A note on Fibonacci Sequences of Random Variables

TL;DR

This paper investigates the distributional and limit properties of Fibonacci-like sequences generated by random initial variables, providing insights into their probabilistic behavior and convergence characteristics.

Contribution

It introduces a framework for analyzing Fibonacci sequences of random variables, focusing on their distributional and limit properties based on initial joint distributions.

Findings

Distributional properties derived for Fibonacci random sequences

Limit behavior and convergence results established

Framework applicable to various initial distributions

Abstract

The focus of this paper is the random sequences in the form $\{X_{0},X_{1},$ $X_{n}=X_{n-2}+X_{n-1},n=2,3,..\dot{\}},$ referred to as Fibonacci Random Sequence (FRS). The initial random variables $X_{0}$ and $X_{1}$ are assumed to be absolutely continuous with joint probability density function (pdf) $f_{X_{0},X_{1}}.$ The FRS is completely determined by $X_{0}$ and $X_{1}$ and the members of Fibonacci sequence $\digamma \equiv\{0,1,1,2,3,5,8,13,21,34,55,89,144,...\}.$ We examine the distributional and limit properties of the random sequence $X_{n},n=0,1,2,...$ .

Equations

$$f_{X_n}(x)=\frac{1}{a_n{a}_{n-1}}\underset{-\infty }{\overset{\infty }{\int }}{f}_{X_0,X_1}(\frac{x-t}{a_{n-1}},\frac{t}{a_n})dt.$$

$$f_{X_n}(x)=\frac{1}{a_n{a}_{n-1}}\underset{-\infty }{\overset{\infty }{\int }}{f}_{X_0}(\frac{x-t}{a_{n-1}})f_{X_1}(\frac{t}{a_n})dt.$$

$$f_{X_n}(x)$$

$$f_{X_2}(x)$$

$$E{X}_n$$

$$Var(X_n)$$

$$\underset{n\to \infty }{\lim }\frac{M_n}{M_{n+1}}$$

$$\underset{n\to \infty }{\lim }\frac{Var(X_{n+1})}{Var(X_n)}$$

$$\phi \equiv \frac{1-\sqrt{5}}{2}=1,\mathrm{6180339887...}$$

$$\frac{d}{dx}{f}_{X_n}(x)=\frac{(-\frac{x}{a_{n-1}})(e^{\frac{x{a}_{n-2}}{a_{n-1}{a}_n}}-1)}{a_{n-2}{a}_{n-1}}+\frac{e^{-\frac{x}{a_{n-1}}}{e}^{\frac{x{a}_{n-2}}{a_{n-1}{a}_n}}}{a_{n-1}{a}_n}=0.$$

$$x_n^{\ast }=\frac{a_{n-1}{a}_n\ln \left(\frac{a_n}{a_n-a_{n-2}}\right)}{a_{n-2}}.$$

$$M_n$$

$$M_{n+1}$$

$$\underset{n\to \infty }{\lim }\frac{a_{n+\alpha }}{a_n}={\phi }^{\alpha }$$

$$\underset{n\to \infty }{\lim }\frac{M_n}{M_{n+1}}=\frac{x_{n+1}^{\ast }}{x_n^{\ast }}=\phi .$$

$$f_{X_n}(x)$$

$$Z_n(\omega )=\frac{X_{n+1}(\omega )}{X_n(\omega )}=\frac{a_{n+1}{X}_1(\omega )+a_n{X}_0(\omega )}{a_n{X}_1(\omega )+a_{n-1}{X}_0(\omega )}=\frac{\frac{a_{n+1}}{a_n}{X}_1(\omega )+X_0(\omega )}{X_1(\omega )+\frac{a_{n-1}}{a_n}{X}_0(\omega )}=\frac{\frac{a_{n+1}}{a_n}{X}_1(\omega )+X_0(\omega )}{X_1(\omega )+\frac{1}{\frac{a_n}{a_{n-1}}}{X}_0(\omega )}.$$

$$Z_n(\omega )$$

$${\Omega }_1$$

$$Y_n(\omega )\equiv Y_n=\frac{X_n-E(X_n)}{\sqrt{Var(X_n)}}=\frac{X_0+\frac{a_n}{a_{n-1}}{X}_1-({\mu }_0+\frac{a_n}{a_{n-1}}{\mu }_1)}{\sqrt{{\sigma }_0^2+\frac{a_n^2}{a_{n-1}^2}{\sigma }_1^2}},\omega \in \Omega .$$

$$Y_n\to Y\equiv \frac{X_0+\phi X_1-({\mu }_0+\phi {\mu }_1)}{\sqrt{{\sigma }_0^2+{\phi }^2{\sigma }_1^2}},\text{ as }n\to \infty \text{ for all }\omega \in \Omega .$$

$$P\{Y\le x\}=P\{X_0+\phi X_1\le x\sqrt{{\sigma }_0^2+{\phi }^2{\sigma }_1^2}+({\mu }_0+\phi {\mu }_1)\}.$$

$$f_{X_0+\phi X_1}(x)=\frac{1}{\phi }\underset{0}{\overset{\infty }{\int }}{f}_{X_0}(x-t)f_{X_1}(\frac{t}{\phi })dt.$$

$$f_Y(x)=\sqrt{{\sigma }_0^2+{\phi }^2{\sigma }_1^2}{f}_{X_0+\phi X_1}(x\sqrt{{\sigma }_0^2+{\phi }^2{\sigma }_1^2}+({\mu }_0+\phi {\mu }_1))$$

$$f_{X_0+\phi X_1}(x)$$

$$P\{Y$$

$$f_Y(x)$$

$$f_{X_{0}+\varphi X_{1}}(x)=\left\{\begin{array}[]{ccc}\frac{x}{\varphi},&0\leq x\leq 1&\\ \frac{1}{\varphi},&1\leq x\leq\varphi&\\ \frac{1-x}{\varphi}+1,&\varphi\leq x\leq 1+\varphi&\\ 0,&elsehwere&\end{array}\right..$$

$$P\{Y\le x\}=P\{X_0+\phi X_1\le x\sqrt{{\sigma }_0^2+{\phi }^2{\sigma }_1^2}+({\mu }_0+\phi {\mu }_1)\}$$

$${\mu }_0$$

$$a$$

$$F_Y(x)$$

$$f_{Y}(x)=\left\{\begin{array}[]{cccc}0,&x<-\frac{b}{a}\text{ \ or }x>\frac{1+\varphi-b}{a}&&\\ \frac{(ax+b)a}{\varphi},&-\frac{b}{a}<x\leq\frac{1-b}{a}&&\\ \frac{a}{\varphi}&\frac{1-b}{a}<x\leq\frac{\varphi-b}{a}&&\\ \frac{a(1+\varphi-b-ax)}{\varphi}&\frac{\varphi-b}{a}<x\leq\frac{1+\varphi-b}{a}&&\end{array}\right..$$

Full text

Key words. Random variable, distribution function, probability density function,  sequence of random variables.

1 Introduction

Let $\{\Omega,\digamma,P\}$ be a probability space and $X_{i}\equiv X_{i}(\omega),\omega\in\Omega,i=0,1$ be absolutely continuous random variables defined on this probability space with joint probability density function (pdf) $f_{X_{0},X_{1}}(x,y).$ Consider a sequence of random variables $X_{n}\equiv X_{n}(\omega),n\geq 1$ given in $\{\Omega,\digamma,P\}$ defined as $\{X_{0},X_{1},$ $X_{n}=X_{n-2}+X_{n-1},$ $n=2,3,..\}.$ We call this sequence  ”the Fibonacci Sequence of Random Variables”.  It is clear that $X_{2}=X_{0}+X_{1},$ $X_{3}=X_{0}+2X_{1},...$ and for any $n=0,1,2,...$ we have $X_{n}=a_{n-1}X_{0}+a_{n}X_{1},$ where  $\ \{a_{n}=a_{n-2}+a_{n-1},n=2,3,...;a_{0}=0,a_{1}=1,a_{2}=1$ $\}$ is the Fibonacci sequence $\digamma\equiv\{0,1,1,2,3,5,8,13,21,34,55,89,144,...\}.$ It is also clear that the Fibonacci Sequence of Random Variables (FSRV) $X_{n},n=0,1,2,...$ is the sequence of dependent random variables based on initial random variables $X_{0}$ and $X_{1},$ which fully defined by the members of the Fibonacci sequence $\digamma$.  We are interested in the behavior of FSRV, i.e. the distributional properties of $X_{n}$ and joint distributions of $X_{n}$ and $X_{n+k\text{ }}$ for any $n$ and $k.$  In the Appendix Figure A1 and Figure A2, we present some examples of realizations of FSRV in the case of independent random variables $X_{0}$ and $X_{1}$ having * Uniform(0,1)* distribution and Standard 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 $X_{0}$ and $X_{1}$ have exponential and uniform distributions, respectively.  Then, there is an investigation of limit behavior of ratios of some characteristics of  pdf of $X_{n}$ for large $n.$ In the considered examples, the ratio of maximums of the pdfs, modes and expected values of consecutive elements of FSRV converge to golden ratio $\varphi\equiv\frac{1-\sqrt{5}}{2}=1,6180339887...$ .  The ratio $X_{n+1}/X_{n}$ and normalized sums  of $X_{n}$’s for large $n$ are discussed in Section 3. In Section 4,  the focus is on the joint distributions of  $X_{n}$ and $X_{n+k},$ for $2\leq k\leq n$ and on the prediction of $X_{n+k}$ given $X_{n}.$

2 Distributions

Consider $\ X_{n}=a_{n-1}X_{0}+a_{n}X_{1},$ $n=0,1,2,...$ , where $X_{0}$ and $X_{1}$ are absolutely continuous random variables with joint pdf $f_{X_{0},X_{1}}(x,y),$ $(x,y)\in\mathbb{R}^{2}$ and $a_{n},n=0,1,2,...$ is the Fibonacci sequence.  Denote by $f_{0}$ and $f_{1}$ the marginal pdf’s of $X_{0}$ and $X_{1},$ respectively.

Theorem 1

The pdf of $X_{n}$ is

[TABLE]

If $X_{0}$ and $X_{1}$ are independent, then

[TABLE]

Proof. Equations (1) and (2)  are straightforward results  of distributions of linear functions of random variables (see eg., Feller (1971), Ross (2002), Gnedenko (1978), Skorokhod (2005))

Case 1

Exponential distribution. Let  $X_{0}$ and $X_{1}$  be independent and identically distributed (iid) random variables having exponential distribution with parameter $\lambda=1.$ Then the pdf of $X_{n}$ is

[TABLE]

Below in Figure 1, the graphs of $f_{X_{n}}(x)$ for different values of $n$ are presented.

[TABLE]

The expected value of $X_{n}$ is

[TABLE]

and variance is

[TABLE]

Theorem 2

Let $M_{n}=\underset{0<x<\infty}{\max}f_{X_{n}}(x)$ and $x_{n}^{\ast}=\underset{0<x<\infty}{\arg\max}f_{X_{n}}(x)$ be the maximum of $f_{X_{n}}(x)$ and mode of $\ X_{n},$ $n=2,3,...,$ respectively. Then

[TABLE]

where

[TABLE]

is the golden ratio.

Proof. The following can easily be verified

[TABLE]

The equation (4) has unique solution

[TABLE]

Therefore $X_{n}$ is unimodal and we have

[TABLE]

and using

[TABLE]

we obtain

[TABLE]

Case 2

Uniform distribution. Let $X_{0}$ and $X_{1}$ be iid with $Uniform(0,1)$ distribution. Then from (2) we obtain

[TABLE]

[TABLE]

It can be easily verified that $E(X_{n})=\frac{a_{n-1}+a_{n}}{2}=\frac{a_{n+1}}{2}$ and $var(X_{n})=\frac{a_{n-1}^{2}+a_{n}^{2}}{12}.$

One can observe that    $f_{X_{n}}(x)$ is not unimodal, $f_{X_{n}}(x)$ is constant in the interval $(a_{n-1},a_{n})$ and $\ M_{n}=\underset{0<x<1}{\max}f_{X_{n}}(x)=\frac{1}{a_{n-1}}$ , $\ \underset{0<x<1}{\inf\arg\min}f_{X_{n}}(x)=a_{n-1},$ $\underset{0<x<1}{\sup\arg\min}f_{X_{n}}(x)=a_{n}$.

It is not difficult to observe that the similar to Theorem 1 results hold also in this case.

3 Large $n$ and normalized Fibonacci sequence of random variables

Let $\ X_{n}=a_{n-1}X_{0}+a_{n}X_{1},$ $n=0,1,2,...$  be FSRV, where $X_{0}$ and $X_{1}$ are absolutely continuous random variables with joint pdf $f_{X_{0},X_{1}}(x,y),$ $(x,y)\in\mathbb{R}^{2}.$ Consider the sequence of random variables $Z_{n}\equiv\frac{X_{n+1}}{X_{n}},n=1,2,...$ . One has

[TABLE]

Since  $\underset{n\rightarrow\infty}{\lim}\frac{a_{n+1}}{a_{n}}=\varphi,$ ($\varphi=\frac{1-\sqrt{5}}{2}=1,6180339887...$ is the golden ratio), it follows that

[TABLE]

For the normalized FSRV, the following limit relationship is valid.

Theorem 3

Let $E(X_{i})=\mu_{i},Var(X_{i})=\sigma_{i}^{2},i=0,1$ and

[TABLE]

Then,

[TABLE]

The limiting random variable $Y\equiv Y(\omega)$  has distribution function (cdf)

[TABLE]

It is clear that the pdf of $X_{0}+\varphi X_{1}$ is

[TABLE]

and the pdf of $Y$ is then

[TABLE]

Example 1

Let $\ X_{0}$ and $X_{1}$ be iid random variables having exponential distribution with parameter $\lambda=1,$ then from (11) we have

[TABLE]

Therefore,

[TABLE]

where $c(x)=x\sqrt{\sigma_{0}^{2}+\varphi^{2}\sigma_{1}^{2}}+(\mu_{0}+\varphi\mu_{1}).$  And the pdf is

[TABLE]

[TABLE]

Example 2

Let $X_{0}$ and $X_{1}$ be independent random variables with $Uniform(0,1)$ distribution. Then from (11) we have

[TABLE]

*This is a trapezoidal pdf with graph given below in Figure 4.

[TABLE]

To find the distribution of limiting random variable $Y,$ we consider

[TABLE]

It is clear that

[TABLE]

and the cdf of $Y$ is

[TABLE]

The pdf of $Y$ is

[TABLE]

3.1 Limits of normalized sums of Fibonacci sequence of random

variables

Here we are interested in the limiting behavior of sums of members of FSRV. Consider $S_{n}=\sum\limits_{i=0}^{n}X_{i}$.  We have

[TABLE]

Since

[TABLE]

Therefore

[TABLE]

The pdf of $S_{n}$ is

[TABLE]

Theorem 4

Under conditions of Theorem 3 for a sequence $X_{0},X_{1},$ $X_{n}=a_{n-1}X_{0}+a_{n}X_{1},$ $n=2,3,...$     we have

[TABLE]

[TABLE]

where $Y$ has cdf  (10).

Proof. Indeed,

[TABLE]

Example 3

Let  $X_{0}$ and $X_{1}$ be iid exponential(1) random variables. Then the pdf of $S_{n}$ is

[TABLE]

4 Joint distributions of $X_{n}$ and $X_{n+k}$

Next, we focus on the joint distributions of $X_{n}=a_{n-1}X_{0}+a_{n}X_{1}$ and $X_{n+k}=a_{n+k-1}X_{0}+a_{n+k}X_{1},$ for $k\geq 1.$

Theorem 5

The joint pdf of $X_{n}$ and $X_{n+k}$ is

[TABLE]

Proof. Let

[TABLE]

The Jacobian of this linear transformation is  $J=a_{n-1}a_{n+k}-a_{n}a_{n+k-1}$ and the solution of the system of equations (20) is

[TABLE]

Therefore, the joint pdf of $X_{n}$ and $X_{n+k}$ is

[TABLE]

Using the d’Ocagne’s identity   (see e.g. Dickson (1966)) $\ a_{m}a_{n+1}-a_{m+1}a_{n}=(-1)^{n}a_{m-n}$ we have  $J=a_{n-1}a_{n+k}-a_{n}a_{n+k-1}=-(a_{n+k-1}a_{n}-a_{n+k}a_{n-1})=(-1)^{n}a_{k}.$ Therefore,

[TABLE]

Corollary 1

If $X_{0}$ and $X_{1}$ are independent then

[TABLE]

Example 4

Let $X_{0}$ and $X_{1}$ be iid exponential(1) random variables, $n=4,k=3.$ Then  $a_{n+k}=a_{7}=13,$ $a_{n+k-1}=a_{6}=8,$ $a_{n-1}=a_{3}=2,$ $a_{n}=a_{4}=3$   and $a_{k}=a_{3}=2.$ Then from (22)

[TABLE]

*The marginal pdf’s are *

[TABLE]

and

[TABLE]

Example 5

Let $X_{0}$ and $X_{1}$ be independent uniform(0,1) random variables. Again, let $n=4,k=3.$ Then  $a_{n+k}=a_{7}=13,$ $a_{n+k-1}=a_{6}=8,$ $a_{n-1}=a_{3}=2,$ $a_{n}=a_{4}=3$   and $a_{k}=a_{3}=2.$ Then

[TABLE]

(To check whether (34) is a pdf,  we need to show $\int\limits_{0}^{1}\int\limits_{0}^{1}f_{X_{n},X_{n+k}}(x,y)dxdy=1.$ Indeed,

[TABLE]

For $n=4$ and $k=3,$ the

[TABLE]

5 Prediction of future values

It is well known that with respect to squared error loss, the best unbiased predictor of $X_{n+k},$ given $X_{n}$  is

[TABLE]

Let

[TABLE]

then $E\{X_{n+k}\mid X_{n}\}=g(X_{n}).$ Using (1) and (19)    from (41) one can easily calculate the best predictor of $X_{n+k},$ given $X_{n}.$

Example 6

Let $X_{0}$ and $X_{1}$ be independent exponential(1) random variables. Let $n=4,k=3.$ Then  $a_{n+k}=a_{7}=13,$ $a_{n+k-1}=a_{6}=8,$ $a_{n-1}=a_{3}=2,$ $a_{n}=a_{4}=3$   and $a_{k}=a_{3}=2$ as in Example 4. Then from (30) we can write

[TABLE]

Therefore,

[TABLE]

Conclusion 1

In this note, we considered the sequence of random variables $\{X_{0},X_{1},$ $X_{n}=X_{n-2}+X_{n-1},$ $n=2,3,..\}$ which is equivalent to $\left\{X_{0},X_{1},X_{n}=\ a_{n-1}X_{0}+a_{n}X_{1},n=2,3,...\right\},$ where $X_{0}$ and $X_{1}$ are  absolutely continuous random variables with joint pdf $f_{X_{0},X_{1}},$ and $a_{n}=a_{n-1}+a_{n-2},$ $n=2,3,...$ $(a_{0}=0,$ $a_{1}=1)$ is the Fibonacci sequence. In the paper, the sequence $X_{n},n=0,1,2,...$  is referred to as the Fibonacci Sequence of Random  Variables. We investigated the limiting properties of some ratios and normalizing sums of this sequence. For exponential and uniform distribution cases, we derived some interesting limiting properties that reduce to the golden ratio and also investigated the joint distributions of $X_{n}$ and $X_{n+k}.$ The considered random sequence has benefical properties and may be worthy of attention associated with random sequences and autoregressive models.

6 Appendix

For illustration of the behaviour of FSRV, the simulated values of random variables $X_{0}$ and $X_{1}$ from uniform (0,1) and standard normal distribution are obtained. The corresponding codes in R are also given. 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 (i in 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$<-$seq(1:10); plot(c,z,col=”red”,bg=”yellow”,pch=22,bty=”l”);