Fibonacci congruences and applications

René Blacher (1)

1 IPS - Inférence Processus Stochastiques

When we consider a congruence T (x) ≡ ax modulo m as a pseudo random number generator, there are several means of ensuring the independence of two successive num- bers. In this report, we show that this dependence depends on the continued fraction expansion of m/a. We deduce that the congruences such that m and a are two successive elements of Fibonacci sequences are those having the weakest dependence. We will use this result to obtain truly random number sequences xn . For that purpose, we will use non-deterministic sequences yn such that the conditional probabilities have Lipschitz coefficients not too large. They are transformed using Fibonacci congruences and we will get by this way sequences xn . These sequences xn admit the IID model for correct model.

Mots clés: Correct models; Fibonacci sequence; Random numbers; Congruence; Dependence; Correct models.

Domaines: Probabilités [math.PR]; Statistiques [math.ST]; Théorie [stat.TH]; Modélisation et simulation

René Blacher. Fibonacci congruences and applications. Open Journal of Statistics, 2011, 1 (2), pp.128-138. ⟨10.4236/ojs.2011.12015⟩. ⟨hal-00587108⟩