Um passeio aleatório com catástrofes

Abstract

We discussed upon the results and proofs of the article [3], in which the authors studied a class of models for populations growth with catastrophes. From a mathematical point of view, this kind of model can be understood as a random walk in non-negative integers, whose transition probabilities are state depend. The formidable thing about this class of random walks is their stability. In fact, an even larger class of random walks are stable, see [19]. In accordance with [3]'s script, we proof the positive recurrence of this random walk and explain its stationary distribution. We commented on the strong stability of this process. Soon after, we defined a useful coupling for this Markov chain, which we used extensively to prove many results. We established lower and upper bounds for the distance in total variation, analyzing the behavior of convergence towards stationarity of this process. After this analysis we were able to discuss the main result, according to the authors themselves, of [3]: The class of models presents the phenomenon of cutoff. The main challenge in proving this result is that the defined coupling provides quotas for the distance in total variation only considering the fixed initial states, and the state space of this chain is countable. Finally, we present some other interesting results, such as the logarithmic dependence between the initial state and the time of first extinction of the process.