2020/02/12 by Dariusz Buraczewski, Buraczewski, Dariusz, Alexander Iksanov +3
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #60G42 #60J80. Secondary: 60F05 #FOS: Mathematics #Primary: 60G50 #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2002.05215
openalex publication_date 2020/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We work under the A"ıdékon-Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate nonnegative random variable that we denote by Z. It is shown that 𝔼 Z1_\Z≤ x\=log x+o(log x) as x→∞. Also, we provide necessary and sufficient conditions under which 𝔼 Z1_\Z≤ x\=log x+\rm const+o(1) as x→∞. This more precise asymptotics is a key tool for proving distributional limit theorems which quantify the rate of convergence of the derivative martingale to its limit Z. The methodological novelty of the present paper is a three terms representation of a subharmonic function of at most linear growth for a killed centered random walk of finite variance. This yields the aforementioned asymptotics and should also be applicable to other models.