A simple master Theorem for discrete divide and conquer recurrences

The aim of this note is to provide a Master Theorem for some discrete divide and conquer recurrences: $$X_n=a_n+\sum _{j=1}^m{b}_j{X}_{\lfloor \frac{n}{m_j}\rfloor },$$ where the $m_i$’s are integers with $m_i\ge 2$. The main novelty of this work is there is no assumption of regularity or monotonicity for $(a_n)$. Then, this result can be applied to various sequences of random variables $(a_n{)}_{n\ge 0}$, for example such that ${\sup }_{n\ge 1}\mathbb{E}(\mathrm{\mid }{a}_n\mathrm{\mid })<+\mathrm{\infty }$.