A conditional limit theorem with a random number of summands

S. G. Gushchin

Diskretnaya Matematika, Tome 9 (1997) no. 2, pp. 131-138.

Résumé

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 $$ \sum_{i=1}^m \eta_i = n. $$