Extremes of local times for simple random walks on symmetric trees
| Autor: | Yoshihiro Abe |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
Statistics and Probability
60G70 local times 01 natural sciences Measure (mathematics) Point process Combinatorics 010104 statistics & probability Gumbel distribution Mathematics::Probability 60J55 60J10 60G70 FOS: Mathematics Limit (mathematics) 0101 mathematics Mathematics Probability (math.PR) 010102 general mathematics random multiplicative cascade measure trees derivative martingale Vertex (geometry) Random measure simple random walk 60J10 60J55 Statistics Probability and Uncertainty Multiplicative cascade Constant (mathematics) Mathematics - Probability |
| Zdroj: | Electron. J. Probab. |
| Popis: | We consider local times of the simple random walk on the $b$-ary tree of depth $n$ and study a point process which encodes the location of the vertex with the maximal local time and the properly centered maximum over leaves of each subtree of depth $r_n$ rooted at the $(n-r_n)$ level, where $(r_n)_{n \geq 1}$ satisfies $\lim_{n \to \infty} r_n = \infty$ and $\lim_{n \to \infty} r_n/n \in [0, 1)$. We show that the point process weakly converges to a Cox process with intensity measure $\alpha Z_{\infty} (dx) \otimes e^{-2\sqrt{\log b}~y}dy$, where $\alpha > 0$ is a constant and $Z_{\infty}$ is a random measure on $[0, 1]$ which has the same law as the limit of a critical random multiplicative cascade measure up to a scale factor. As a corollary, we establish convergence in law of the maximum of local times over leaves to a randomly shifted Gumbel distribution. Comment: 48 pages, Update: several minor corrections, new proof for Lemma 2.1 |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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