Geodesic
Convergence of random sequences with the independent random indices.~I
Teoriâ veroâtnostej i ee primeneniâ, Tome 39 (1994) no. 2, pp. 313-333

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Necessary and sufficient conditions are obtained for the weak convergense of arbitrary random sequences with independent random indices under nonrandom centering and normalization. The relationship between these conditions and the concept of identifiability of families of probability distributions is traced back. As particular cases,. the conditions for the convergence of “growing” random sums and maximum partial random sums are given.
Keywords: random sequences with random indices, weak convergence, tightness, location mixtures, scale mixtures, identifiable mixtures, random sums. 
@article{TVP_1994_39_2_a4,
     author = {V. Yu. Korolev},
     title = {Convergence of random sequences with the independent random {indices.~I}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {313--333},
     publisher = {mathdoc},
     volume = {39},
     number = {2},
     year = {1994},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1994_39_2_a4/}
}
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V. Yu. Korolev. Convergence of random sequences with the independent random indices.~I. Teoriâ veroâtnostej i ee primeneniâ, Tome 39 (1994) no. 2, pp. 313-333. http://geodesic.mathdoc.fr/item/TVP_1994_39_2_a4/