Uniform Cramér moderate deviations and Berry-Esseen bounds for a supercritical branching process in a random environment

Cramér moderate deviations and Berry-Esseen bounds for Mandelbrot’s cascade in a random environment

“…In a random environment, analysis of the branching process and Mandelbrot's cascade are closely related fields of study, see Liang and Liu (2010) [17], Li, Liu and Peng (2019) [16]. In this article, we expand the relevant conclusions in (2020) [6] to the Mandelbrot cascade in a random environment.…”

“…Especially, Grama, Liu and Miqueu (2017) [8] constructed the Cramér large deviation and Berry-Esseen bounds for a supercritical branching process in a random environment (BPRE). Fan, Hu and Liu (2020) [6] developed a new method and relaxed the conditions, generalize the results in [8]. In a random environment, analysis of the branching process and Mandelbrot's cascade are closely related fields of study, see Liang and Liu (2010) [17], Li, Liu and Peng (2019) [16].…”

Deviation inequalities for a supercritical branching process in a random environment

“…where C is a positive constant. See also Fan et al [14] with more general conditions. Asymptotic expansions, no matter how precise, do not diminish the need for probability inequalities valid for all n, x.…”

Comparison on the criticality parameters for two supercritical branching processes in random environments

“…For subcritical BPRE, researches focus on the study of the survival probability and conditional limit theorems: see, for instance, Vatutin [23], Afanasyev et al [1], Vatutin and Zheng [24] and Bansaye and Vatutin [5]. While, for supercritical BPRE, a number of researches have studied central limit theorem, moderate and large deviations; see, for instance, Böinghoff and Kersting [8], Bansaye and Böinghoff [4], Huang and Liu [17], Nakashima [20], Böinghoff [7], and Grama et al [15], Fan et al [13] and Gao [14]. See also Wang and Liu [25] and Huang et al [18] for BPRE with immigrations.…”