The impact of periodicity on the zero-crossings of random functions

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Wilson, Lorna Rachel Maven (2015) The impact of periodicity on the zero-crossings of random functions. PhD thesis, University of Nottingham.

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Abstract

Continuous random processes are used to model a huge variety of real world phenomena. In particular, the zero-crossings of such processes find application in modelling processes of diffusion, meteorology, genetics, finance and applied probability. Understanding the zero-crossings behaviour improves prediction of phenomena initiated by a threshold crossing, as well as extremal problems where the turning points of the process are of interest. To identify the Probability Density Function (PDF) for the times between successive zero-crossings of a stochastic process is a challenging problem with a rich history.

This thesis considers the effect of an oscillatory auto-correlation function on the zero-crossings of a Gaussian process. Examining statistical properties of the number of zeros in a fixed time period, it is found that increasing the rate of oscillations in the auto-correlation function results in more ‘deterministic’ realisations of the process. The random interval times between successive zeros become more regular, and the variance is reduced. Accurate calculation of the variance is achieved through analysing the correlation between intervals,which numerical simulations show can be anti-correlated or correlated, depending on the rate of oscillations in the auto-correlation function. The persistence exponent describes the tail of the inter-event PDF, which is steeper where zero-crossings occur more regularly. It exhibits a complex phenomenology, strongly influenced by the oscillatory nature of the auto-correlation function.

The interplay between random and deterministic components of a system governs its complexity. In an ever-more complex world, the potential applications for this scale of ‘regularity’ in a random process are far reaching and powerful.

Item Type: Thesis (University of Nottingham only) (PhD)
Supervisors: Hopcraft, Keith
Jakeman, E.
Subjects: Q Science > QA Mathematics > QA273 Probabilities
Faculties/Schools: UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID: 30472
Depositing User: Wilson, Lorna
Date Deposited: 14 Jan 2016 11:17
Last Modified: 19 Oct 2017 14:56
URI: https://eprints.nottingham.ac.uk/id/eprint/30472

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