Branching Random Walks Conditioned on Particle Numbers

In this paper, we consider a pruned Galton–Watson tree conditioned to have k particles in generation n, i.e. we take a Galton–Watson tree satisfying Zn=k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_n=k$$\end{document}, and delete all branches that die before generation n. We show that with k fixed and n→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow \infty $$\end{document}, the first n generations of this tree can be described by an explicit probability measure Pkst\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {P}}^{st}_k$$\end{document}. As an application, we study a branching random walk (Vu)u∈T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(V_u)_{u\in T}$$\end{document} indexed by such a pruned Galton–Watson tree T, and give the asymptotic tail behavior of the span and gap statistics of its k particles in generation n, (Vu)|u|=n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(V_u)_{|u|=n}$$\end{document}. This is the discrete version of Ramola et al. (Chaos Solitons Fractals 74:79–88, 2015), generalized to arbitrary offspring and displacement distributions with moment constraints.