Jolanta Misiewicz

Simulating Stable, Substable and Weakly Stable Random Vectors Jolanta Misiewicz

We describe in this book stable, sub-stable and the most general class of weakly stable distributions and processes, which seem to be good candidates for use in stochastic modeling in many areas. In the firs chapter we recall basic properties of characteristic functions for random variables and vectors. By norm-dependent characteristic functions we describe norm-dependent positive definite matrices, which are extensively used in stochastic modeling random vectors with fixed correlation structure. The second chapter contains five definitions of stable variables together with proofs of their equivalence. Next we give few representation involving much simpler random variables. Fourth chapter contains a description of stable vectors and their spectral representation. We recall covariation function and covariation ratio - parameters describing the dependence structure of symmetric stable vectors. In the last chapter we talk about much less known weakly stable vectors. We give detailed proofs of their properties. In section 6 we talk about elliptically symmetric and 1-dependent vectors together with the Cambanis, Keener and Simons construction.