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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.

Small deviation probabilities of weighted sums with fast decreasing weights

Rozovsky L. V.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 1

161--178

We examine small deviation probabilities of weighted sums of i.i.d. positive random variables whose distribution function is regularly varying at zero provided that weights are decreasing fast enough.

Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities

Jaroś J., Takaśi K.

Opuscula Mathematica

2015

Vol. 35, no. 1

47--69

We consider n-dimensional cyclic systems of second order differential equations [formula] (*) under the assumption that the positive constants α and β satisfy α1...αn > β1...βn and pi(t) and qi(t) are regularly varying functions, and analyze positive strongly increasing solutions of system (*) in the framework of regular variation. We show that the situation for the existence of regularly varying solutions of positive indices for (*) can be characterized completely, and moreover that the asymptotic behavior of such solutions is governed by the unique formula describing their order of growth precisely. We give examples demonstrating that the main results for (*) can be applied to some classes of partial differential equations with radial symmetry to acquire accurate information about the existence and the asymptotic behavior of their radial positive strongly increasing solutions.