## Thesis etd-02092008-223901 |

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Thesis type

Tesi di dottorato di ricerca

Author

CASTELLUCCI, ANNALISA

URN

etd-02092008-223901

Thesis title

φ-Subgaussian random processes: properties and their applications

Academic discipline

MAT/06

Course of study

MATEMATICA

Supervisors

**Relatore**Giuliano, Rita

Keywords

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

Graduation session start date

28/01/2008

Availability

Full

Summary

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.

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.

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.

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.

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.

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.

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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