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Probability distribution of functional random variables
 Dolník, Viktor ; Hlubinka, Daniel (advisor) ; Hlávka, Zdeněk (referee)
We describe basic notions of functional random elements and the space of functions L2 [0, 1]. We discuss the non-existence of a probability density functional and the re- quirements for integrating in a functional space. In Chapter 2, we define distribution functionals and introduce a goodness-of-fit test which utilises them. The concept of char- acteristic functionals follows in Chapter 3, along with the latest test for Gaussianity of functional random elements. We conclude the chapter with our own new goodness-of- fit test, where we prove the distribution of its test statistic under the alternative, then under the null hypothesis, and lastly the distribution of the bootstrapped test statistic. Finally, we illustrate the theory on a simulation study of the empirical significance level and power of the goodness-of-fit tests. 1
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