Geodesic
A conditional limit theorem with a random number of summands
Diskretnaya Matematika, Tome 9 (1997) no. 2, pp. 131-138 
S. G. Gushchin. A conditional limit theorem with a random number of summands. Diskretnaya Matematika, Tome 9 (1997) no. 2, pp. 131-138. http://geodesic.mathdoc.fr/item/DM_1997_9_2_a13/
@article{DM_1997_9_2_a13,
     author = {S. G. Gushchin},
     title = {A conditional limit theorem with a random number of summands},
     journal = {Diskretnaya Matematika},
     pages = {131--138},
     year = {1997},
     volume = {9},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1997_9_2_a13/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

For a sequence of independent identically distributed random vectors with integer-valued non-negative components $(\xi_1^{(i)},\ldots,\xi_s^{(i)},\eta_i)$, $i=1,2,\dots$, we prove a limit theorem for the joint distribution of the sums $$ \sum_{i=1}^m \xi_j^{(i)}, \qquad j=1,\dots,s, $$ for $n\to\infty$ and the random $m$ determined by the condition $$ \sum_{i=1}^m \eta_i = n. $$