Viswanath's constant
Encyclopedia
In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation
fn = fn−1 ± fn−2, where the signs + or − are chosen at random with equal probability 1/2, independently for different n. By a theorem of Harry Kesten
and Hillel Furstenberg, random recurrent sequences of this kind grow at a certain exponential rate
, but it is difficult to compute the rate explicitly. In 1999, Divakar Viswanath showed that the growth rate of the random Fibonacci sequence is equal to 1.1319882487943…, a mathematical constant
that was later named Viswanath's constant.
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:
Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....
fn = fn−1 ± fn−2, where the signs + or − are chosen at random with equal probability 1/2, independently for different n. By a theorem of Harry Kesten
Harry Kesten
Harry Kesten is an American mathematician best known for his work in probability, most notably on random walks and percolation theory.- Biography :...
and Hillel Furstenberg, random recurrent sequences of this kind grow at a certain exponential rate
Exponential growth
Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value...
, but it is difficult to compute the rate explicitly. In 1999, Divakar Viswanath showed that the growth rate of the random Fibonacci sequence is equal to 1.1319882487943…, a mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...
that was later named Viswanath's constant.
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 relationA run of the random Fibonacci sequence starts with 1,1 and the value of the each subsequent term is determined by a fair coin
Fair coin
In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair 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:
-
Growth rate
Johannes KeplerJohannes KeplerJohannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...
discovered that as n increases, the ratio of the successive terms of the Fibonacci sequence {Fn} approaches the golden ratioGolden ratioIn mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
which is approximately 1.61803. In 1765, Leonhard EulerLeonhard EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
published an explicit formula, known today as the Binet formula,
It demonstrates that the Fibonacci numbers grow at an exponential rate equal to the golden ratio φ.
In 1960, Hillel Furstenberg and Harry KestenHarry KestenHarry Kesten is an American mathematician best known for his work in probability, most notably on random walks and percolation theory.- Biography :...
showed that for a general class of random matrix products, the norm grows as λn, where n is the number of factors. Their results apply to a broad class of random sequence generating processes that includes the random Fibonacci sequence. As a consequence, the nth root of |fn| converges to a constant value almost surelyAlmost surelyIn probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...
, or with probability one:
An explicit expression for this constant was found by Divakar Viswanath in 1999. It uses Furstenberg's formula for the Lyapunov exponentLyapunov exponentIn mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories...
of a random matrix product and integration over a certain fractal measureFractalA fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
on the Stern–Brocot tree. Moreover, Viswanath computed the numerical value above using floating pointFloating pointIn computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16...
arithmetics validated by an analysis of the rounding error.
Related work
The Embree–Trefethen constant describes the qualitative behavior of the random sequence with the recurrence relation
for different values of β. -
-
