Mathematica is an extremely powerful and flexible symbolic computer algebra system that enables the user to deal with complicated algebraic tasks. It can also easily handle the numerical and graphical sides. One such task is the derivation of moment generating functions (MGF) and characteristic functions (CF), demonstrably effective tools to characterize a distribution. In this paper, we define some rules in Mathematica to help in computing the MGF and CF for linear combination of independent random variables. These commands utilizes pattern-matching code that enhances Mathematica's ability to simplify expressions involving the product of algebraic terms. This enhancement to Mathematica's functionality can be of particular benefit for MGF and CF. Applications of these rules to determine mean, variance and distribution are illustrated for various independent random variables.

The novelty and originality of the work consists in the fact that in order to obtain asymptotic expansions with optimal values of the remainder terms in the zone of large deviations, along with the cumulant method the classical method of characteristic functions has to be used. In addition, when solving the problems stated in the work, other than the well known results in the problems of limit theorems of the probability theory and mathematical statistics, we have to estimate constants. Technically it is frequently rather a complicated task. The results obtained in the work have good opportunities to be applied in probability theory, mathematical statistics, econometric, etc. That is illustrated in the last section of the work in which theorems of large deviations are proved in the summation of weighted random variables with weights as well as discounted limit theorems.

Nagrinėjama silpnai priklausomų atsitiktinių dydžių statistika. Šio magistrinio darbo tikslas, turimą sumą, suvesti į nepriklausomų atsitiktinių dydžių sumą, kuri leistų tolimesniam tyrimui, pritaikyti žinomas teoremas, skaičiavimus bei rezultatus iš nepriklausomų atsitiktinių dydžių teorijos. / We analyze of weakly dependent random variables statistics. The objective of this master thesis is to deduce sum to independent random variables sum, which will be useful for applaying known theorems, calculations and results from the theory in independent random variables.

Une classe générale de statistiques de rangs basées sur la fonction caractéristique est introduite afin de tester l'hypothèse composite d'appartenance à une famille de copules multidimensionnelles. Ces statistiques d'adéquation sont définies comme des distances fonctionnelles de type L_2 pondérées entre une version non paramétrique et une version semi-paramétrique de la fonction caractéristique que l'on peut associer à une copule. Il est démontré que ces statistiques de test se comportent asymptotiquement comme des V-statistiques dégénérées d'ordre quatre et que leurs lois limites s'expriment en termes de sommes pondérées de variables khi-deux indépendantes. La convergence des tests sous des alternatives générales est établie, de même que la validité du bootstrap paramétrique pour le calcul de valeurs critiques. Le comportement des nouveaux tests sous des tailles d'échantillons faibles et modérées est étudié à l'aide de simulations et est comparé à celui d'un test concurrent fondé sur la copule empirique. La méthodologie est finalement illustrée sur un jeu de données à plusieurs dimensions.

The probability distribution of a random variable created by summing a random number of the independent and identically distributed random variables is called a compound probability distribution. In this work is described a compound distribution as well as a calculation of its characteristics. Especially, the thesis is focused on studying a special case of compound distribution where each addend has the log-normal distribution and their number has the negative binomial distribution. Here are also described some approaches to estimate the parameters of LN and NB distribution. Further, the impact of these estimates on the final compound distribution is analyzed.

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Neste trabalho propomos a utilização do método da função característica empírica (ECF - empirical characteristic function), para estimação do modelo de duração condicional estocástica (SCD - stochastic conditional duration). Para determinação das variáveis latentes do processo utilizamos três alternativas: um filtro de Kalman, um filtro obtido por integração numérica e um filtro baseado na expansão de Gram-Charlier até 4ª ordem. Os resultados são então aplicados em séries de duração da GE, Microsoft e USD/EUR. / We propose the use of the empirical characteristic function (ECF) method to estimate the parameters of the stochastic conditional duration (SCD) model. In order to estimate the latent variables we propose the use of three alternatives: a Kalman filter, a filter based on numerical integration (quadrature) and a filter based on the 4th-order Gram-Charlier expansion. The results are applied to the estimation of the parameters of the duration process for GE, Microsoft and USD/EUR.

Neste trabalho propomos a utilização do método da função característica empírica (ECF - empirical characteristic function), para estimação do modelo de duração condicional estocástica (SCD - stochastic conditional duration). Para determinação das variáveis latentes do processo utilizamos três alternativas: um filtro de Kalman, um filtro obtido por integração numérica e um filtro baseado na expansão de Gram-Charlier até 4ª ordem. Os resultados são então aplicados em séries de duração da GE, Microsoft e USD/EUR. / We propose the use of the empirical characteristic function (ECF) method to estimate the parameters of the stochastic conditional duration (SCD) model. In order to estimate the latent variables we propose the use of three alternatives: a Kalman filter, a filter based on numerical integration (quadrature) and a filter based on the 4th-order Gram-Charlier expansion. The results are applied to the estimation of the parameters of the duration process for GE, Microsoft and USD/EUR.