# On the tail distribution of the solution to some law equation

@article{Chen2019OnTT, title={On the tail distribution of the solution to some law equation}, author={Xinxing Chen and Chunhua Ma}, journal={arXiv: Probability}, year={2019} }

We consider a distribution equation which was initially studied by Bertoin \cite{Bertoin}: \[M \stackrel{d}{=} \max\{\widetilde{\nu}, \max_{1\leq k\leq \nu}M_k\}.\] where $\{M_k\}_{k\geq 1}$ are i.i.d. copies of $M$ and independent of $(\widetilde{\nu}, \nu)\in\mathbb{R}_+\times\mathbb{N}$. We obtain the tail behaviour of the solution of a generalised equation in a different but direct method by considering the joint tail of $(\widetilde{\nu}, \nu)$.

## One Citation

### Maximal local time of randomly biased random walks on a Galton-Watson tree

- Mathematics
- 2020

We consider a recurrent random walk on a rooted tree in random environment given by a branching random walk. Up to the first return to the root, its edge local times form a Multi-type Galton-Watson…

## References

SHOWING 1-10 OF 13 REFERENCES

### The Survival Probability of a Critical Branching Process in a Random Environment

- Mathematics
- 2001

In this paper we determine the asymptotic behavior of the survival probability of a critical branching process in a random environment. In the special case of independent identically distributed…

### Sub-Gaussian tail bounds for the width and height of conditioned Galton--Watson trees

- Computer Science
- 2010

Under this conditioning, sub-Gaussian tail bounds are derived for both the width and height of a Galton--Watson tree with offspring distribution B satisfying E(B)=1, 0 < Var(B) < infinity, conditioned on having exactly n nodes.

### On distribution tails and expectations of maxima in critical branching processes

- MathematicsJournal of Applied Probability
- 1996

We derive the limit behaviour of the distribution tail of the global maximum of a critical Galton–Watson process and also of the expectations of partial maxima of the process, when the offspring law…

### Sub-exponential tail bounds for conditioned stable Bienaymé–Galton–Watson trees

- Mathematics
- 2015

We establish uniform sub-exponential tail bounds for the width, height and maximal outdegree of critical Bienaymé–Galton–Watson trees conditioned on having a large fixed size, whose offspring…

### Heavy-Tail Phenomena: Probabilistic and Statistical Modeling

- Mathematics
- 2006

Crash Courses.- Crash Course I: Regular Variation.- Crash Course II: Weak Convergence Implications for Heavy-Tail Analysis.- Statistics.- Dipping a Toe in the Statistical Water.- Probability.- The…

### On the Maximal Offspring in a Critical Branching Process with Infinite Variance

- MathematicsJournal of Applied Probability
- 2011

We investigate the maximal number M k of offspring amongst all individuals in a critical Galton-Watson process started with k ancestors. We show that when the reproduction law has a regularly varying…

### Extreme Values, Regular Variation, and Point Processes

- Mathematics
- 1987

Contents: Preface * Preliminaries * Domains of Attraction and Norming Constants * Quality of Convergence * Point Processes * Records and Extremal Processes * Multivariate Extremes * References *…

### On the Maximum of a Branching Process

- Mathematics
- 2016

The pupose of this paper is to estimate P(M> n), where M is the maximum size of a Galton-Watson branching process. Partially, use is made of the fact that such a process may be imbedded in a certain…