Left-tail expansions for Schr\"oder branching processes with explicit convergence rates

Abstract

In previous work, the density of the martingale limit in Schr\"oder branching processes was expressed as a convergent double power-law series with oscillatory terms. The proof relied on two assumptions, one of which imposed a geometric restriction on the critical angle of the Julia set of the offspring generating function near $1$. In this paper, we show that both assumptions can be removed. We derive explicit bounds on the expansion coefficients, which imply locally uniform convergence of the double series. The coefficients decay exponentially in one summation index, with the rate explicitly determined by the critical angle, and super-exponentially in the other. Finally, we investigate the behavior of the critical angle and describe regimes in which it approaches its minimum. This analysis shows how the geometry of the Julia set influences the magnitude of the oscillatory corrections in the left-tail expansion.