Convolutions Induced Discrete Probability Distributions and a New Fibonacci Constant

TL;DR

This paper explores probability distributions derived from Fibonacci sequences through convolution, revealing new constants and variance behaviors that deepen understanding of Fibonacci-related stochastic processes.

Contribution

It introduces a novel Fibonacci-related constant based on convolution-induced distributions and analyzes their variance convergence properties.

Findings

Variance of standard Fibonacci convolution converges to 8.4721359.

Variance of symmetrized Fibonacci convolution converges to approximately 17.1942.

Symmetrized Fibonacci convolution variance is roughly twice that of standard Fibonacci convolution.

Abstract

This paper proposes another constant that can be associated with Fibonacci sequence. In this work, we look at the probability distributions generated by the linear convolution of Fibonacci sequence with itself, and the linear convolution of symmetrized Fibonacci sequence with itself. We observe that for a distribution generated by the linear convolution of the standard Fibonacci sequence with itself, the variance converges to 8.4721359... . Also, for a distribution generated by the linear convolution of symmetrized Fibonacci sequences, the variance converges in an average sense to 17.1942 ..., which is approximately twice that we get with common Fibonacci sequence.