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arXiv · 1902.10600

A simple master Theorem for discrete divide and conquer recurrences

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

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Olivier Garet. 2019-02-27. A simple master Theorem for discrete divide and conquer recurrences. https://arxiv.org/abs/1902.10600

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