# Time reversal dualities for some random forests

* Corresponding author
1 SMILE - Stochastic models for the inference of life evolution
LPMA - Laboratoire de Probabilités et Modèles Aléatoires, CIRB - Centre interdisciplinaire de recherche en biologie
Abstract : We consider a random forest $\mathcal{F}^*$, defined as a sequence of i.i.d. birth-death (BD) trees, each started at time 0 from a single ancestor, stopped at the first tree having survived up to a fixed time $T$. We denote by $\left(\xi^*_t,\ 0\leq t\leq T\right)$ the population size process associated to this forest, and we prove that if the BD trees are supercritical, then the time-reversed process $\left(\xi^*_{T-t},\ 0\leq t\leq T\right)$, has the same distribution as $\left(\widetilde\xi^*_t,\ 0\leq t\leq T\right)$, the corresponding population size process of an equally defined forest $\widetilde{\mathcal{F}}^*$, but where the underlying BD trees are subcritical, obtained by swapping birth and death rates or equivalently, conditioning on ultimate extinction. We generalize this result to splitting trees (i.e. life durations of individuals are not necessarily exponential), provided that the i.i.d. lifetimes of the ancestors have a specific explicit distribution, different from that of their descendants. The results are based on an identity between the contour of these random forests truncated up to $T$ and the duality property of Lévy processes. This identity allows us to also derive other useful properties such as the distribution of the population size process conditional on the reconstructed tree of individuals alive at $T$, which has potential applications in epidemiology.
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Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-01067958
Contributor : Miraine Davila Felipe <>
Submitted on : Wednesday, September 24, 2014 - 2:24:27 PM
Last modification on : Wednesday, June 17, 2020 - 4:20:04 PM

### Identifiers

• HAL Id : hal-01067958, version 1
• ARXIV : 1409.6040

### Citation

Miraine Dávila Felipe, Amaury Lambert. Time reversal dualities for some random forests. 2014. ⟨hal-01067958⟩

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