R > 1 times following (7):…”

“…Often, the main contribution to the asymptotic behavior of ϕ n comes from the singularity of Φ(z) at z = 1. Substituting ansatz Φ(z) ≈ A(1 − z) α into the main equation (7), we obtain the equation for α: R…”

“…Because P ′ r (1) ⩾ 1 and p 1r are non-negative, the real parts of solutions α are bounded from below. Note that if α = α 1 + iα 2 is complex then the corresponding term in (32) gives the long-phase logarithmic oscillation C α n −α 1 −1 exp(−iα 2 ln n) (plus the corresponding terms related to the complex conjugate α) similar to that in the asymptotic of the classical Galton-Watson process with one generating function, see, e.g., [7] and [10]. However, in the mixed case of many generating functions (7), the derivation of complete asymptotic series for power coefficients of Φ(z) is, generally speaking, an open problem.…”