Geodesic
Some limit theorems for functionals of random walks in the domain of attraction of the Cauchy law
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 130-140

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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. 
@article{ZNSL_1985_142_a13,
     author = {M. V. Petrova},
     title = {Some limit theorems for functionals of random walks in the domain of attraction of the {Cauchy} law},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {130--140},
     publisher = {mathdoc},
     volume = {142},
     year = {1985},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a13/}
}
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M. V. Petrova. Some limit theorems for functionals of random walks in the domain of attraction of the Cauchy law. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 130-140. http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a13/