  We consider the moving particle process in Rd which is defined in the
following way. There are two independent sequences (Tk) and (dk) of random
variables. The variables Tk are non negative and form an increasing sequence,
while variables dk form an i.i.d sequence with common distribution concentrated
on the unit sphere. The values dk are interpreted as the directions, and Tk as
the moments of change of directions. A particle starts from zero and moves in
the direction d1 up to the moment T1 . It then changes direction to d2 and
moves on within the time interval T2 minus T1 , etc. The speed is constant at
all sites. The position of the particle at time t is denoted by X(t). We
suppose that the points (Tk) form a non homogeneous Poisson point process and
we are interested in the global behavior of the process (X(t)), namely, we are
looking for conditions under which the processes (Y(T,t), T is non negative),
Y(T,t) is X(tT) normalized by B(T), t in (0, 1), weakly converges in C(0, 1) to
some process Y when T tends to infinity. In the second part of the paper the
process X(t) is considered as a Markov chain. We construct diffusion
approximations for this process and investigate their accuracy. The main tool
in this part is the paramertix method.
