We study non-stationary random length sequences of regularly varying distributed random variables. The results by Goldaeva 2013 that are valid for non-stationary random sequences with deterministic length are extended to random length sequences. Namely, we deal with a doubly-indexed array of regularly varying r.v.s $Y_{n,i}$ in which the "row index" $n$ corresponds to time, and the "column index" $i$ corresponds to the level. On the same probability space, one assumes the existence of a sequence of non-negative integer-valued r.v.s $\{N_n: n\ge 1\}$ such that $EN_n