AA note on Fibonacci Sequences of Random Variables
Ismihan BayramogluDepartment of Mathematics, Izmir University of Economics, Izmir, TurkeyE-mail: [email protected] 27, 2019
Abstract
The focus of this paper is the random sequences in the form { X , X , X n = X n − + X n − , n =2 , , .. ˙ } , referred to as Fibonacci Sequence of Random Variables (FSRV). The initial random variables X and X are assumed to be absolutely continuous with joint probability density function (pdf) f X ,X . The FSRV is completely determined by X and X and the members of Fibonacci sequence (cid:122) ≡ { , , , , , , , , , , , , , ... } . We examine the distributional and limit properties ofthe random sequence X n , n = 0 , , , ... .Key words. Random variable, distribution function, probability density function, sequence ofrandom variables. Let { Ω , (cid:122) , P } be a probability space and X i ≡ X i ( ω ) , ω ∈ Ω , i = 0 , f X ,X ( x, y ) . Consider a sequence of random variables X n ≡ X n ( ω ) , n ≥ { Ω , (cid:122) , P } defined as { X , X ,X n = X n − + X n − , n = 2 , , .. } . We call this sequence ”the Fibonacci Sequence of Random Variables”.It is clear that X = X + X , X = X +2 X , ... and for any n = 0 , , , ... we have X n = a n − X + a n X , where { a n = a n − + a n − , n = 2 , , ... ; a = 0 , a = 1 , a = 1 } is the Fibonacci sequence (cid:122) ≡{ , , , , , , , , , , , , , ... } . It is also clear that the Fibonacci Sequence of Random Variables(FSRV) X n , n = 0 , , , ... is the sequence of dependent random variables based on initial random variables X and X , which fully defined by the members of the Fibonacci sequence (cid:122) . We are interested in the1 a r X i v : . [ s t a t . O T ] F e b ehavior of FSRV, i.e. the distributional properties of X n and joint distributions of X n and X n + k forany n and k. In the Appendix Figure A1 and Figure A2, we present some examples of realizations ofFSRV in the case of independent random variables X and X having Uniform(0,1) distribution andStandard normal distribution with the R codes provided.This paper is organized as follows. In Section 2, the probability density function of X n is considered,followed by a discussion of two cases where X and X have exponential and uniform distributions,respectively. Then, there is an investigation of limit behavior of ratios of some characteristics of pdfof X n for large n. In the considered examples, the ratio of maximums of the pdfs, modes and expectedvalues of consecutive elements of FSRV converge to golden ratio ϕ ≡ −√ = 1 , ... . Theratio X n +1 /X n and normalized sums of X n ’s for large n are discussed in Section 3. In Section 4, thefocus is on the joint distributions of X n and X n + k , for 2 ≤ k ≤ n and on the prediction of X n + k given X n . Consider X n = a n − X + a n X , n = 0 , , , ... , where X and X are absolutely continuous randomvariables with joint pdf f X ,X ( x, y ) , ( x, y ) ∈ R and a n , n = 0 , , , ... is the Fibonacci sequence. Denoteby f and f the marginal pdf’s of X and X , respectively. Theorem 1
The pdf of X n is f X n ( x ) = 1 a n a n − ∞ (cid:90) −∞ f X ,X ( x − ta n − , ta n ) dt. (1)If X and X are independent, then f X n ( x ) = 1 a n a n − ∞ (cid:90) −∞ f X ( x − ta n − ) f X ( ta n ) dt. (2) Proof.
Equations (1) and (2) are straightforward results of distributions of linear functions of randomvariables (see eg., Feller (1971), Ross (2002), Gnedenko (1978), Skorokhod (2005))
Case 1
Exponential distribution. Let X and X be independent and identically distributed (iid) randomvariables having exponential distribution with parameter λ = 1 . Then the pdf of X n is f X n ( x ) = 1 a n − (cid:26) exp (cid:18) xa n − a n − a n (cid:19) − (cid:27) exp( − xa n − ) , x ≥ , n = 3 , , ... (3) f X ( x ) = x exp( − x ) , x ≥ . elow in Figure 1, the graphs of f X n ( x ) for different values of n are presented.Figure 1. Graphs of f X n ( x ) , n = 2 , , , , , , , given in (3) The expected value of X n is EX n = 1 a n − ∞ (cid:90) x exp (cid:18) − x (cid:18) a n − a n − a n − a n (cid:19)(cid:19) dx − ∞ (cid:90) x exp( − xa n − ) dx = 1 a n − (cid:18) a n a n − ( a n − a n − ) − a n − (cid:19) = a n +1 . and variance is V ar ( X n ) = 1 a n − ∞ (cid:90) x exp (cid:18) − x (cid:18) a n − a n − a n − a n (cid:19)(cid:19) dx − ∞ (cid:90) x exp( − xa n − ) dx = a n − . Theorem 2
Let M n = max 52 = 1 , ... is the golden ratio. Proof. The following can easily be verified ddx f X n ( x ) = ( − xa n − )( e xan − an − an − a n − a n − + e − xan − e xan − an − an a n − a n = 0 . (4)3he equation (4) has unique solution x ∗ n = a n − a n ln (cid:16) a n a n − a n − (cid:17) a n − . Therefore X n is unimodal and we have M n = f X n ( x ∗ n ) = 1 a n − a n − (cid:18) a n a n − a n − (cid:19) − anan − M n +1 = f X n +1 ( x ∗ ) = 1 a n +1 − a n − (cid:18) a n +1 a n +1 − a n − (cid:19) − an +1 an − and using lim n →∞ a n + α a n = ϕ α we obtain lim n →∞ M n M n +1 = x ∗ n +1 x ∗ n = ϕ. Case 2 Uniform distribution. Let X and X be iid with U nif orm (0 , distribution. Then from (2) weobtain f X n ( x ) = 1 a n a n − a n (cid:90) f X ( x − ta n − ) dt = a n − a n a n − a n (cid:90) dF X ( x − ta n − )= 1 a n (cid:26) F X ( xa n − ) − F X ( x − a n a n − ) (cid:27) = , x < and x > a n + a n − xa n a n − ≤ x ≤ a n − a n a n − ≤ x ≤ a n a n (1 − x − a n a n − ) a n ≤ x ≤ a n + a n − . (5)4 igure 2. Graphs of f X n ( x ) , n = 5 , , , , , given in (5)It can be easily verified that E ( X n ) = a n − + a n = a n +1 and var ( X n ) = a n − + a n . One can observe that f X n ( x ) is not unimodal, f X n ( x ) is constant in the interval ( a n − , a n ) and M n = max Theorem 3 Let E ( X i ) = µ i , V ar ( X i ) = σ i , i = 0 , and Y n ( ω ) ≡ Y n = X n − E ( X n ) (cid:112) V ar ( X n ) = X + a n a n − X − ( µ + a n a n − µ ) (cid:114) σ + a n a n − σ , ω ∈ Ω . Then, Y n → Y ≡ X + ϕX − ( µ + ϕµ ) (cid:112) σ + ϕ σ , as n → ∞ for all ω ∈ Ω . The limiting random variable Y ≡ Y ( ω ) has distribution function (cdf ) P { Y ≤ x } = P { X + ϕX ≤ x (cid:113) σ + ϕ σ + ( µ + ϕµ ) } . (6)It is clear that the pdf of X + ϕX is f X + ϕX ( x ) = 1 ϕ ∞ (cid:90) f X ( x − t ) f X ( tϕ ) dt. (7)and the pdf of Y is then f Y ( x ) = (cid:113) σ + ϕ σ f X + ϕX ( x (cid:113) σ + ϕ σ + ( µ + ϕµ )) (8) Example 1 Let X and X be iid random variables having exponential distribution with parameter λ = 1 , then from (7) we have f X + ϕX ( x ) = 1 ϕ x (cid:90) exp( − x − t ) exp( t − tϕ ) dt = exp( − x ) ϕ − x (1 − ϕ )) − . Therefore, P { Y ≤ x } = P { X + ϕX ≤ x (cid:113) σ + ϕ σ + ( µ + ϕµ ) } = c ( x ) (cid:90) (cid:26) exp( − t ) ϕ − t (1 − ϕ )) − (cid:27) dt, here c ( x ) = x (cid:112) σ + ϕ σ + ( µ + ϕµ ) . And the pdf is f Y ( x ) = (cid:112) σ + ϕ σ (cid:110) exp( − c ( x )) ϕ − (cid:104) exp (cid:16) c ( x )(1 − ϕ ) (cid:17) − (cid:105)(cid:111) , x ≥ − ( µ + ϕµ ) √ σ + ϕ σ Otherwise = (cid:112) ϕ (cid:110) exp( − c ( x )) ϕ − (cid:104) exp (cid:16) c ( x )(1 − ϕ ) (cid:17) − (cid:105)(cid:111) , x ≥ − ϕ √ ϕ Otherwise . Figure 3. The graph of pdf f Y ( x ) Example 2 Let X and X be independent random variables with U nif orm (0 , distribution. Thenfrom (7) we have f X + ϕX ( x ) = xϕ , ≤ x ≤ ϕ , ≤ x ≤ ϕ − xϕ + 1 , ϕ ≤ x ≤ ϕ , elsehwere . This is a trapezoidal pdf with graph given below in Figure 4. igure 4. The graph of f X + ϕX ( x ) To find the distribution of limiting random variable Y, we consider P { Y ≤ x } = P { X + ϕX ≤ x (cid:113) σ + ϕ σ + ( µ + ϕµ ) } It is clear that µ = µ = 1 / , σ = σ = 1 / ,a = (cid:113) σ + ϕ σ = (cid:114) ϕ , b = µ + ϕµ = 1 + ϕ and the cdf of Y is F Y ( x ) = P { Y ≤ x } = P { X + ϕX ≤ ax + b } = x ≤ − ba ϕ ax + b (cid:82) udu = ( ax + b ) ϕ , − ba ≤ x ≤ − ba ϕ + ϕ ax + b (cid:82) du = ϕ + ax + b − ϕ , − ba ≤ x ≤ ϕ − ba ϕ + ϕ + ϕ ax + b (cid:82) ϕ ( − uϕ + 1) du = ax +2 b +2 axϕ +2 bϕ − a x − axb − b − ϕ − ϕ ϕ − ba ≤ x ≤ ϕ − ba x ≥ ϕ − ba . he pdf of Y is f Y ( x ) = , x < − ba or x > ϕ − ba ( ax + b ) aϕ , − ba < x ≤ − baaϕ − ba < x ≤ ϕ − baa (1+ ϕ − b − ax ) ϕ ϕ − ba < x ≤ ϕ − ba . Here we are interested in the limiting behavior of sums of members of FSRV. Consider S n = n (cid:80) i =0 X i . Wehave S n = X + X + · · · + X n = X + X + n (cid:88) i =2 X i = X + X + n (cid:88) i =2 ( a i − X + a i X )= X + X + X n (cid:88) i =2 a i − + X n (cid:88) i =2 a i = X + X + X n − (cid:88) i =1 a i + X ( n (cid:88) i =1 a i − a )= X + X + X ( a n +1 − 1) + X ( a n +2 − − a )= a n +1 X + ( a n +2 − X. Since n (cid:88) i =1 a i = a n +2 − . Therefore S n = X + X + · · · + X n = a n +1 X + ( a n +2 − X . S n is f S n ( x ) = 1 a n +1 ( a n +1 − ∞ (cid:90) −∞ f X ( x − ta n +1 ) f X ( ta n +2 − 1) ) dt. (9) Theorem 4 Under conditions of Theorem 3 for a sequence X , X , X n = a n − X + a n X , n = 2 , , ... we have ES n = a n +1 µ + ( a n +2 − µ V ar ( S n ) = a n +1 σ + ( a n +2 − σ S n − E ( S n ) (cid:112) var ( S n ) → Y as n → ∞ , for all ω ∈ Ω , where Y has cdf (6). Proof. Indeed, S n − E ( S n ) (cid:112) var ( S n )= X + ( a n +2 a n +1 − a n +1 ) X − ( µ + ( a n +2 a n +1 − a n +1 ) µ (cid:113) σ + ( a n +2 a n +1 − a n +1 ) σ , → X + ϕX − ( µ + ϕµ ) (cid:112) σ + ϕ σ = Y, as n → ∞ . Example 3 Let X and X be iid exponential(1) random variables. Then the pdf of S n is f S n ( x ) = 1 a n +1 ( a n +2 − x (cid:90) exp( x − ta n +1 ) exp( ta n +2 − 1) ) dt = exp( − xa n +1 ) a n +1 − a n +2 + 1 (cid:18) − exp( − x (cid:18) a n +2 − − a n +1 (cid:19)(cid:19) . (10) X n and X n + k Next, we focus on the joint distributions of X n = a n − X + a n X and X n + k = a n + k − X + a n + k X , for k ≥ . heorem 5 The joint pdf of X n and X n + k is f X n ,X n + k ( y , y )= 1 a k f X ,X ( a n + k y − y a n ( − n a k , a n − y − a n + k − y ( − n a k ) . (11) Proof. Let y = a n − x + a n x y = a n + k − x + a n + k x . (12)The Jacobian of this linear transformation is J = a n − a n + k − a n a n + k − and the solution of the systemof equations (12) is x = ( a n + k y − y a n ) / ( a n − a n + k − a n a n + k − ) x = ( a n − y − a n + k − y ) / ( a n − a n + k − a n a n + k − ) . Therefore, the joint pdf of X n and X n + k is f X n ,X n + k ( y , y )= 1 | a n − a n + k − a n a n + k − | f X ,X ( a n + k y − y a n a n − a n + k − a n a n + k − ,a n − y − a n + k − y a n − a n + k − a n a n + k − ) . (13)Using the d’Ocagne’s identity (see e.g. Dickson (1966)) a m a n +1 − a m +1 a n = ( − n a m − n we have J = a n − a n + k − a n a n + k − = − ( a n + k − a n − a n + k a n − ) = ( − n a k . Therefore, f X n ,X n + k ( y , y )= 1 a k f X ,X ( a n + k y − y a n ( − n a k , a n − y − a n + k − y ( − n a k ) . Corollary 1 If X and X are independent then f X n ,X n + k ( x, y )= 1 a k f X (cid:18) a n + k x − ya n ( − n a k (cid:19) f X (cid:18) a n − y − a n + k − x ( − n a k (cid:19) . (14)11 xample 4 Let X and X be iid exponential(1) random variables, n = 4 , k = 3 . Then a n + k = a = 13 ,a n + k − = a = 8 , a n − = a = 2 , a n = a = 3 and a k = a = 2 . Then from (14) f X ,X ( x, y )= exp( − (13 / x + (3 / y ) × exp( − y + 4 x ) , x ≥ and x ≤ y ≤ / x otherwise . = exp( − (5 / x ) exp( y/ x ≥ and x ≤ y ≤ / x otherwise (15) The marginal pdf ’s are f X ( x ) = e − x − e − x , x ≥ , otherwise (16) and f X ( x ) = (cid:0) e − x − e − x (cid:1) , x ≥ , otherwise . Example 5 Let X and X be independent uniform(0,1) random variables. Again, let n = 4 , k = 3 . Then a n + k = a = 13 , a n + k − = a = 8 , a n − = a = 2 , a n = a = 3 and a k = a = 2 . Then f X n ,X n + k ( x, y )= 1 a k f X (cid:18) a n + k x − ya n ( − n a k (cid:19) f X (cid:18) a n − y − a n + k − x ( − n a k (cid:19) = a k ≤ a n + k x − ya n ( − n a k ≤ , ≤ a n − y − a n + k − x ( − n a k ≤ , otherwise (17)12 To check whether (17) is a pdf, we need to show (cid:82) (cid:82) f X n ,X n + k ( x, y ) dxdy = 1 . Indeed, (cid:90) (cid:90) f X n ,X n + k ( x, y ) dxdy = 1 a k (cid:90) (cid:90) ≤ an + kx − yan ( − nak ≤ , ≤ an − y − an + k − x ( − nak ≤ dxdy = a n + k x − ya n = t, a n − y − a n + k − x = sx = ta n − + sa n ( − n a k , y = sa n + k + ta n + k − ( − n a k t ≤ ( − n a k , s ≤ ( − n a k J = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a n − ( − n a k a n ( − n a k a n + k − ( − n a k a n + k ( − n a k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = a n − a n + k − a n a n + k − ( − n a k = ( − n a k ( − n a k = 1 a k ( − n a k (cid:90) − n a k (cid:90) | ( − n a k | dxdy = 1 . ) For n = 4 and k = 3 , the f X ,X ( x, y )= 12 f X (cid:18) x − y (cid:19) f X (cid:18) y − x (cid:19) = , ≤ x − y ≤ , ≤ y − x ≤ , otherwise . (18) It is well known that with respect to squared error loss, the best unbiased predictor of X n + k , given X n is E { X n + k | X n } . Let g ( x ) = E { X n + k | X n = x } = 1 f X n ( x ) ∞ (cid:90) −∞ yf X n ,X n + k ( x, y ) dy, (19)then E { X n + k | X n } = g ( X n ) . Using (1) and (11) from (19) one can easily calculate the best predictorof X n + k , given X n . xample 6 Let X and X be independent exponential(1) random variables. Let n = 4 , k = 3 . Then a n + k = a = 13 , a n + k − = a = 8 , a n − = a = 2 , a n = a = 3 and a k = a = 2 as in Example 4. Thenfrom (15) we can write g ( x ) = 1 e − x − e − x / x (cid:90) x y 12 exp( − (5 / x ) exp( y/ dy = 13 12 e − x/ − e − x/ + 6 e − x/ − e − x/ e − x/ − e − x/ = 4 x − − x e − x/ − , Therefore, X (cid:39) X − − X e − X / − . . Conclusion 1 In this note, we considered the sequence of random variables { X , X , X n = X n − + X n − ,n = 2 , , .. } which is equivalent to { X , X , X n = a n − X + a n X , n = 2 , , ... } , where X and X areabsolutely continuous random variables with joint pdf f X ,X , and a n = a n − + a n − , n = 2 , , ... ( a = 0 ,a = 1) is the Fibonacci sequence. In the paper, the sequence X n , n = 0 , , , ... is referred to as theFibonacci Sequence of Random Variables. We investigated the limiting properties of some ratios andnormalizing sums of this sequence. For exponential and uniform distribution cases, we derived someinteresting limiting properties that reduce to the golden ratio and also investigated the joint distributionsof X n and X n + k . The considered random sequence has benefical properties and may be worthy of attentionassociated with random sequences and autoregressive models. References [1] Dickson. L. E. (1966) History of the Theory of Numbers Volume 1, New York: Chelsea.[2] Gnedenko, B.V. (1978) The Theory of Probability , Mir Publishers, Moscow.[3] Feller, W. (1971) An Introduction to Probability Theory and Its Applications , Volume 2, John Wiley& Sons Inc. , New York, London, Sydney.[4] Melham, R.S. and Shannon, A.G. (1995) A generalization of the Catalan identity and some conse-quences , The Fibonacci Quarterly 33, 82–84, 1995.145] Ross, S. (2016) A First Course in Probability . Prentice-Hall Inc. , NJ.[6] Skorokhod, A.V. (2005) Basic Principles and Applications of Probability Theory , Springer. For illustration of the behaviour of FSRV, the simulated values of random variables X and X fromuniform (0,1) and standard normal distribution are obtained. The corresponding codes in R are alsogiven. The corresponding code in R for uniform(0,1) distribution is: > a < − seq(1:10); for (i in 3:10) a[i]=a[i-1]+a[i-2]; x < − runif(10); y < − runif(10); z < − numeric(10); for (iin 2:10) z[i]=a[i]*x[i]+a[i-1]*x[i-1]; c < − seq(1:10); plot(c,z,col=”red”,bg=”yellow”,pch=22,bty=”l”);The corresponding code in R for standard normal distribution is: > a < − seq(1:10); for (i in 3:10) a[i]=a[i-1]+a[i-2]; x < − rnorm(10); y < − rnorm(10); z < − numeric(10); for(i in 2:10) z[i]=a[i]*x[i]+a[i-1]*x[i-1]; c < −−