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}.

Read more about this topic:  Random Fibonacci Sequence

Famous quotes containing the word description:

    “A sound mind in a sound body, is a short, but full description of a happy state in this World: he that has these two, has little more to wish for; and he that wants either of them, will be little the better for anything else.”
    —John Locke (1632–1704)

    “As they are not seen on their way down the streams, it is thought by fishermen that they never return, but waste away and die, clinging to rocks and stumps of trees for an indefinite period; a tragic feature in the scenery of the river bottoms worthy to be remembered with Shakespeare’s description of the sea-floor.”
    —Henry David Thoreau (1817–1862)

    “I fancy it must be the quantity of animal food eaten by the English which renders their character insusceptible of civilisation. I suspect it is in their kitchens and not in their churches that their reformation must be worked, and that Missionaries of that description from [France] would avail more than those who should endeavor to tame them by precepts of religion or philosophy.”
    —Thomas Jefferson (1743–1826)