Barbara H. Jasiulis-Gołdyn

Abstract: The paper deals with a new class of random walks strictly connected with the Pareto
distribution. We consider stochastic processes in the sense of generalized convolution or weak
generalized convolution. The processes are Markov processes in the usual sense. Their
structure is similar to perpetuity or autoregressive model. We prove the theorem which
describes the magnitude of the fluctuations of random walks generated by generalized
convolutions.

We give a construction and basic properties of random walks with respect to
the Kendall convolution. We show that they are not classical Lévy processes. The
paper proposes a new technique to cumulate the Pareto-type distributions using a
modification of the Williamson transform and contains many new properties of weakly
stable probability measure connected with the Kendall convolution. It seems that the
Kendall convolution produces a new class of heavy tailed distributions of Pareto-type.

2000 AMS Mathematics Subject Classification: Primary: 60G50; Secondary: 60J05,
44A35, 60E10.

Keywords and phrases: Random walk, generalized convolution, weakly stable
distribution, Kendall convolution, Pareto distribution, Markov process, Williamson
transform.