Fractional non-homogeneous Poisson and Pólya-Aeppli processes of order kk and beyond

We introduce two classes of point processes: a fractional non-homogeneous Poisson process of order k and a fractional non-homogeneous Pólya-Aeppli process of order k. We characterize these processes by deriving their non-local governing equations. We further study the covariance structure of the processes and investigate the long-range dependence property

Fractional non-homogeneous Poisson and Pólya-Aeppli processes of order k and beyond

We introduce two classes of point processes: a fractional non-homogeneous Poisson process of order k and a fractional non-homogeneous Pólya-Aeppli process of order k: We characterize these processes by deriving their non-local governing equations. We further study the covariance structure of the processes and investigate the long-range dependence property

On the moment characteristics for the univariate compound poisson and bivariate compound poisson processes with applications

The univariate and bivariate compound Poisson process (CPP and BCPP,respectively) ensure a better description than the homogeneous Poisson processfor clustering of events. In this paper, new explicit representations ofthe moment characteristics (general, central, factorial, binomial and ordinarymoments, factorial cumulants) and some covariance structures are derivedfor the CPP and BCPP. Then, the skewness and kurtosis of the univariateCPP are obtained for the first time and special cases of the CPP are studiedin detail. Applications to two real data sets are given to illustrate the usageof these processes

Rare Events for the Manneville-Pomeau map

We prove a dichotomy for Manneville-Pomeau maps $f:[0,1]\to [0, 1]$: given any point $\zeta\in [0,1]$, either the Rare Events Point Processes (REPP), counting the number of exceedances, which correspond to entrances in balls around $\zeta$, converge in distribution to a Poisson process; or the point $\zeta$ is periodic and the REPP converge in distribution to a compound Poisson process. Our method is to use inducing techniques for all points except 0 and its preimages, extending a recent result by Haydn, Winterberg and Zweim\"uller, and then to deal with the remaining points separately. The preimages of 0 are dealt with applying recent results by Ayta\c{c}, Freitas and Vaienti. The point $\zeta=0$ is studied separately because the tangency with the identity map at this point creates too much dependence, which causes severe clustering of exceedances. The Extremal Index, which measures the intensity of clustering, is equal to 0 at $\zeta=0$, which ultimately leads to a degenerate limit distribution for the partial maxima of stochastic processes arising from the dynamics and for the usual normalising sequences. We prove that using adapted normalising sequences we can still obtain non-degenerate limit distributions at $\zeta=0$

Testing a Random Number Generator: formal properties and automotive application

L'elaborato analizza un metodo di validazione dei generatori di numeri casuali (RNG), utilizzati per garantire la sicurezza dei moderni sistemi automotive. Il primo capitolo fornisce una panoramica della struttura di comunicazione dei moderni autoveicoli attraverso l'utilizzo di centraline (ECU): vengono riportati i principali punti di accesso ad un automobile, assieme a possibili tipologie di hacking; viene poi descritto l'utilizzo dei numeri casuali in crittografia, con particolare riferimento a quella utilizzata nei veicoli. Il secondo capitolo riporta le basi di probabilità necessarie all'approccio dei test statistici utilizzati per la validazione e riporta i principali approcci teorici al problema della casualità. Nei due capitoli centrali, viene proposta una descrizione dei metodi probabilistici ed entropici per l'analisi di dati reali utilizzati nei test. Vengono poi descritti e studiati i 15 test statistici proposti dal National Institute of Standards and Technology (NIST). Dopo i primi test, basati su proprietà molto semplici delle sequenze casuali, vengono proposti test più sofisticati, basati sull'uso della trasformata di Fourier (per testare eventuali comportamenti periodici), dell'entropia (strettamente connessi con la comprimibilità della sequenza), o sui random path. Due ulteriori test, permettono di valutare il buon funzionamento del generatore, e non solo delle singole sequenze generate. Infine, il quinto capitolo è dedicato all'implementazione dei test al fine di testare il TRNG delle centraline

A Quasi-Likelihood Approach to Zero-Inflated Spatial Count Data

The increased accessibility of data that are geographically referenced and correlated increases the demand for techniques of spatial data analysis. The subset of such data comprised of discrete counts exhibit particular difficulties and the challenges further increase when a large proportion (typically 50% or more) of the counts are zero-valued. Such scenarios arise in many applications in numerous fields of research and it is often desirable to infer on subtleties of the process, despite the lack of substantive information obscuring the underlying stochastic mechanism generating the data. An ecological example provides the impetus for the research in this thesis: when observations for a species are recorded over a spatial region, and many of the counts are zero-valued, are the abundant zeros due to bad luck, or are aspects of the region making it unsuitable for the survival of the species? In the framework of generalized linear models, we first develop a zero-inflated Poisson generalized linear regression model, which explains the variability of the responses given a set of measured covariates, and additionally allows for the distinction of two kinds of zeros: sampling ("bad luck" zeros), and structural (zeros that provide insight into the data-generating process). We then adapt this model to the spatial setting by incorporating dependence within the model via a general, leniently-defined quasi-likelihood strategy, which provides consistent, efficient and asymptotically normal estimators, even under erroneous assumptions of the covariance structure. In addition to this advantage of robustness to dependence misspecification, our quasi-likelihood model overcomes the need for the complete specification of a probability model, thus rendering it very general and relevant to many settings. To complement the developed regression model, we further propose methods for the simulation of zero-inflated spatial stochastic processes. This is done by deconstructing the entire process into a mixed, marked spatial point process: we augment existing algorithms for the simulation of spatial marked point processes to comprise a stochastic mechanism to generate zero-abundant marks (counts) at each location. We propose several such mechanisms, and consider interaction and dependence processes for random locations as well as over a lattice