  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 following the idea
given in [1]. The processes are Markov processes in the usual sense. Their
structure is similar to perpetuity or autoregressive model. We prove 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 L\'evy 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 classes of
heavy tailed distributions of Pareto-type.
