Relative growth of the partial sums of certain random Fibonacci-like sequences

TL;DR

This paper studies the asymptotic distribution of normalized partial sums of random Fibonacci-like sequences, revealing Pareto-like tail behavior influenced by the noise strength, which acts as a measure of the noise's singularity.

Contribution

It introduces a new probabilistic analysis of Fibonacci-like sequences with noise, showing convergence to a Pareto-like distribution and characterizing noise strength through tail behavior.

Findings

Normalized partial sums converge in distribution to a Pareto-like variable.

The tail index s measures the noise strength and decreases monotonically.

Heavy-tailed distribution indicates the noise is 'singular' and 'big' even with slight perturbations.

Abstract

We consider certain Fibonacci-like sequences $(X_n)_{n\geq 0}$ perturbed with a random noise. Our main result is that $\frac{1}{X_n}\sum_{k=0}^{n-1}X_k$ converges in distribution, as $n$ goes to infinity, to a random variable $W$ with Pareto-like distribution tails. We show that $s=\lim_{x\to \infty} \frac{-\log P(W>x)}{\log x}$ is a monotonically decreasing characteristic of the input noise, and hence can serve as a measure of its strength in the model. Heuristically, the heavy-taliped limiting distribution, versus a light-tailed one with $s=+\infty,$ can be interpreted as an evidence supporting the idea that the noise is "singular" in the sense that it is "big" even in a "slightly" perturbed sequence.