## Tesi etd-02092008-223901 |

Thesis type

Tesi di dottorato di ricerca

Author

CASTELLUCCI, ANNALISA

URN

etd-02092008-223901

Title

φ-Subgaussian random processes: properties and their applications

Settore scientifico disciplinare

MAT/06

Corso di studi

MATEMATICA

Commissione

**Relatore**Giuliano, Rita

Parole chiave

- Orlicz spaces
- law of iterated logarithm
- φ-Subgaussian random variables
- R-Correlograms

Data inizio appello

28/01/2008;

Consultabilità

completa

Riassunto analitico

In this thesis φ-Subgaussian random variables are studied and used to solve some classical problems as, for example, an estimation of the correlation function of a Gaussian stationary process, and some topics about the behaviour of random process are tackled.<br><br>More precisely, we deal with a weaker formulation of the Law of Iterated Logarithm: it is studied for some kind of φ- subgaussian martingales and stochastic integrals. We first prove that a process (X<sub>t</sub>)<sub>t</sub>, under φ-subgaussinity assumptions, verifies an analogous of the law of itherated logarithm.Then we find some hypotheses on the integrand processes which force the stochastic integral to be a φ-Subgaussian martingale. Main interest of these results is that they can be applied also to stochastic integrals with unbounded integrand processes.<br><br>We also study the asymptotic behaviour of the maxima of a φ-Subgaussian random sequence and the convergence of some series connected with this maxima. In this case the relationship between these results and the complete convergences is mostly interesting. <br><br><br>Finally φ-Subgaussian theory is used to study a continuous time estimator ℜ=(ℜ<sub>t</sub>)<sub>t</sub> of the Relay Correlation Function (a modification of the well known Correlation Function) of a Gaussian stationary process. In particular, proving that tha random variable ℜ<sub>t</sub> is φ-Subgaussian we find a pointwise confidence interval. A metric entropy approach is used to obtain an uniform confidence intervals.

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