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

Abstract

We consider certain Fibonacci-like sequences $(X_n{)}_{n\ge 0}$ 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.

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• Random recurrence equations and ruin in a Markov-dependent stochastic economic environment
• One-dimensional linear recursions with Markov-dependent coefficients
• Almost-sure Growth Rate of Generalized Random Fibonacci sequences
• Weak Disorder in Fibonacci Sequences