Random Fibonacci Sequence - Description

Description

The random Fibonacci sequence is an integer random sequence {fn}, where f1 = f2 = 1 and the subsequent terms are determined from the random recurrence relation

f_n = \begin{cases} f_{n-1}+f_{n-2}, & \text{ with probability 1/2}; \\ f_{n-1}-f_{n-2}, & \text{ with probability 1/2}. \end{cases}

A run of the random Fibonacci sequence starts with 1,1 and the value of the each subsequent term is determined by a fair coin toss: given two consecutive elements of the sequence, the next element is either their sum or their difference with probability 1/2, independently of all the choices made previously. If in the random Fibonacci sequence the plus sign is chosen at each step, the corresponding run is the Fibonacci sequence {Fn},

If the signs alternate in minus-plus-plus-minus-plus-plus-... pattern, the result is the sequence

However, such patterns occur with vanishing probability in a random experiment. In a typical run, the terms will not follow a predictable pattern:

1, 1, 2, 3, 1, -2, -3, -5, -2, -3, \ldots \text{ for the signs } +, +, -, -, -, +, -, -, \ldots.

Similarly to the deterministic case, the random Fibonacci sequence may be profitably described via matrices:

where the signs are chosen independently for different n with equal probabilities for + or −. Thus

where {Mk} is a sequence of independent identically distributed random matrices taking values A or B with probability 1/2:

A=\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}, \quad B=\begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix}.

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