A simple master Theorem for discrete divide and conquer recurrences
Olivier Garet
TL;DR
This paper develops a discrete master theorem for divide-and-conquer recurrences of the form $X_n=a_n+\sum_{j=1}^m b_j X_{\lfloor n/m_j \rfloor}$, allowing general (potentially random) input sequences $a_n$ without regularity assumptions. By identifying a critical exponent $s_0$ as the positive root of $\sum_{j=1}^m b_j m_j^{-s}=1$ and imposing a non-lattice condition on the $m_j$, the authors prove that $X_n/n^{s_0}$ converges to a linear functional $L=\sum_{j=0}^{\infty} \ell_j a_j$, with explicit Tauberian expressions for the coefficients $\ell_j$ and an affine decomposition $X_n=\sum_{j=0}^n K_n^j a_j$ having $K_n^j/n^{s_0}\to \ell_j$. The results extend to random $a_n$, yielding almost-sure convergence to $L$ under mild moment or tail conditions; under independence and non-atomicity, $L$ is non-degenerate, and with appropriate exponential moment assumptions, $L$ (and thus $X_n/n^{s_0}$) inherits exponential moments. Overall, the work generalizes classical master theorems to irregular, possibly random inputs, providing precise limiting behavior for a broad class of discrete divide-and-conquer recursions and enabling probabilistic analyses of randomized algorithms.
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 $\sup_{n\ge 1}\mathbb{E}(|a_n|)<+\infty$.