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On Adjoint Additive Processes

Evans Kristian P., Jacob Niels

Probability and Mathematical Statistics

2020

Vol. 40, Fasc. 2

205--223

Starting with an additive process (Yt)t≥0, it is in certain cases possible to construct an adjoint process (Xt)t≥0 which is itself additive. Moreover, assuming that the transition densities of (Yt)t≥0 are controlled by a natural pair of metrics dψ;t and δψ;t, we can prove that the transition densities of (Xt)t≥0 are controlled by the metrics δ ψ,1/treplacing dψ,t and dψ,/treplacing δψ,t.

On a limit structure of the Galton-Watson branching processes with regularly varying generating functions

Imomov Azam A.

Probability and Mathematical Statistics

2019

Vol. 39, Fasc. 1

61--73

We investigate limit properties of discrete time branching processes with application of the theory of regularly varying functions in the sense of Karamata. In the critical situation we suppose that the offspring probability generating function has an infinite second moment but its tail regularly varies. In the noncritical case, the finite moment of type E [x ln x] is required. The lemma on the asymptotic representation of the generating function of the process and its differential analogue will underlie our conclusions.