<p>We investigate mathematical tools, Edgeworth series expansion and the saddle point method, which are approximation techniques that help us to estimate the distribution function for the standardized mean of independent identical distributed random variables where we will take into consideration the lattice case. Later on we will describe one important application for these mathematical tools where game developing companies can use them to reduce the amount of time needed to satisfy their standard requests before they approve any game</p>

We investigate mathematical tools, Edgeworth series expansion and the saddle point method, which are approximation techniques that help us to estimate the distribution function for the standardized mean of independent identical distributed random variables where we will take into consideration the lattice case. Later on we will describe one important application for these mathematical tools where game developing companies can use them to reduce the amount of time needed to satisfy their standard requests before they approve any game

The aggregate interference distribution in cognitive radio networks is studied in a rigorous and analytical way using the popular Poisson point process model. While a number of results are available for this model for non-cognitive radio networks, cognitive radio networks present extra levels of difficulties for the analysis, mainly due to the exclusion region around the primary receiver, which are typically addressed via various ad-hoc approximations (e.g., based on the interference cumulants) or via the large-deviation analysis. Unlike the previous studies, we do not use here ad-hoc approximations but rather obtain the asymptotic interference distribution in a systematic and rigorous way, which also has a guaranteed level of accuracy at the distribution tail. This is in contrast to the large deviation analysis, which provides only the (exponential) order of scaling but not the outage probability itself. Unlike the cumulant-based analysis, our approach provides a guaranteed level of accuracy at the distribution tail. Additionally, our analysis provides a number of novel insights. In particular, we demonstrate that there is a critical transition point below which the outage probability decays only polynomially but above which it decays super-exponentially. This provides a solid analytical foundation to the earlier empirical observations in the literature and also reveals what are the typical ways outage events occur in different regimes. The analysis is further extended to include interference cancelation and fading (from a broad class of distributions). The outage probability is shown to scale down exponentially in the number of canceled nearest interferers in the below-critical region and does not change significantly in the above-critical one. The proposed asymptotic expressions are shown to be accurate in the non-asymptotic regimes as well.

In this thesis, we address the statistics of mutual information of amplify-and-forward (AF) multiple-input multiple-output (MIMO) two-hop relay channels, where the source terminal (ST), relay terminal (RT) and destination terminal (DT) are equipped with a number of correlated antennas and there is a line-of-sight (LOS) component (Rician fading) of each link. To the best of our knowledge, deriving analytical expressions for the statistics of mutual information of the relay channel is difficult and still unsolvable. To circumvent the mathematical difficulties, we consider this problem under the large-system regimen in which the numbers of antennas at the transmitter and receiver go to infinity with a fixed ratio. In the large-system regimen, this thesis has made the following contributions: 1) We get the mean and the variance of the mutual information of the concerned relay channel. 2) We show that the mutual information distribution converges to the Gaussian distribution. The analytical results are derived by mean of two powerful tools developed in the context of theoretical physics: emph{saddle-point approximation} and emph{superanalysis}. The derived analytical results are very general and can degenerate to several previously results as special cases. From a degenerated case, we realize that the previous result by Wagner {em et al.} cite{Wag-08} is wrong and thus we provide the corrected result. Finally, Simulation results demonstrate that even for a moderate number of antennas at each end, the proposed analytical results provide undistinguishable results as those obtained by Monte-Carlo results.

The aggregate interference distribution in cognitive radio networks is studied in a rigorous and analytical way using the popular Poisson point process model. While a number of results are available for this model for non-cognitive radio networks, cognitive radio networks present extra levels of difficulties for the analysis, mainly due to the exclusion region around the primary receiver, which are typically addressed via various ad-hoc approximations (e.g., based on the interference cumulants) or via the large-deviation analysis. Unlike the previous studies, we do not use here ad-hoc approximations but rather obtain the asymptotic interference distribution in a systematic and rigorous way, which also has a guaranteed level of accuracy at the distribution tail. This is in contrast to the large deviation analysis, which provides only the (exponential) order of scaling but not the outage probability itself. Unlike the cumulant-based analysis, our approach provides a guaranteed level of accuracy at the distribution tail. Additionally, our analysis provides a number of novel insights. In particular, we demonstrate that there is a critical transition point below which the outage probability decays only polynomially but above which it decays super-exponentially. This provides a solid analytical foundation to the earlier empirical observations in the literature and also reveals what are the typical ways outage events occur in different regimes. The analysis is further extended to include interference cancelation and fading (from a broad class of distributions). The outage probability is shown to scale down exponentially in the number of canceled nearest interferers in the below-critical region and does not change significantly in the above-critical one. The proposed asymptotic expressions are shown to be accurate in the non-asymptotic regimes as well.

The aggregate interference distribution in cognitive radio networks is studied in a rigorous and analytical way using the popular Poisson point process model. While a number of results are available for this model for non-cognitive radio networks, cognitive radio networks present extra levels of difficulties for the analysis, mainly due to the exclusion region around the primary receiver, which are typically addressed via various ad-hoc approximations (e.g., based on the interference cumulants) or via the large-deviation analysis. Unlike the previous studies, we do not use here ad-hoc approximations but rather obtain the asymptotic interference distribution in a systematic and rigorous way, which also has a guaranteed level of accuracy at the distribution tail. This is in contrast to the large deviation analysis, which provides only the (exponential) order of scaling but not the outage probability itself. Unlike the cumulant-based analysis, our approach provides a guaranteed level of accuracy at the distribution tail. Additionally, our analysis provides a number of novel insights. In particular, we demonstrate that there is a critical transition point below which the outage probability decays only polynomially but above which it decays super-exponentially. This provides a solid analytical foundation to the earlier empirical observations in the literature and also reveals what are the typical ways outage events occur in different regimes. The analysis is further extended to include interference cancelation and fading (from a broad class of distributions). The outage probability is shown to scale down exponentially in the number of canceled nearest interferers in the below-critical region and does not change significantly in the above-critical one. The proposed asymptotic expressions are shown to be accurate in the non-asymptotic regimes as well.

<div>The Subword Complexity of a character string refers to the number of distinct substrings of any length that occur as contiguous patterns in the string. The kth Subword Complexity in particular, refers to the number of distinct substrings of length k in a string of length n. In this work, we evaluate the expected value and the second factorial moment of the kth Subword Complexity for the binary strings over memory-less sources. We first take a combinatorial approach to derive a probability generating function for the number of occurrences of patterns in strings of finite length. This enables us to have an exact expression for the two moments in terms of patterns' auto-correlation and correlation polynomials. We then investigate the asymptotic behavior for values of k=a log n. In the proof, we compare the distribution of the kth Subword Complexity of binary strings to the distribution of distinct prefixes of independent strings stored in a trie. </div><div>The methodology that we use involves complex analysis, analytical poissonization and depoissonization, the Mellin transform, and saddle point analysis.</div>

Submitted by (ana.araujo@ufrpe.br) on 2016-08-10T13:00:13Z No. of bitstreams: 1 Rejane dos Santos Brito.pdf: 1642561 bytes, checksum: 084711a62c79f703133a032643c8d19f (MD5) / Made available in DSpace on 2016-08-10T13:00:13Z (GMT). No. of bitstreams: 1 Rejane dos Santos Brito.pdf: 1642561 bytes, checksum: 084711a62c79f703133a032643c8d19f (MD5) Previous issue date: 2009-03-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / We make a review about Edgeworth, Lugannani-Rice, Daniels and Cordeiro-Ferrari asymptotic approximations. We use the Cordeiro-Ferrari asymptotic approximation to approximate the gamma distribution G(m;f ) by the exponential distribution with mean a. In a further application, based on the statistical proposed by them, we approximate the t-Student distribution with n degrees of freedom using the normal standard distribution. Moreover, we realize a study about the functionalities of the beta generalized distributions. We obtain moments of the generalized beta distributions using the Lauricella and Kampé de Fériet generalized functions. Beyond this, we propose a new generalized beta distribution called beta power. Finally, we realize some applications in regression models by logistic regression and further more using discriminant analysis. / Inicialmente, realiza-se uma revisão literária sobre as expansões assintóticas de Daniels, Edgeworth, Lugannani-Rice e Cordeiro-Ferrari. Mediante uso da expansão de Cordeiro- Ferrari, torna-se possível realizar um estudo correspondente a aproximação da distribuição gama G(m;f ) em função da distribuição exponencial com média a. E, ainda, numa outra aplicação, faz-se a aproximação da distribuição t-Student com n graus de liberdade em função da distribuição normal padrão. Além disso, apresenta-se um estudo correspondente às funcionalidades das distribuições beta generalizadas e, ainda, a obtenção dos momentos das distribuições beta generalizadas mediante as funções de Lauricella e generalizada de Kampé de Fériet. Propõe-se, ainda, a generalização da distribuição power como sendo uma nova distribuição beta generalizada. Por fim, realizam-se algumas aplicações em modelos de regressão, mediante regressão logística, bem como em modelos de análise discriminante.

Soit q ⩾ 2 une puissance première fixe. L’objectif principal de cette thèse est d’étudier le comportement asymptotique de la fonction arithmétique Π_q(n,k) comptant le nombre de polynômes moniques de degré n et ayant exactement k facteurs irréductibles (avec multiplicité) sur le corps fini F_q. Warlimont et Car ont montré que l’objet Π_q(n,k) est approximativement distribué de Poisson lorsque 1 ⩽ k ⩽ A log n pour une constante A > 0. Plus tard, Hwang a étudié la fonction Π_q(n,k) pour la gamme complète 1 ⩽ k ⩽ n. Nous allons d’abord démontrer une formule asymptotique pour Π_q(n,k) en utilisant une technique analytique classique développée par Sathe et Selberg. Nous reproduirons ensuite une version simplifiée du résultat de Hwang en utilisant la formule de Sathe-Selberg dans le champ des fonctions. Nous comparons également nos résultats avec ceux analogues existants dans le cas des entiers, où l’on étudie tous les nombres naturels jusqu’à x avec exactement k facteurs premiers. En particulier, nous montrons que le nombre de polynômes moniques croît à un taux étonnamment plus élevé lorsque k est un peu plus grand que logn que ce que l’on pourrait supposer en examinant le cas des entiers. Pour présenter le travail ci-dessus, nous commençons d’abord par la théorie analytique des nombres de base dans le contexte des polynômes. Nous introduisons ensuite les fonctions arithmétiques clés qui jouent un rôle majeur dans notre thèse et discutons brièvement des résultats bien connus concernant leur distribution d’un point de vue probabiliste. Enfin, pour comprendre les résultats clés, nous donnons une discussion assez détaillée sur l’analogue de champ de fonction de la formule de Sathe-Selberg, un outil récemment développé par Porrit et utilisons ensuite cet outil pour prouver les résultats revendiqués. / Let q ⩾ 2 be a fixed prime power. The main objective of this thesis is to study the asymptotic behaviour of the arithmetic function Π_q(n,k) counting the number of monic polynomials that are of degree n and have exactly k irreducible factors (with multiplicity) over the finite field F_q. Warlimont and Car showed that the object Π_q(n,k) is approximately Poisson distributed when 1 ⩽ k ⩽ A log n for some constant A > 0. Later Hwang studied the function Π_q(n,k) for the full range 1 ⩽ k ⩽ n. We will first prove an asymptotic formula for Π_q(n,k) using a classical analytic technique developed by Sathe and Selberg. We will then reproduce a simplified version of Hwang’s result using the Sathe-Selberg formula in the function field. We also compare our results with the analogous existing ones in the integer case, where one studies all the natural numbers up to x with exactly k prime factors. In particular, we show that the number of monic polynomials grows at a surprisingly higher rate when k is a little larger than logn than what one would speculate from looking at the integer case. To present the above work, we first start with basic analytic number theory in the context of polynomials. We then introduce the key arithmetic functions that play a major role in our thesis and briefly discuss well-known results concerning their distribution from a probabilistic point of view. Finally, to understand the key results, we give a fairly detailed discussion on the function field analogue of the Sathe-Selberg formula, a tool recently developed by Porrit and subsequently use this tool to prove the claimed results.