Geodesic
Local limit theorems for functionals in the conditional invariance principle
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part X, Tome 158 (1987), pp. 32-38

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We prove the strong convergence of distributions of a large class of functionals of random polygonal paths constructed from independent, identically distributed variables. 
@article{ZNSL_1987_158_a2,
     author = {P. Breier},
     title = {Local limit theorems for functionals in the conditional invariance principle},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {32--38},
     publisher = {mathdoc},
     volume = {158},
     year = {1987},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1987_158_a2/}
}
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P. Breier. Local limit theorems for functionals in the conditional invariance principle. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part X, Tome 158 (1987), pp. 32-38. http://geodesic.mathdoc.fr/item/ZNSL_1987_158_a2/