# Fractional Randomness and the Brownian Bridge

Charles Tapiero, Pierre Vallois

Research output: Contribution to journalArticle

### Abstract

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.

Original language English (US) 835-843 9 Physica A: Statistical Mechanics and its Applications 503 https://doi.org/10.1016/j.physa.2018.02.097 Published - Aug 1 2018

### Fingerprint

Brownian Bridge
Randomness
Fractional
Uniform distribution
finance
calculus
Fractional Calculus
Statistical Modeling
Fractional Brownian Motion
Exponential distribution
Finance
Insurance
theorems
Central limit theorem
scaling
Scaling
Imply

### ASJC Scopus subject areas

• Statistics and Probability
• Condensed Matter Physics

### Cite this

Fractional Randomness and the Brownian Bridge. / Tapiero, Charles; Vallois, Pierre.

In: Physica A: Statistical Mechanics and its Applications, Vol. 503, 01.08.2018, p. 835-843.

Research output: Contribution to journalArticle

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