The study of the statistics of records in a time series is of capital importance in a wide variety of systems, including climate studies, finance and economics, hydrology, sports and others. Recently, the study of records has found a renewed interest and applications in diverse complex systems such as the evolution of the thermo-remanent magnetization in spin-glasses, evolution of the vortex density with increasing magnetic field in type-II disordered superconductors, avalanches of elastic lines in a disordered medium, the evolution of fitness in biological populations, and in models of growing networks, amongst others.
Interestingly, despite record statistics has been extensively studied in the past 50 years, there are still non-trivial questions to be addressed, even for the case of independent and identically distributed (i.i.d.) random variables. While the statistics of the total number of records and record ages (how long a record survives before it gets broken by the next one) in a given sequence of size N have been thoroughly studied, both for uncorrelated time series, as well as for correlated time series such as a random walk, there is hardly any study of the statistics of the label of the longest record M.
In this seminar, I will present analytical results for the statistics of M for i.i.d. random variables, each drawn from a continuous probability distribution function, and use them to introduce a new quantity that measures the distance between the last record and the most longevous one. Remarkably, all the results for the statistics of M are shown to be universal, i.e. independent of the distribution of the random variables.