Convolutions Induced Discrete Probability Distributions and a New Fibonacci Constant
aa r X i v : . [ m a t h . P R ] M a y CONVOLUTIONS INDUCED DISCRETE PROBABILITYDISTRIBUTIONS AND A NEW FIBONACCI CONSTANT
ARULALAN RAJAN, JAMADAGNI, VITTAL RAO AND ASHOK RAO
Abstract.
This paper proposes another constant that can be associ-ated with Fibonacci sequence. In this work, we look at the probabil-ity distributions generated by the linear convolution of Fibonacci se-quence with itself, and the linear convolution of symmetrized Fibonaccisequence with itself. We observe that for a distribution generated bythe linear convolution of the standard Fibonacci sequence with itself,the variance converges to 8.4721359. . . . Also, for a distribution gener-ated by the linear convolution of symmetrized Fibonacci sequences, thevariance converges in an average sense to 17.1942. . . , which is approxi-mately twice that we get with common Fibonacci sequence. Introduction
Fibonacci sequence is known to have some fascinating properties. Themost classical of these is the convergence of the ratio f [ n ] f [ n − to the goldenmean ϕ [1],[2]. Viswanath [3], proved that with certain randomness intro-duced in Fibonacci sequence, the n th root of the absolute value of the n th term in the sequence converges,with a probability of 1 (i.e. with extremelyrare exceptions, almost surely), to another constant 1.13198824 . . . . In thispaper, we propose yet another constant that can be associated with the Fi-bonacci sequence. This constant, corresponds to the limit variance of thedistribution generated by the convolution of Fibonacci sequence with itself.The variance of such a distribution asymptotically approaches to 8.4721359. . . .The paper is organized as follows:Section 2 of the paper discusses, in brief, some of the known constants as-sociated with the Fibonacci sequence. In section 3 we propose the use ofFibonacci sequence convolution to generate discrete probability distribu-tions. In section 4 we discuss the results followed by a few observations insection 5, with conclusions in section 6.2. Some Constants Associated with Fibonacci Sequence
It is well known that the classical Fibonacci sequence is generated by therecursion given in eq.1 below[4] f [ n ] = f [ n −
1] + f [ n −
2] where f [0] = 0; f [1] = 1 . (1) A closed form expression for generating Fibonacci sequence is given by theBinet’s formula, as in Eq.2, below, that relates golden mean ϕ to the se-quence. f [ n ] = ϕ n − (1 − ϕ ) n √ ϕ n − ( − /ϕ ) n √ ϕ = √ . This is one of the most important and well known con-stants that has been associated with the Fibonacci sequence ever since Ke-pler showed that the golden ratio is the limit of the ratios of successiveterms of the Fibonacci sequence. A decade ago, Viswanath [3] computedanother constant that explained how fast the random Fibonacci sequencesgrow. In his work, he introduced randomness in the recurrence that gener-ates Fibonacci sequence. Introducing randomness in Fibonacci sequence,we get a random Fibonacci sequence that is defined by the recurrence f [ n ] = f [ n − ± f [ n −
2] with signs chosen as given below in Eq. 3, f [ n ] = ( f [ n −
1] + f [ n − , with probability 0.5; f [ n − − f [ n − , with probability 0.5 . (3)Viswanath showed that in the case of the above random Fibonacci sequence, n p | f [ n ] | → . . . . with probability 1, as n → ∞ . (4)In this work, we propose yet another constant that is closely related to theFibonacci sequence.3. Convolution of Fibonacci Sequence and DiscreteProbability Distributions
The motivation for this work comes from attempts to exploit the discretenature of integer sequences for generating discrete probability distributions.In [5], the authors have looked at slow growing sequences and their convolu-tions, that approximates a Gaussian distribution with a mean squared errorof about 10 − or even less. In this section, we look at the discrete probabilitydistributions generated by a single convolution of Fibonacci sequences.Let x [ n ] and x [ n ] be two discrete sequences. Then the linear convolutionof the two sequences is defined as, in Eq.(5),below. y [ n ] = ∞ X k = −∞ x [ k ] x [ n − k ] (5)If x [ n ] and x [ n ] are two finite length sequences of length L and M respec-tively, then the length of y [ n ] is L + M −
1. If the two sequences to beconvolved, have the same length L , then the length of y [ n ] is 2 L −
1. In thiswork, both x [ n ] and x [ n ] are finite length Fibonacci sequences of the samelength, L . The convolution result is taken as the profile of the discrete prob-ability distribution. The set of indices, n , namely 1 , , , . . . , L + M −
1, of
IBONACCI SEQUENCE: SOME OBSERVATIONS AND A NEW CONSTANT 3 y [ n ] = x [ n ] ∗ x [ n ], is considered the set of values that a discrete randomvariable X can take. The probability of X = n is defined by P ( X = n ) = y [ n ]sum( y [ n ]) , where, sum( y [ n ]) = L + M − X n =1 y [ n ] (6)Figure 1 illustrates the convolution profiles obtained for different lengthsof the classical Fibonacci sequence. Figure 2 gives the convolution of twoFibonacci sequences of the same length L , with symmetry employed at L/ L/ L . Figure 2 also compares the convolutionresult with an estimated normal probability density function with the samevariance as the convolution result. Length c onv (f i bon acc i , f i bon acc i ) Figure 1.
Linear Convolution of Fibonacci Sequences4.
Results
In the previous section, we looked at the linear convolution of Fibonaccisequence and the discrete distributions generated. We find that, increasingthe length of Fibonacci Sequence and employing symmetry results in a profilethat is similar to a very narrow Gaussian profile. For these distributions, weplot the variance and standard deviation as functions of the length of thesequence. From figure 3, we observe that the variance of the distributionsmoothly converge to a constant. This is due to the asymptotic and rapidgrowth of the Fibonacci sequence for large values of n . We also find fromfigure 3, that the convergence, in case of Fibonacci sequence with symmetryemployed, is only in the average sense. ARULALAN RAJAN, JAMADAGNI, VITTAL RAO AND ASHOK RAO x f ( x ) Estimated PDFObserved PDF
Figure 2.
Fibonacci sequence (symmetry employed) convo-lution and narrow Gaussian distribution
X: 38Y: 8.472
Length L V a r i a n ce ( (cid:86) ) X: 68Y: 8.472 X: 89Y: 8.472X: 32Y: 17.07 X: 57Y: 16.94
Symmetrized Fibonacci SequenceGeneral Fibonacci Sequence
Figure 3.
Variation of Variance with Length of Fibonacci Sequence5.
Observations
We give below, some of the interesting properties, associated with Fi-bonacci sequence, that we have observed. Though, we have established afew of them mathematically, we are currently working on the analysis of therest of them. • Observation 1. With the standard Fibonacci sequence itself be-ing considered as a distribution function, the variance converges to4.23606797750108.
IBONACCI SEQUENCE: SOME OBSERVATIONS AND A NEW CONSTANT 5
X= 97Y= 192 X= 138Y= 274X= 52Y= 102 X= 200Y= 398X= 5Y= 8
Length of Fibonacci Sequence L o ca ti on o f t h e m a x i m u m e l e m e n t i n t h e li n ea r c onvo l u ti on o f e qu a l l e ng t h F i bon acc i S e qu e n ce s Figure 4.
Plot of Index of Absolute Maximum for convolu-tion of Fibonacci Sequences • Observation 2. When we convolve (linear) standard Fibonacci se-quence with itself and take the resulting sequence as a distributionfunction, the variance converges to 8.47213595500216. • Observation 3. The value mentioned in observation (2) is twice thevalue mentioned in observation (1). • Observation 4. Let S be the standard Fibonacci Sequence. Let S be the sequence S in the reverse order. Let S be the linear convolu-tion of S and S . The variance of S converges to 8.47213595500216. • Observation 5. Let S be the standard Fibonacci Sequence. Let S be the sequence S in the reverse order. Let S be the linear con-volution of S and S . Let S be a sequence obtained by linearlyconvolving S with S or S . The variance in either cases saturatesto 12.7081989582623. This value is 3 times the value listed in obser-vation (1). • Observation 6: The maximum value in the sequence resulting fromthe linear convolution of two standard Fibonacci sequences of length L occurs at 2 L − • Observation 7: Let S be a standard Fibonacci, L length sequence.Let S be the sequence that is symmetrically extended version of S . S has a length 2 L . Convolving S with itself yields anothersequence S , of length 4 L −
1. Considering S as a distribution, thevariance of S converges in an average sense to 17.19423665579735.The swing is between 17.4442399455347 and 16.9442333660600.From figure 1, we find that the sequence, resulting from the linear con-volution of two increasing Fibonacci sequences of the same length, has itsmaximum at the extreme. From figure 4, we find that the maximum is ARULALAN RAJAN, JAMADAGNI, VITTAL RAO AND ASHOK RAO located at 2 L −
2. Mathematically, one can prove this by many ways. How-ever, we follow the approach of proof by contradiction. For this, it is enoughto show that(I) y [2 L − < y [2 L − y [2 L − < y [2 L − y [2 L −
1] is the last element of the sequence y [ n ] of length 2 L −
1. Thisis because the linear convolution of two monotonically increasing functionswill have only one absolute maximum.
Proof.
First we prove y [2 L − < y [2 L − y [2 L −
1] = ( f [ L ]) , (7) ⇒ = ( f [ L −
1] + f [ L − (8) y [2 L −
1] = ( f [ L − + ( f [ L − + 2 f [ L − f [ L −
2] (9)Now, we look at y [2 L − y [2 L −
2] = f [ L − f [ L ] + f [ L − f [ L ] (10)= 2 f [ L − f [ L ] (11)We need to prove that Eq.(9) is less than Eq.(10)Let us assume that y [2 L − ≥ y [2 L −
2] i.e., ( f [ L − + ( f [ L − + 2 f [ L − f [ L − ≥ f [ L − f [ L ] (12) ≥ f [ L − • ( f [ L −
1] + f [ L − ≥ f [ L − + 2 f [ L − f [ L −
2] (14) ( f [ L − + ( f [ L − + 2 f [ L − f [ L − ≥ f [ L − + 2 f [ L − f [ L −
2] (15)
From (15) we find that, ( f [ L − ≥ ( f [ L − (16)This implies that f [ L − > f [ L −
1] which is a contradiction arising due toour assumption that y [2 L − ≥ y [2 L − y [2 L − < y [2 L − ∀ L > y [2 L − < y [2 L − y [2 L − ≥ y [2 L −
2] i.e., y [2 L −
3] = 2 f [ L − f [ L ] + ( f [ L − (18) ⇒ f [ L − f [ L ] + ( f [ L − ≥ f [ L − f [ L ] (19)Substituting for f [ L ] = f [ L −
1] + f [ L − f [ L − + 2 f [ L − f [ L −
2] + ( f [ L − ≥ f [ L − + 2 f [ L − f [ L −
2] (20) f [ L − ≥ ( f [ L − (21) ⇒ √ ≥ f [ L − f [ L −
2] (22)
IBONACCI SEQUENCE: SOME OBSERVATIONS AND A NEW CONSTANT 7
This is a contradiction, as the ratio of f [ L − f [ L − approaches the golden ratio ϕ which is greater than √
2. Thus we find that on convolving Fibonaccisequence of length L , ( L ≥ y [ n ], thathas its maximum at n = 2 L − (cid:3) From the previous section 4 we note that the variance of a discrete distri-bution generated by the linear convolution of Fibonacci sequence with itself,saturates to a constant of value 8.4721359. . . . For a distribution generatedby the linear convolution of symmetrized Fibonacci sequences, the variancesaturates in an average sense to 17.1942. . . , which is approximately twotimes 8.4721359. . . . 6.
Conclusion
This work proposes another constant 8.4721359. . . that can be associatedwith Fibonacci sequence. The constant corresponds to the value to whichthe variance, of the discrete distribution generated by linear convolutionof Fibonacci sequences of the same length, converges to. Further, it isinteresting to observe that on performing linear convolution of a Fibonaccisequence of length L with itself, the maximum value always occurs at 2 L − References [1] J. Kepler,
The Six-Cornered Snowflake , Oxford University Press, 1966.[2] R. Knott,
Fibonacci Numbers and the Golden Section
The On-Line Encyclopedia of Integer Sequences
Integer Sequence based Discrete Gaussian andDiscrete Random Number Generator , Proc. of the 17th Intl. Conf. on Advanced Com-puting and Communication, Dec.2009, Bangalore, India.
Centre for Electronics Design and Technology, Indian Institute of Sci-ence, Bangalore, India
E-mail address : [email protected]@cedt.iisc.ernet.in