Modern Stochastics: Theory and Applications

This paper provides a multivariate extension of Bertoin’s pathwise construction of a Lévy process conditioned to stay positive or negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged.

We study the frequency process $f_{1}$ of the block of 1 for a Ξ-coalescent Π with dust. If Π stays infinite, $f_{1}$ is a jump-hold process which can be expressed as a sum of broken parts from a stick-breaking procedure with uncorrelated, but in general non-independent, stick lengths with common mean. For Dirac-Λ-coalescents with $\varLambda =\delta _{p}$, $p\in [\frac{1}{2},1)$, $f_{1}$ is not Markovian, whereas its jump chain is Markovian. For simple Λ-coalescents the distribution of $f_{1}$ at its first jump, the asymptotic frequency of the minimal clade of 1, is expressed via conditionally independent shifted geometric distributions.