The random Fibonacci sequence is defined by t_1 = t_2 = 1 and
t_n = ± t_{n–1} + t_{n–2} ,
for n ? 3, where each ± sign is chosen at random with P(+) = P(–) = 1/2. We can think of all possible such sequences as forming a binary tree T. Viswanath has shown that almost all random Fibonacci sequences grow exponentially at the rate 1.13198824.... He was only able to find 8 decimal places of this constant through the use of random matrix theory and a fractal measure, although Bai has extended the constant by 5 decimal places. Numerical experimentation is inefficient because the convergence is so slow. We will discuss a new computation of Viswanath's constant which is based on a formula due to Kalmár-Nagy, and uses an interesting reduction R of the tree T developed by Rittaud.
Also, we will focus on the growth rate of the expected value of a random Fibonacci sequence, which was studied by Rittaud. This differs from the almost sure growth rate in that we first find an expression for the average of the n^th term in a sequence, and then calculate its growth. We will derive this growth rate using a slightly different and more simplified method than Rittaud, using the tree R and a Pascal-like array of numbers, for which we can further give an explicit formula.
We will also consider what happens to random Fibonacci sequences when we remove the randomness. Specifically, we will choose coefficients which belong to the set {1, –1} and form periodic cycles. By rewriting our recurrences using matrix products, we will analyze sequence growth and develop criteria based on eigenvalue, trace and order, for determining whether a given sequence is bounded, grows linearly or grows exponentially. Further, we will introduce an equivalence relation on the coefficient cycles such that each equivalence class has a common growth rate, and consider the number of such classes for a given cycle length. Lastly we will look at two ways to completely characterize the trace, given the coefficient cycle, by breaking the matrix product up into blocks.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:NSHD.ca#10222/15381 |
| Date | 20 August 2012 |
| Creators | McLellan, Karyn Anne |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
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