A simple master Theorem for discrete divide and conquer recurrences

TL;DR

This paper introduces a straightforward Master Theorem for a class of discrete divide and conquer recurrences that does not require regularity or monotonicity assumptions on the sequence terms, broadening applicability.

Contribution

It provides a novel Master Theorem applicable to recurrences with non-regular, non-monotonic sequences, including those involving bounded expected values of random variables.

Findings

Applicable to sequences with bounded expected values

Handles non-regular, non-monotonic sequences

Extends divide and conquer analysis to broader cases

Abstract

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 $\sup_{n\ge 1}\mathbb{E}(|a_n|)<+\infty$.