A Probabilistic Approach to Generalized Zeckendorf Decompositions

TL;DR

This paper introduces a probabilistic framework for analyzing generalized Zeckendorf decompositions, linking them to Markov chains to derive laws of large numbers, CLTs, and gap properties.

Contribution

It presents a novel Markov chain-based approach to study the distribution and properties of generalized Zeckendorf decompositions.

Findings

Established laws of large numbers for digit sums.

Proved central limit theorems for digit distributions.

Analyzed gap (zero) statistics in expansions.

Abstract

Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. The simplest cases are base-$b$ expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence. The expansions are finite sequences of nonnegative integer coefficients (satisfying certain technical conditions to guarantee uniqueness of the decomposition) and which can be viewed as analogs of sequences of variable-length words made from some fixed alphabet. In this paper we present a new approach and construction for uniform measures on expansions, identifying them as the distribution of a Markov chain conditioned not to hit a set. This gives a unified approach that allows us to easily recover results on the expansions from analogous results for Markov chains, and in this paper we focus on laws of large numbers, central limit theorems for sums of digits, and statements on gaps (zeros) in expansions. We expect the approach to prove useful in other similar contexts.