On a Probabilistic Property of the Fibonacci Sequence

Let 77l5..., 77w,... be a sequence of Independent integer-valued random variables. Let SnT]l + -~ + 7jn,Ari = ESn, B* = varS„,P„(m) = P(Sn = m), and f(t,rfj) denote the characteristic function of the random variable 77 •. The local limit theorem (LLT) is formulated as Pn(m) = (27rBy -exp{-(m^f /2B} + o(B~) when n-^00 uniformly for m. The first results on the normal approximation of binomial distributions belong to de M oivre, Laplace, and Poisson. Very general theorems on the LLT were obtained by von Mises in [1], Assuming additionally that the summands are i.i.d. and have a finite variance, B. Gnedenko [2] derived necessary and sufficient conditions for the LLT. The next step, for not i.i.d. but uniformly bounded variables, was made by Yu. V. Prohorov in [3]. Besides those mentioned above, the LLT problem was investigated by W. Feller [4] and C. Stone [5]. More complete bibliographical information can be found in [6]. It is well known that for uniformly distributed random variables the LLT is equivalent to the central limit theorem [9], [10]. Hence, it is reasonable to ask whether this holds in general. The answer is negative. Using the Fibonacci sequence, we will construct below another sequence of independent asymptotically uniformly distributed random variables which satisfies the central limit theorem, has the uniform asymptotic negligibility (UAN) property but for which the local limit theorem fails to be valid. Let [1; 1,..., 1,...] be a continued fraction representation of the number <p (1 + j5)/2. Denote by Pj I Qj the convergents of the continued fraction of <p, which can^be represented by the table below. j