Recursive random variables with subgaussian distributions

with K, n0 ≥ 1, (X n )n≥0 identically distributed as (Xn)n≥0 for r = 1, . . . , K , a random vector I (n) = (I (n) 1 , . . . , I (n) K ) of integers in {0, . . . , n − 1} and a random bn such that (X n )n≥0, . . . , (X ) n )n≥0, (I (n), bn) are independent. The symbol d = denotes equality in distribution. In applications, the I (n) r are random subgroup sizes, bn is a toll function specifying the particular quantity of a combinatorial structure and (X n )n≥0 are copies of the quantity (Xn)n≥0, that correspond to the contribution of subgroup r. Typical parameters Xn range from the depths, sizes and path lengths of trees, the number of various sub-structures or components of combinatorial structures, the number