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
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:
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:
Read more about this topic: Random Fibonacci Sequence
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