  The paper is devoted to the relationship between the continuous Markovian
description of Levy flights developed previously and their equivalent
representation in terms of discrete steps of a wandering particle, a certain
generalization of continuous time random walks. Our consideration is confined
to the one-dimensional model for continuous random motion of a particle with
inertia. Its dynamics governed by stochastic self-acceleration is described as
motion on the phase plane {x,v} comprising the position x and velocity v=dx/dt
of the given particle. A notion of random walks inside a certain neighbourhood
L of the line v=0 (the x-axis) and outside it is developed. It enables us to
represent a continuous trajectory of particle motion on the plane {x,v} as a
collection of the corresponding discrete steps. Each of these steps matches one
complete fragment of the velocity fluctuations originating and terminating at
the "boundary" of L. As demonstrated, the characteristic length of particle
spatial displacement is mainly determined by velocity fluctuations with large
amplitude, which endows the derived random walks along the x-axis with the
characteristic properties of Levy flights. Using the developed classification
of random trajectories a certain parameter-free core stochastic process is
constructed. Its peculiarity is that all the characteristics of Levy flights
similar to the exponent of the Levy scaling law are no more than the parameters
of the corresponding transformation from the particle velocity v to the related
variable of the core process. In this way the previously found validity of the
continuous Markovian model for all the regimes of Levy flights is explained.
