Longest increasing subsequences of random walks and correlated time series

Abstract

The goal of this project is to continue the study of the longest increasing sequences (LIS) of random walks initiated in the scope of the FAPESP grant BPE 2017/22166-9. Previous results suggest some questions about the universality of the distribution of the LIS of simple (step increments -1, +1) and lazy (step increments -1, 0, +1) random walks as well as the possible existence of a phase transition in the asymptotic behavior of the LIS of random walks with heavy-tailed distribution of step increments. From the applied point of view, we intend to explore the possibility of employing some LIS statistics to the characterization of correlated time series, in particular through the relationship between the asymptotic behavior of the length of the LIS of the time series with the tail index of the underlying distribution of its increments. This proposal also intends to obtain funds to acquire a medium-sized workstation and to upgrade an equipment acquired in a former project funded by FAPESP (2015) to expand its capacity and extend its lifetime. (AU)