Stationary distributions of the simplest dynamical systems in \(R^ n\) perturbed by generalized Poisson process (Q2266527)

Let z(t) be a compound Poisson process having as parameters \(\lambda >0\) and the n-dimensional distribution F; let A be an \(n\times n\) matrix. The author solves \(x(t)=x_ 0+\int^{t}_{0}\) \(Ax(u)du+z(t)\), expresses the infinitely divisible distribution of x(t) and shows that it has a limit (explicitly given), nondepending on \(x_ 0\), for \(t\to \infty\) iff Re \(\mu\) \(<0\) for all eigenvalues \(\mu\) of A and \(\int \log (1+| y|)dF(y)<\infty.\) For a given t, (x(t)-a(\(\lambda)\))/b(\(\lambda)\), for some nonrandom \(a(\lambda),b(\lambda)>0\), converges in distribution when \(\lambda\) \(\to \infty\) to a nondegenerate one iff: b(\(\lambda)\) is regularly varying with exponent 1/\(\alpha\), the limit \(G_ 0\) is stable n-dimensional, F is in the domain of attraction of a stable law G, both with exponent \(\alpha\), \(G,G_ 0\) being bijectively related by a formula expressing \(G_ 0\) as an integral in u of G transformed by exp(-Au) for \(\alpha <2\), etc. If \(A=T\Lambda T^{-1}\), T real, \(\Lambda =(\lambda_ j)\) diagonal, \(\zeta =T^{-1}x(t)\), etc, and \(\lambda\) \(\to 0\) then sgn \(\zeta\) and \((| \zeta_ i|^{-\lambda /\lambda_ i})\), asymptotically, are independent and the second has the distribution of (\(\beta\),...,\(\beta)\) with \(\beta\) uniform on (0,1), etc.