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arXiv · 1709.04936

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

Alexander Roitershtein·Zhirou Zhou
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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.

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Alexander Roitershtein, Zhirou Zhou. 2017-09-14. Relative growth of the partial sums of certain random Fibonacci-like sequences. https://doi.org/10.1080/10236198.2017.1378353

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