Geodesic
Local limit theorems for conditionally independent random variables controlled by a finite Markov chain
Teoriâ veroâtnostej i ee primeneniâ, Tome 44 (1999) no. 1, pp. 143-148

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The sufficient conditions for the validity of local theorems for conditionally independent random variables controlled by a finite homogeneous regular Markov chain are given in terms of conditional distributions of summands.
Keywords: conditionally independent random variables, local theorems for densities, local theorem for lattice distributions, normal attraction to the stable law. 
@article{TVP_1999_44_1_a14,
     author = {T. Shervashidze},
     title = {Local limit theorems for conditionally independent random variables controlled by a finite {Markov} chain},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {143--148},
     publisher = {mathdoc},
     volume = {44},
     number = {1},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1999_44_1_a14/}
}
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T. Shervashidze. Local limit theorems for conditionally independent random variables controlled by a finite Markov chain. Teoriâ veroâtnostej i ee primeneniâ, Tome 44 (1999) no. 1, pp. 143-148. http://geodesic.mathdoc.fr/item/TVP_1999_44_1_a14/