The Schröder case of the generalized Delta conjecture

2019 ◽  
Vol 81 ◽  
pp. 58-83 ◽  
Author(s):  
Michele D’Adderio ◽  
Alessandro Iraci ◽  
Anna Vanden Wyngaerd
Keyword(s):  
2016 ◽  
Vol 48 (3) ◽  
pp. 672-690 ◽  
Author(s):  
Hui He

Abstract Given a supercritical Galton‒Watson process {Zn} and a positive sequence {εn}, we study the limiting behaviors of ℙ(SZn/Zn≥εn) with sums Sn of independent and identically distributed random variables Xi and m=𝔼[Z1]. We assume that we are in the Schröder case with 𝔼Z1 log Z1<∞ and X1 is in the domain of attraction of an α-stable law with 0<α<2. As a by-product, when Z1 is subexponentially distributed, we further obtain the convergence rate of Zn+1/Zn to m as n→∞.


1988 ◽  
Vol 25 (A) ◽  
pp. 215-228 ◽  
Author(s):  
N. H. Bingham

LetWbe the usual almost-sure limit random variable in a supercritical simple branching process; we study its tail behaviour. For the left tail, we distinguish two cases, the ‘Schröder' and ‘Böttcher' cases; both appear in work of Harris and Dubuc. The Schröder case is related to work of Karlin and McGregor on embeddability in continuous-time (Markov) branching processes. New results are obtained for the Böttcher case; there are links with recent work of Barlow and Perkins on Brownian motion on a fractal. The right tail is also considered. Use is made of recent progress in Tauberian theory.


1988 ◽  
Vol 25 (A) ◽  
pp. 215-228 ◽  
Author(s):  
N. H. Bingham

Let W be the usual almost-sure limit random variable in a supercritical simple branching process; we study its tail behaviour. For the left tail, we distinguish two cases, the ‘Schröder' and ‘Böttcher' cases; both appear in work of Harris and Dubuc. The Schröder case is related to work of Karlin and McGregor on embeddability in continuous-time (Markov) branching processes. New results are obtained for the Böttcher case; there are links with recent work of Barlow and Perkins on Brownian motion on a fractal. The right tail is also considered. Use is made of recent progress in Tauberian theory.


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