Branching random walk with infinite progeny mean: a tale of two tails
| Autor: | Ray, Souvik, Hazra, Rajat Subhra, Roy, Parthanil, Soulier, Philippe |
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| Rok vydání: | 2019 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We study the extremes of branching random walks under the assumption that the underlying Galton-Watson tree has infinite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the regularly varying case, it is shown that the point process sequence of normalized extremes converges to a Poisson random measure. We study the asymptotics of the scaled position of the rightmost particle in the $n$-th generation when the tail of the displacement behaves like $\exp(-K(x))$, where either $K$ is a regularly varying function of index $r> 0$, or $K$ has an exponential growth. We identify the exact scaling of the maxima in all cases and show the existence of a non-trivial limit when $r> 1$. Comment: 33 pages. Improved version and contains many new results. Section 5 and section 6 added on very rapidly varying tails and cloudspeed respectively. The proofs are streamlined and many new arguments added |
| Databáze: | arXiv |
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