Date(s) : 09/04/2024 iCal
14 h 30 min - 15 h 30 min
Abstract: Consider a stationary sequence $X=(X_n)$ of integer-valued random variables with mean $m in [-infty, infty]$. Let $S=(S_n)$ be the stochastic process with increments $X$ and such that $S_0=0$. For each time $i$, draw an edge from $(i,S_i)$ to $(j,S_j)$, where $j>i$ is the smallest integer such that $S_j geq S_i$, if such a $j$ exists. This defines the record graph of $X$.
We show that if $X$ is ergodic, then its record graph exhibits the following phase transitions when $m$ ranges from $-infty$ to $infty$. For $m<0$, the record graph has infinitely many connected components which are all finite trees. At $m=0$, it is either a one-ended tree or a two-ended tree. For $m>0$, it is a two-ended tree.
We analyse the distribution of the component of $0$ in the record graph when $X$ is an i.i.d. sequence of random variables whose common distribution is supported on ${-1,0,1,ldots}$, making $S$ a skip-free to the left random walk. For this random walk, if $m<0$, then the component of $0$ is a unimodular typically re-rooted Galton-Watson Tree. If $m=0$, then the record graph rooted at $0$ is a one-ended unimodular random tree, specifically, it is a unimodular Eternal Galton-Watson Tree. If $m>0$, then the record graph rooted at $0$ is a unimodularised bi-variate Eternal Kesten Tree.
A unimodular random directed tree is said to be record representable if it is the component of $0$ in the record graph of some stationary sequence. We show that every infinite unimodular ordered directed tree with a unique succession line is record representable. In particular, every one-ended unimodular ordered directed tree has a unique succession line and is thus record representable.
Emplacement
Salle de séminaire de l'I2M à St Charles
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