Some limit theorems for functionals of random walks in the domain of attraction of the Cauchy law

For different types of random walks in the domain of attraction of the Cauchy law one proves a series of theorems on the weak convergence of the random polygons $\nu_n(t)$ with the nodes $\Big(\frac kn,\frac{\pi}{\log n}\sum_{i=1}^kf(\zeta_i)\Big)$, $k=1,\dots,n$, $\nu_n(0)=0$ in the space $C[0,1]$ to a certain degenerate process.