Limit theorems on convergence to generalized Cauchy type processes

We prove a limit theorem on convergence of mathematical expectations of functionals of sums of independent random variables to a Cauchy problem solution for an evolution equation $\frac{\partial{u}}{\partial{t}}=(-1)^m\mathcal{A}_mu$ where $\mathcal{A}_m$ is a convolution operator with a generalized function $|x|^{-2m-2}, m\in\mathbf{N}$.