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 $$ \frac{\partial u}{\partial t}={\left(-1\right)}^m{\mathcal{A}}_{\mu }, $$ where $$ {\mathcal{A}}_m $$ is a convolution operator with a generalized function |x|−2m − 2, m ∈ N.