# A note on Fibonacci Sequences of Random Variables

## TL;DR

This paper investigates the distributional and limit properties of Fibonacci-like sequences generated by random initial variables, providing insights into their probabilistic behavior and convergence characteristics.

## Contribution

It introduces a framework for analyzing Fibonacci sequences of random variables, focusing on their distributional and limit properties based on initial joint distributions.

## Key findings

- Distributional properties derived for Fibonacci random sequences
- Limit behavior and convergence results established
- Framework applicable to various initial distributions

## Abstract

The focus of this paper is the random sequences in the form $\{X_{0},X_{1},$ $X_{n}=X_{n-2}+X_{n-1},n=2,3,..\dot{\}},$ referred to as Fibonacci Random Sequence (FRS). The initial random variables $X_{0}$ and $X_{1}$ are assumed to be absolutely continuous with joint probability density function (pdf) $f_{X_{0},X_{1}}.$ The FRS is completely determined by $X_{0}$ and $X_{1}$ and the members of Fibonacci sequence $\digamma \equiv\{0,1,1,2,3,5,8,13,21,34,55,89,144,...\}.$ We examine the distributional and limit properties of the random sequence $X_{n},n=0,1,2,...$ .