Geodesic
Asymptotic distribution of sequences with random indices
Matematičeskie zametki, Tome 6 (1969) no. 6, pp. 705-712 Cet article a éte moissonné depuis la source Math-Net.Ru

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A general form is determined for the limit distribution function of a sequence of random vectors with random indices $(S_1(N_n^{(1)},\dots,S_r(N_n^{(r)}))$ in the case when sequences $(S_1(n),\dots,S_r(n))$ and $(N_n^{(1)},\dots,N_n^{(r)})$ for appropriate normalization have a nonsingular joint limit distribution. 
@article{MZM_1969_6_6_a6,
     author = {L. A. Zolotuhina},
     title = {Asymptotic distribution of sequences with random indices},
     journal = {Matemati\v{c}eskie zametki},
     pages = {705--712},
     year = {1969},
     volume = {6},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1969_6_6_a6/}
}
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L. A. Zolotuhina. Asymptotic distribution of sequences with random indices. Matematičeskie zametki, Tome 6 (1969) no. 6, pp. 705-712. http://geodesic.mathdoc.fr/item/MZM_1969_6_6_a6/