### Book chapter

## Random walks in non homogeneous Poisson environment

In the first part of the paper we consider a "random flight" process in \(R^d\) and obtain the weak limits under different transformations of the Poissonian switching times. In the second part we construct diffusion approximations for this process and investigate their accuracy. To prove the weak convergence result we use the approach of Stroock and Varadhan (1979). We consider more general model which may be called "random walk over ellipsoids in \(R^d\)". For this model we establish the Edgeworth type expansion. The main tool in this part is the parametrix method (Konakov (2012), Konakov and Mammen (2009)).