|
'''Recursive step :'''
:<math>
f(n)=\left\{\begin{matrix} fpf+(n), & \mbox{with probability 0.5}\\ fmf-(n), & \mbox{with probability 0.5}\end{matrix}\right.
</math>
where,
:<math>fpf+(n)=f(n-1) + f(n-2)</math>
:<math>fmf-(n)=f(n-1) - f(n-2)</math>
or in other words, the decision whether to add or subtract the previous two elements of the sequence to get the next element, is taken at random with a probability of 0.5 favouring each decision (Saysay with a toss of a fair coin.)
In a sequence, thus constructed, with a probability of 1, (i.e. with extremely rare exceptions), [[almost surely]]) the ratio of the absolute values of successive terms converges to the value of the constant, for large values of ''n''.
==Explication==
The constant was discovered by [[Divakar Viswanath]] in 1999.
[[Johannes Kepler]] had shown that for normal FiboancciFibonacci sequences, (where the randomness of the sign does not occur), the ratio of the successive numbers converged to the [[golden mean]]. Thus, for any large ''n'', the golden mean constant raised to the power of ''n'' yields the nth term of the sequence, with astonishing accuracy.
Though it seems surprising that a similar ratio could be obtained for a series of elements obtained by randomly chosen signs, a little thoughtful intuition wouldthought showshows that there are extremely rare cases where this ratio does not hold. For example, consider the following series
1,1,0,1,-1,0,...
This series is not allowed to "grow beyond" 1 or -1, only because the oscillating signs of + and - appear in a systematic pattern. As long, as the series is constrained by this pattern, Viswanath's constant will never seem to hold for the elements of this sequence. However,but in a perfectly random experiment, the chances that such patterns of + and - are obtained are extremely negligible.
==Significance==
In 1960, [[Hillel Furstenberg]] and [[Harry Kesten]] had shown that for a a general class of random-sequence generating processes that includes the random Fibonacci sequence, the absolute value of the nth''n''th term converges to a power of of a fixed constant. This seminal proof was highly significant toin advances in [[laser]] technology and the study of glasses[[glass]]es. The [[Nobel Prize for Physics]] in [[1977]] went to [[Philip Warren Anderson]] of [[Bell Laboratories]], [[Sir Nevill Francis Mott]] of [[Cambridge University]] in [[England]], and [[John Hasbrouck van Vleck ]] of [[Harvard]] "for their fundamental theoretical investigations of the electronic structure of magnetic and disordered systems". These inverstiagtionsinvestigations were largely dependent on Furstenberg's and Kesten's proof. By specifying the exact value of the constant, Viswanath has given the proof a solid finish.
Viswanath's constant is expected to be of great significance to the study of probabilistic sequences. For example, it may suitably explain the case of rabbits randomly allowed to prey on each other. (See [[Fibonacci sequence]] for the original statement of the rabbit problem.) This step, would allow closer simulation of real world scenarios in various applications.
==External Links==
|
