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A Version of the Random Directed Forest and its Convergence to the Brownian Web

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Abstract

Several authors have studied convergence in distribution to the Brownian web under diffusive scaling of systems of Markovian random walks. In a paper by R. Roy, K. Saha and A. Sarkar, convergence to the Brownian web is proved for a system of coalescing random paths—the random directed forest—which are not Markovian. Paths in the random directed forest do not cross each other before coalescence. Here, we study a variation of the random directed forest where paths can cross each other and prove convergence to the Brownian web. This provides an example of how the techniques to prove convergence to the Brownian web for systems allowing crossings can be applied to non-Markovian systems.

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Acknowledgements

We would like to thank Maria Eulalia Vares, Leandro Pimentel and Luiz Renato Fontes for useful comments and suggestions. We would also like to thank the anonymous referee for useful feedback and suggestions regarding the paper.

Funding

G. Valle was supported by CNPq Grant 308006/2018-6 and FAPERJ Grant E-26/202.636/2019. L. Zuaznábar was supported by CAPES

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Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, 21945-970, Brazil

    Glauco Valle

  2. Universidade de São Paulo, R. do Matão, 1010 - Butantã, São Paulo, SP, 05508-090, Brazil

    Leonel Zuaznábar

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  1. Glauco Valle
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  2. Leonel Zuaznábar
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Correspondence to Glauco Valle.

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Appendix A. A Technical Estimate

Appendix A. A Technical Estimate

Lemma A.1

Let N be some positive integer random variable and \((\zeta _n)_{n\ge 1}\) a nonnegative sequence of identically distributed random variables. If for some \(k\ge 1, \delta>0 \text { and } l>\frac{(k+2)(1+\delta )}{\delta }\), we have \(\mathbb {E}[\zeta _1^{k(1+\delta )}]\) and \(\mathbb {E}[N^l]\) finite, then for \(S:=\sum _{n=1}^{N}\zeta _n\) we get that \(\mathbb {E}[S^k]\) is also finite.

Proof

We have that \(0\le S\le N\max _{1\le j\le N}\zeta _j\) what implies that

$$\begin{aligned} S^k\le N^k\max _{1\le j\le N}\zeta ^k_j\le N^k\sum _{j=1}^{N}\zeta ^k_j. \end{aligned}$$

Hence,

$$\begin{aligned} \mathbb {E}\big [S^k\big ]&\le \mathbb {E}\Big [N^k\sum _{j=1}^{N}\zeta _j^k\Big ]=\sum _{n=1}^{\infty }n^k\sum _{j=1}^{n}\mathbb {E}\big [\mathbbm {1}_{\{N=n\}}\zeta ^k_j\big ]. \end{aligned}$$

Applying Hölder inequality, we get

$$\begin{aligned} \mathbb {E}\big [S^k\big ]\le \sum _{n=1}^{\infty }n^k\sum _{j=1}^{n}\mathbb {E}\big [\zeta _j^{k(1+\delta )}\big ]^{\frac{1}{1+\delta }}\mathbb {P}[N=n]^\frac{\delta }{1+\delta }=\mathbb {E}\big [\zeta _1^{k(1+\delta )}\big ]^{\frac{1}{1+\delta }}\sum _{n=1}^{\infty }n^{k+1}\mathbb {P}[N=n]^{\frac{\delta }{1+\delta }}. \end{aligned}$$

Applying Chebyshev inequality, we get that \(\mathbb {E}[S^k]\) is bounded above by

$$\begin{aligned} \mathbb {E}\big [\zeta _1^{k(1+\delta )}\big ]^{\frac{1}{1+\delta }}\sum _{n=1}^{\infty }n^{k+1}\frac{\mathbb {E}[N^l]^\frac{\delta }{1+\delta }}{n^{\frac{l\delta }{1+\delta }}}=\mathbb {E}\big [\zeta _1^{k(1+\delta )}\big ]^{\frac{1}{1+\delta }}\mathbb {E}[N^l]^{\frac{\delta }{1+\delta }}\sum _{n=1}^{\infty }\frac{1}{n^{\frac{l\delta }{1+\delta }-(k+1)}}<\infty . \end{aligned}$$

\(\square \)

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Valle, G., Zuaznábar, L. A Version of the Random Directed Forest and its Convergence to the Brownian Web. J Theor Probab 36, 948–1002 (2023). https://doi.org/10.1007/s10959-022-01202-z

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