A note on Fibonacci Sequences of Random Variables

The focus of this paper is the random sequences in the form $\{X_0,X_1,$ $X_n=X_{n-2}+X_{n-1},n=2,3,\mathrm{.}\mathrm{.}\dot{\}},$ 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) $f_{X_0,X_1}\mathrm{.}$ 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 $X_n,n=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.}$ .