Fractional Randomness and the Brownian Bridge

This paper introduces a statistical approach to fractional randomness based on the Central Limit Theorem. We show under general conditions that fractional noise-randomness defined relative to a uniform distribution, implies as well a fractional Brownian Bridge randomness rather than a Fractional Brownian Motion. We analyze further their fractional properties, namely, their variance and covariance and obtain specific results for particular distributions including the fractional uniform distribution and an exponential distribution. The results we obtain have both practical and theoretical implications to the many applications of fractional calculus and in particular, when they are applied to modeling statistical problems where time scaling and randomness prime. This is the case in finance, insurance and risk models as well as in other areas of interest.