Lyapunov exponent and variance in the CLT for products of random matrices related to random Fibonacci sequences

Rajeshwari Majumdar; Phanuel Mariano; Hugo Panzo; Lowen Peng; Anthony Sisti

doi:10.3934/dcdsb.2020126

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2020, Volume 25, Issue 12: 4779-4799. Doi: 10.3934/dcdsb.2020126

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Lyapunov exponent and variance in the CLT for products of random matrices related to random Fibonacci sequences

1.
Department of Politics, New York University, New York, NY 10012, USA
2.
Department of Mathematics, Union College, Schenectady, NY 12308, USA
3.
Faculties of Electrical Engineering and Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel
4.
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA
5.
Department of Biostatistics, Brown University, Providence, RI 02912, USA

^* Corresponding author: Phanuel Mariano
^* Corresponding author: Phanuel Mariano

† Research was supported in part by NSF Grant DMS-1262929.
‡ Research was supported in part by NSF Grant DMS-1405169, 1712427.
** Research was supported in part by the UConn Mathematics Department and a Zuckerman fellowship.

Received: April 2019

Revised: December 2019

Early access: April 2020

Published: December 2020

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Abstract

We consider three matrix models of order 2 with one random entry $ \epsilon $ and the other three entries being deterministic. In the first model, we let $ \epsilon\sim \rm{Bernoulli}\left(\frac{1}{2}\right) $. For this model we develop a new technique to obtain estimates for the top Lyapunov exponent in terms of a multi-level recursion involving Fibonacci-like sequences. This in turn gives a new characterization for the Lyapunov exponent in terms of these sequences. In the second model, we give similar estimates when $ \epsilon\sim \rm{Bernoulli}\left(p\right) $ and $ p\in [0, 1] $ is a parameter. Both of these models are related to random Fibonacci sequences. In the last model, we compute the Lyapunov exponent exactly when the random entry is replaced with $\xi\epsilon$ where $\epsilon$ is a standard Cauchy random variable and $\xi$ is a real parameter. We then use Monte Carlo simulations to approximate the variance in the CLT for both parameter models.

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Mathematics Subject Classification: Primary: 37H15; Secondary: 37M25, 60B20, 60B15, 11B39.

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Figure 1. Histogram

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Figure 2. CDF

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Figure 3. $ n = 1\, 000\, 000 $

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Figure 4. $ \lambda(\xi) $ vs. $ \xi $

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Figure 5. $ k = 0.01 $, $ n = 1000 $, $ m = 1\, 000\, 000 $

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Figure 6. $ k = 0.25 $, $ n = 1000 $, $ m = 5\, 000\, 000 $

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Figure 7. $ k = 0.01 $, $ n = 1000 $, $ m = 1\, 000\, 000 $

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