We study a random walk on a point process given by an ordered array of points $( omega_k, , k in Z)$ on the real line. The distances $omega_{k+1} - omega_k$ are i.i.d. random variables in the domain of attraction of a $eta$-stable law, with $eta in (0,1) cup (1,2)$. The random walk has i.i.d. jumps such that the transition probabilities between $omega_k$ and $omega_ell$ depend on $ell-k$ and are given by the distribution of a $Z$-valued random variable in the domain of attraction of an $alpha$-stable law, with $alpha in (0,1) cup (1,2)$. Since the defining variables, for both the random walk and the point process, are heavy-tailed, we speak of a Lévy flight on a Lévy random medium. For all combinations of the parameters $alpha$ and $eta$, we prove the annealed functional limit theorem for the suitably rescaled process, relative to the optimal Skorokhod topology in each case. When the limit process is not càdlàg, we prove convergence of the finite-dimensional distributions. When the limit process is deterministic, we also prove a limit theorem for the fluctuations, again relative to the optimal Skorokhod topology.
Limit theorems for Lévy flights on a 1D Lévy random medium
Samuele Stivanello;Alessandra BIANCHI;Elena Magnanini
2021
Abstract
We study a random walk on a point process given by an ordered array of points $( omega_k, , k in Z)$ on the real line. The distances $omega_{k+1} - omega_k$ are i.i.d. random variables in the domain of attraction of a $eta$-stable law, with $eta in (0,1) cup (1,2)$. The random walk has i.i.d. jumps such that the transition probabilities between $omega_k$ and $omega_ell$ depend on $ell-k$ and are given by the distribution of a $Z$-valued random variable in the domain of attraction of an $alpha$-stable law, with $alpha in (0,1) cup (1,2)$. Since the defining variables, for both the random walk and the point process, are heavy-tailed, we speak of a Lévy flight on a Lévy random medium. For all combinations of the parameters $alpha$ and $eta$, we prove the annealed functional limit theorem for the suitably rescaled process, relative to the optimal Skorokhod topology in each case. When the limit process is not càdlàg, we prove convergence of the finite-dimensional distributions. When the limit process is deterministic, we also prove a limit theorem for the fluctuations, again relative to the optimal Skorokhod topology.File | Dimensione | Formato | |
---|---|---|---|
21-EJP626.pdf
accesso aperto
Descrizione: Articolo principale
Tipologia:
Published (publisher's version)
Licenza:
Creative commons
Dimensione
583.89 kB
Formato
Adobe PDF
|
583.89 kB | Adobe PDF | Visualizza/Apri |
Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.