We study the asymptotic tail behavior of the conditional probability distributions of rt+k and rt+1 + · · · + rt+k when (rt )t∈N is a GARCH(1, 1) process. As an application, we examine the relation between the extreme lower quantiles of these random variables.

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We study a stochastic version of the classical Becker-Döring model, well-known kinetic model for cluster formation that predicts existence long-lived metastable state before thermodynamically unfavorable nucleation occurs, leading to phase transition phenomena. This continuous-time Markov chain has received little attention, compared its deterministic differential equations counterpart. show fo...

Abstract We determine the distributions of some random variables related to a simple model an epidemic with contact tracing and cluster isolation. This enables us apply general limit theorems for super-critical Crump–Mode–Jagers branching processes. Notably, we compute explicitly asymptotic proportion isolated clusters given size amongst all clusters, conditionally on survival epidemic. Somewha...