A Markov chain Fibonacci model (Q1001163)

The authors suggest to take the first \(n\) Fibonacci numbers and look for Markov chains (MC) whose limiting (steady state) probabilities \(\pi=(\pi_1,\dots,\pi_n)\) are proportional to these numbers. This is done for discrete time MC and if this property is satisfied for an \(n\times n\) transition matrix \(P\), it is shown that there is a class of matrices, which is explicitly described, all with the same limiting probabilities \(\pi\). One of the results covers MC with limiting probabilities \[ \pi= (\pi_1,\pi_2,\dots,\pi_n)= \bigg(\frac{F_{2n-1}}{F_{2n}}, \frac{F_{2n-3}}{F_{2n}},\dots, \frac{F_1}{F_{2n}}\bigg), \] where \(F_k\) is the \(k\) Fibonacci number. Then the authors make an interesting parallel with continuous time MC. Now the goal is to use the infinitesimal, or the \(Q\)-matrix, of the MC, and achieve the same limiting distribution \(\pi\), which was obtained for discrete time MC. Finally, the authors show how to deal with MC whose state space is infinite and such that the limiting probabilities tt involve the Fibonacci numbers in one or another way. It is more than curious to see a complete description of MC for which \(\pi=(L,L^3,L^5,\dots)\) or \(\pi=(L^2,L^3,L^4,\dots)\). Here \(L= (\sqrt{5}-1)/2\) is the reciprocal of the golden ratio \(\varphi= (\sqrt{5}+1)/2\). A few related results are presented, e.g. using the so-called Tribonacci number matrix. It is worth mentioning that the MC of the types treated in this paper can be used to model the dynamics of a population with special birth and death properties.