Ratios of ordered points of point processes with regularly varying intensity measures

We study limiting properties of ratios of ordered points of point processes whose intensity measures have regularly varying tails, giving a systematic treatment which points the way to “large-trimming” properties of extremal processes and a variety of applications. Our point process approach facilitates a connection with the negative binomial process of Gregoire (1984) and consequently to certain generalised versions of the Poisson–Dirichlet distribution.     

Discrete-time trawl processes

We introduce a class of discrete time stationary trawl processes taking real or integer values and written as sums of past values of independent ‘seed’ processes on shrinking intervals (‘trawl heights’). Related trawl processes in continuous time were studied in Barndorff-Nielsen et al. (2011, 2014).In the case when the trawl function decays as a power function of the lag with exponent $1<\alpha <2$, the trawl process exhibits long memory and its covariance function is non-summable. We show that under general conditions on generic seed process, the normalized partial sums of such trawl process may tend either to a fractional Brownian motion or to an $\alpha$-stable Lévy process. Moreover if the trawl function admits a faster decay rate, then the classical Donsker’s invariance principle holds true.     

A stable Langevin model with diffusive-reflective boundary conditions

In this note, we consider the construction of a one-dimensional stable Langevin type process confined in the upper half-plane and submitted to diffusive-reflective boundary conditions whenever the particle position hits 0. We show that two main different regimes appear according to the values of the chosen parameters. We then use this study to construct the law of a (free) stable Langevin process conditioned to stay positive, thus extending earlier works on integrated Brownian motion. This construction further allows to obtain the exact asymptotics of the persistence probability of the integrated stable Lévy process. In addition, the paper is concluded by solving the associated trace problem in the symmetric case.     

Nonparametric inference of gradual changes in the jump behaviour of time-continuous processes

In applications the properties of a stochastic feature often change gradually rather than abruptly, that is: after a constant phase for some time they slowly start to vary. In this paper we discuss statistical inference for the detection and the localization of gradual changes in the jump characteristic of a discretely observed Ito semimartingale. We propose a new measure of time variation for the jump behaviour of the process. The statistical uncertainty of a corresponding estimate is analysed by deriving new results on the weak convergence of a sequential empirical tail integral process and a corresponding multiplier bootstrap procedure.